id	sid	tid	token	lemma	pos
ap-1688	1	1	acta	acta	PROPN
ap-1688	1	2	polytechnica	polytechnica	PROPN
ap-1688	1	3	vol	vol	NOUN
ap-1688	1	4	.	.	PROPN
ap-1688	2	1	52	52	NUM
ap-1688	2	2	no	no	NOUN
ap-1688	2	3	.	.	PUNCT
ap-1688	3	1	6/2012	6/2012	NUM
ap-1688	4	1	the	the	DET
ap-1688	4	2	asymptotic	asymptotic	ADJ
ap-1688	4	3	properties	property	NOUN
ap-1688	4	4	of	of	ADP
ap-1688	4	5	turbulent	turbulent	ADJ
ap-1688	4	6	solutions	solution	NOUN
ap-1688	4	7	to	to	ADP
ap-1688	4	8	the	the	DET
ap-1688	4	9	navier	navier	NOUN
ap-1688	4	10	-	-	PUNCT
ap-1688	4	11	stokes	stoke	NOUN
ap-1688	4	12	equations	equation	NOUN
ap-1688	4	13	zdeněk	zdeněk	VERB
ap-1688	4	14	skalák	skalák	VERB
ap-1688	4	15	faculty	faculty	NOUN
ap-1688	4	16	of	of	ADP
ap-1688	4	17	civil	civil	ADJ
ap-1688	4	18	engineering	engineering	NOUN
ap-1688	4	19	,	,	PUNCT
ap-1688	4	20	czech	czech	PROPN
ap-1688	4	21	technical	technical	PROPN
ap-1688	4	22	university	university	PROPN
ap-1688	4	23	in	in	ADP
ap-1688	4	24	prague	prague	PROPN
ap-1688	4	25	,	,	PUNCT
ap-1688	4	26	thákurova	thákurova	PROPN
ap-1688	4	27	7	7	NUM
ap-1688	4	28	,	,	PUNCT
ap-1688	4	29	166	166	NUM
ap-1688	4	30	29	29	NUM
ap-1688	4	31	prague	prague	NOUN
ap-1688	4	32	,	,	PUNCT
ap-1688	4	33	czech	czech	PROPN
ap-1688	4	34	republic	republic	PROPN
ap-1688	4	35	corresponding	correspond	VERB
ap-1688	4	36	author	author	NOUN
ap-1688	4	37	:	:	PUNCT
ap-1688	4	38	skalak@mat.fsv.cvut.cz	skalak@mat.fsv.cvut.cz	NOUN
ap-1688	4	39	abstract	abstract	NOUN
ap-1688	4	40	in	in	ADP
ap-1688	4	41	this	this	DET
ap-1688	4	42	paper	paper	NOUN
ap-1688	4	43	we	we	PRON
ap-1688	4	44	study	study	VERB
ap-1688	4	45	the	the	DET
ap-1688	4	46	large	large	ADJ
ap-1688	4	47	time	time	NOUN
ap-1688	4	48	behavior	behavior	NOUN
ap-1688	4	49	of	of	ADP
ap-1688	4	50	solutions	solution	NOUN
ap-1688	4	51	to	to	ADP
ap-1688	4	52	the	the	DET
ap-1688	4	53	navier	navier	NOUN
ap-1688	4	54	-	-	PUNCT
ap-1688	4	55	stokes	stoke	NOUN
ap-1688	4	56	equations	equation	NOUN
ap-1688	4	57	.	.	PUNCT
ap-1688	5	1	we	we	PRON
ap-1688	5	2	present	present	VERB
ap-1688	5	3	a	a	DET
ap-1688	5	4	brief	brief	ADJ
ap-1688	5	5	survey	survey	NOUN
ap-1688	5	6	of	of	ADP
ap-1688	5	7	results	result	NOUN
ap-1688	5	8	concerning	concern	VERB
ap-1688	5	9	energy	energy	NOUN
ap-1688	5	10	decay	decay	NOUN
ap-1688	5	11	,	,	PUNCT
ap-1688	5	12	and	and	CCONJ
ap-1688	5	13	discuss	discuss	VERB
ap-1688	5	14	a	a	DET
ap-1688	5	15	related	related	ADJ
ap-1688	5	16	phenomenon	phenomenon	NOUN
ap-1688	5	17	of	of	ADP
ap-1688	5	18	the	the	DET
ap-1688	5	19	large	large	ADJ
ap-1688	5	20	time	time	NOUN
ap-1688	5	21	energy	energy	NOUN
ap-1688	5	22	concentration	concentration	NOUN
ap-1688	5	23	in	in	ADP
ap-1688	5	24	the	the	DET
ap-1688	5	25	frequency	frequency	NOUN
ap-1688	5	26	space	space	NOUN
ap-1688	5	27	occurring	occur	VERB
ap-1688	5	28	in	in	ADP
ap-1688	5	29	any	any	DET
ap-1688	5	30	turbulent	turbulent	ADJ
ap-1688	5	31	solution	solution	NOUN
ap-1688	5	32	.	.	PUNCT
ap-1688	6	1	this	this	PRON
ap-1688	6	2	leads	lead	VERB
ap-1688	6	3	us	we	PRON
ap-1688	6	4	to	to	ADP
ap-1688	6	5	the	the	DET
ap-1688	6	6	study	study	NOUN
ap-1688	6	7	of	of	ADP
ap-1688	6	8	solutions	solution	NOUN
ap-1688	6	9	in	in	ADP
ap-1688	6	10	the	the	DET
ap-1688	6	11	besov	besov	NOUN
ap-1688	6	12	spaces	space	NOUN
ap-1688	6	13	and	and	CCONJ
ap-1688	6	14	to	to	AUX
ap-1688	6	15	proof	proof	NOUN
ap-1688	6	16	that	that	SCONJ
ap-1688	6	17	if	if	SCONJ
ap-1688	6	18	we	we	PRON
ap-1688	6	19	choose	choose	VERB
ap-1688	6	20	a	a	DET
ap-1688	6	21	suitable	suitable	ADJ
ap-1688	6	22	initial	initial	ADJ
ap-1688	6	23	condition	condition	NOUN
ap-1688	6	24	then	then	ADV
ap-1688	6	25	in	in	ADP
ap-1688	6	26	some	some	DET
ap-1688	6	27	besov	besov	NOUN
ap-1688	6	28	spaces	space	VERB
ap-1688	6	29	the	the	DET
ap-1688	6	30	energy	energy	NOUN
ap-1688	6	31	of	of	ADP
ap-1688	6	32	the	the	DET
ap-1688	6	33	associated	associated	ADJ
ap-1688	6	34	solution	solution	NOUN
ap-1688	6	35	does	do	AUX
ap-1688	6	36	not	not	PART
ap-1688	6	37	decrease	decrease	VERB
ap-1688	6	38	asymptotically	asymptotically	ADV
ap-1688	6	39	to	to	ADP
ap-1688	6	40	zero	zero	NUM
ap-1688	6	41	.	.	PUNCT
ap-1688	7	1	keywords	keyword	NOUN
ap-1688	7	2	:	:	PUNCT
ap-1688	7	3	navier	navier	NOUN
ap-1688	7	4	-	-	PUNCT
ap-1688	7	5	stokes	stoke	NOUN
ap-1688	7	6	equations	equation	NOUN
ap-1688	7	7	,	,	PUNCT
ap-1688	7	8	besov	besov	NOUN
ap-1688	7	9	spaces	space	NOUN
ap-1688	7	10	.	.	PUNCT
ap-1688	8	1	1	1	NUM
ap-1688	8	2	introduction	introduction	NOUN
ap-1688	8	3	we	we	PRON
ap-1688	8	4	consider	consider	VERB
ap-1688	8	5	the	the	DET
ap-1688	8	6	navier	navier	NOUN
ap-1688	8	7	-	-	PUNCT
ap-1688	8	8	stokes	stoke	NOUN
ap-1688	8	9	equations	equation	NOUN
ap-1688	8	10	for	for	ADP
ap-1688	8	11	a	a	DET
ap-1688	8	12	viscous	viscous	ADJ
ap-1688	8	13	incompressible	incompressible	ADJ
ap-1688	8	14	fluid	fluid	NOUN
ap-1688	8	15	which	which	PRON
ap-1688	8	16	fills	fill	VERB
ap-1688	8	17	the	the	DET
ap-1688	8	18	whole	whole	ADJ
ap-1688	8	19	threedimensional	threedimensional	ADJ
ap-1688	8	20	space	space	NOUN
ap-1688	8	21	r3	r3	PROPN
ap-1688	8	22	with	with	ADP
ap-1688	8	23	the	the	DET
ap-1688	8	24	absence	absence	NOUN
ap-1688	8	25	of	of	ADP
ap-1688	8	26	external	external	ADJ
ap-1688	8	27	forces	force	NOUN
ap-1688	8	28	:	:	PUNCT
ap-1688	8	29	∂tu+∇	∂tu+∇	PROPN
ap-1688	8	30	·	·	PUNCT
ap-1688	8	31	(	(	PUNCT
ap-1688	8	32	u⊗	u⊗	PROPN
ap-1688	8	33	u	u	NOUN
ap-1688	8	34	)	)	PUNCT
ap-1688	8	35	=	=	SYM
ap-1688	8	36	∆u−∇p	∆u−∇p	PROPN
ap-1688	8	37	,	,	PUNCT
ap-1688	8	38	(	(	PUNCT
ap-1688	8	39	1	1	X
ap-1688	8	40	)	)	PUNCT
ap-1688	8	41	∇	∇	X
ap-1688	8	42	·	·	PUNCT
ap-1688	8	43	u	u	NOUN
ap-1688	8	44	=	=	NOUN
ap-1688	8	45	0	0	NUM
ap-1688	8	46	,	,	PUNCT
ap-1688	8	47	(	(	PUNCT
ap-1688	8	48	2	2	X
ap-1688	8	49	)	)	PUNCT
ap-1688	8	50	u(x	u(x	NOUN
ap-1688	8	51	,	,	PUNCT
ap-1688	8	52	0	0	NUM
ap-1688	8	53	)	)	PUNCT
ap-1688	8	54	=	=	SYM
ap-1688	8	55	u0(x	u0(x	NOUN
ap-1688	8	56	)	)	PUNCT
ap-1688	8	57	.	.	PUNCT
ap-1688	9	1	(	(	PUNCT
ap-1688	9	2	3	3	X
ap-1688	9	3	)	)	PUNCT
ap-1688	9	4	here	here	ADV
ap-1688	9	5	u	u	X
ap-1688	9	6	:	:	PUNCT
ap-1688	9	7	r3	r3	PROPN
ap-1688	9	8	×	×	NOUN
ap-1688	10	1	[	[	X
ap-1688	10	2	0,∞	0,∞	NUM
ap-1688	10	3	)	)	PUNCT
ap-1688	10	4	→	→	SYM
ap-1688	10	5	r3	r3	PROPN
ap-1688	10	6	denotes	denote	VERB
ap-1688	10	7	the	the	DET
ap-1688	10	8	unknown	unknown	ADJ
ap-1688	10	9	velocity	velocity	NOUN
ap-1688	10	10	field	field	NOUN
ap-1688	10	11	and	and	CCONJ
ap-1688	10	12	p	p	NOUN
ap-1688	10	13	:	:	PUNCT
ap-1688	10	14	r3×	r3×	NOUN
ap-1688	11	1	[	[	X
ap-1688	11	2	0,∞)→	0,∞)→	NOUN
ap-1688	11	3	r	r	NOUN
ap-1688	11	4	is	be	AUX
ap-1688	11	5	the	the	DET
ap-1688	11	6	unknown	unknown	ADJ
ap-1688	11	7	pressure	pressure	NOUN
ap-1688	11	8	.	.	PUNCT
ap-1688	12	1	u0	u0	ADJ
ap-1688	12	2	=	=	VERB
ap-1688	12	3	u0(x	u0(x	PROPN
ap-1688	12	4	)	)	PUNCT
ap-1688	12	5	=	=	SYM
ap-1688	12	6	(	(	PUNCT
ap-1688	12	7	u01(x	u01(x	NOUN
ap-1688	12	8	)	)	PUNCT
ap-1688	12	9	,	,	PUNCT
ap-1688	12	10	u02(x	u02(x	PROPN
ap-1688	12	11	)	)	PUNCT
ap-1688	12	12	,	,	PUNCT
ap-1688	12	13	u03(x	u03(x	PROPN
ap-1688	12	14	)	)	PUNCT
ap-1688	12	15	)	)	PUNCT
ap-1688	12	16	is	be	AUX
ap-1688	12	17	a	a	DET
ap-1688	12	18	given	give	VERB
ap-1688	12	19	initial	initial	ADJ
ap-1688	12	20	velocity	velocity	NOUN
ap-1688	12	21	.	.	PUNCT
ap-1688	13	1	the	the	DET
ap-1688	13	2	mathematical	mathematical	ADJ
ap-1688	13	3	theory	theory	NOUN
ap-1688	13	4	of	of	ADP
ap-1688	13	5	the	the	DET
ap-1688	13	6	navier	navier	NOUN
ap-1688	13	7	-	-	PUNCT
ap-1688	13	8	stokes	stoke	NOUN
ap-1688	13	9	equations	equation	NOUN
ap-1688	13	10	has	have	AUX
ap-1688	13	11	been	be	AUX
ap-1688	13	12	developed	develop	VERB
ap-1688	13	13	since	since	SCONJ
ap-1688	13	14	the	the	DET
ap-1688	13	15	pioneering	pioneering	ADJ
ap-1688	13	16	work	work	NOUN
ap-1688	13	17	by	by	ADP
ap-1688	13	18	leray	leray	INTJ
ap-1688	13	19	(	(	PUNCT
ap-1688	13	20	[	[	X
ap-1688	13	21	7	7	NUM
ap-1688	13	22	]	]	NUM
ap-1688	13	23	)	)	PUNCT
ap-1688	13	24	.	.	PUNCT
ap-1688	14	1	plenty	plenty	NOUN
ap-1688	14	2	of	of	ADP
ap-1688	14	3	papers	paper	NOUN
ap-1688	14	4	and	and	CCONJ
ap-1688	14	5	books	book	NOUN
ap-1688	14	6	can	can	AUX
ap-1688	14	7	now	now	ADV
ap-1688	14	8	be	be	AUX
ap-1688	14	9	found	find	VERB
ap-1688	14	10	in	in	ADP
ap-1688	14	11	the	the	DET
ap-1688	14	12	literature	literature	NOUN
ap-1688	14	13	concerning	concern	VERB
ap-1688	14	14	various	various	ADJ
ap-1688	14	15	aspects	aspect	NOUN
ap-1688	14	16	of	of	ADP
ap-1688	14	17	the	the	DET
ap-1688	14	18	theory	theory	NOUN
ap-1688	14	19	,	,	PUNCT
ap-1688	14	20	among	among	ADP
ap-1688	14	21	them	they	PRON
ap-1688	14	22	the	the	DET
ap-1688	14	23	famous	famous	ADJ
ap-1688	14	24	problem	problem	NOUN
ap-1688	14	25	(	(	PUNCT
ap-1688	14	26	still	still	ADV
ap-1688	14	27	unresolved	unresolved	ADJ
ap-1688	14	28	)	)	PUNCT
ap-1688	14	29	whether	whether	SCONJ
ap-1688	14	30	a	a	DET
ap-1688	14	31	solution	solution	NOUN
ap-1688	14	32	of	of	ADP
ap-1688	14	33	the	the	DET
ap-1688	14	34	navier	navier	NOUN
ap-1688	14	35	-	-	PUNCT
ap-1688	14	36	stokes	stoke	NOUN
ap-1688	14	37	equations	equation	NOUN
ap-1688	14	38	with	with	ADP
ap-1688	14	39	smooth	smooth	ADJ
ap-1688	14	40	data	datum	NOUN
ap-1688	14	41	remains	remain	VERB
ap-1688	14	42	regular	regular	ADJ
ap-1688	14	43	for	for	ADP
ap-1688	14	44	all	all	DET
ap-1688	14	45	times	time	NOUN
ap-1688	14	46	or	or	CCONJ
ap-1688	14	47	can	can	AUX
ap-1688	14	48	develop	develop	VERB
ap-1688	14	49	a	a	DET
ap-1688	14	50	blow	blow	NOUN
ap-1688	14	51	-	-	PUNCT
ap-1688	14	52	up	up	NOUN
ap-1688	14	53	in	in	ADP
ap-1688	14	54	a	a	DET
ap-1688	14	55	finite	finite	ADJ
ap-1688	14	56	time	time	NOUN
ap-1688	14	57	.	.	PUNCT
ap-1688	15	1	in	in	ADP
ap-1688	15	2	this	this	DET
ap-1688	15	3	paper	paper	NOUN
ap-1688	15	4	we	we	PRON
ap-1688	15	5	are	be	AUX
ap-1688	15	6	interested	interested	ADJ
ap-1688	15	7	in	in	ADP
ap-1688	15	8	the	the	DET
ap-1688	15	9	large	large	ADJ
ap-1688	15	10	time	time	NOUN
ap-1688	15	11	behavior	behavior	NOUN
ap-1688	15	12	of	of	ADP
ap-1688	15	13	the	the	DET
ap-1688	15	14	solutions	solution	NOUN
ap-1688	15	15	and	and	CCONJ
ap-1688	15	16	we	we	PRON
ap-1688	15	17	start	start	VERB
ap-1688	15	18	with	with	ADP
ap-1688	15	19	the	the	DET
ap-1688	15	20	following	following	ADJ
ap-1688	15	21	basic	basic	ADJ
ap-1688	15	22	question	question	NOUN
ap-1688	15	23	:	:	PUNCT
ap-1688	15	24	does	do	VERB
ap-1688	15	25	the	the	DET
ap-1688	15	26	kinetic	kinetic	ADJ
ap-1688	15	27	energy	energy	NOUN
ap-1688	15	28	of	of	ADP
ap-1688	15	29	the	the	DET
ap-1688	15	30	solutions	solution	NOUN
ap-1688	15	31	decrease	decrease	NOUN
ap-1688	15	32	to	to	ADP
ap-1688	15	33	zero	zero	NUM
ap-1688	15	34	as	as	SCONJ
ap-1688	15	35	time	time	NOUN
ap-1688	15	36	t	t	PROPN
ap-1688	15	37	goes	go	VERB
ap-1688	15	38	to	to	ADP
ap-1688	15	39	infinity	infinity	NOUN
ap-1688	15	40	?	?	PUNCT
ap-1688	16	1	this	this	DET
ap-1688	16	2	question	question	NOUN
ap-1688	16	3	was	be	AUX
ap-1688	16	4	first	first	ADV
ap-1688	16	5	raised	raise	VERB
ap-1688	16	6	by	by	ADP
ap-1688	16	7	leray	leray	ADV
ap-1688	16	8	in	in	ADP
ap-1688	16	9	[	[	X
ap-1688	16	10	7	7	X
ap-1688	16	11	]	]	PUNCT
ap-1688	16	12	in	in	ADP
ap-1688	16	13	1934	1934	NUM
ap-1688	16	14	and	and	CCONJ
ap-1688	16	15	the	the	DET
ap-1688	16	16	intuitive	intuitive	ADJ
ap-1688	16	17	answer	answer	NOUN
ap-1688	16	18	is	be	AUX
ap-1688	16	19	positive	positive	ADJ
ap-1688	16	20	,	,	PUNCT
ap-1688	16	21	since	since	SCONJ
ap-1688	16	22	we	we	PRON
ap-1688	16	23	consider	consider	VERB
ap-1688	16	24	no	no	DET
ap-1688	16	25	external	external	ADJ
ap-1688	16	26	forces	force	NOUN
ap-1688	16	27	here	here	ADV
ap-1688	16	28	.	.	PUNCT
ap-1688	17	1	indeed	indeed	ADV
ap-1688	17	2	,	,	PUNCT
ap-1688	17	3	the	the	DET
ap-1688	17	4	answer	answer	NOUN
ap-1688	17	5	"	"	PUNCT
ap-1688	17	6	yes	yes	INTJ
ap-1688	17	7	"	"	PUNCT
ap-1688	17	8	turns	turn	VERB
ap-1688	17	9	out	out	ADP
ap-1688	17	10	to	to	PART
ap-1688	17	11	be	be	AUX
ap-1688	17	12	correct	correct	ADJ
ap-1688	17	13	,	,	PUNCT
ap-1688	17	14	but	but	CCONJ
ap-1688	17	15	many	many	ADJ
ap-1688	17	16	years	year	NOUN
ap-1688	17	17	passed	pass	VERB
ap-1688	17	18	between	between	ADP
ap-1688	17	19	the	the	DET
ap-1688	17	20	formulation	formulation	NOUN
ap-1688	17	21	of	of	ADP
ap-1688	17	22	the	the	DET
ap-1688	17	23	question	question	NOUN
ap-1688	17	24	and	and	CCONJ
ap-1688	17	25	its	its	PRON
ap-1688	17	26	partial	partial	ADJ
ap-1688	17	27	solution	solution	NOUN
ap-1688	17	28	by	by	ADP
ap-1688	17	29	kato	kato	PROPN
ap-1688	17	30	,	,	PUNCT
ap-1688	17	31	in	in	ADP
ap-1688	17	32	[	[	PUNCT
ap-1688	17	33	6	6	NUM
ap-1688	17	34	]	]	PUNCT
ap-1688	17	35	.	.	PUNCT
ap-1688	18	1	having	having	AUX
ap-1688	18	2	solved	solve	VERB
ap-1688	18	3	the	the	DET
ap-1688	18	4	basic	basic	ADJ
ap-1688	18	5	problem	problem	NOUN
ap-1688	18	6	,	,	PUNCT
ap-1688	18	7	we	we	PRON
ap-1688	18	8	can	can	AUX
ap-1688	18	9	now	now	ADV
ap-1688	18	10	investigate	investigate	VERB
ap-1688	18	11	more	more	ADV
ap-1688	18	12	detailed	detailed	ADJ
ap-1688	18	13	aspects	aspect	NOUN
ap-1688	18	14	of	of	ADP
ap-1688	18	15	energy	energy	NOUN
ap-1688	18	16	decay	decay	NOUN
ap-1688	18	17	.	.	PUNCT
ap-1688	19	1	in	in	ADP
ap-1688	19	2	the	the	DET
ap-1688	19	3	second	second	ADJ
ap-1688	19	4	section	section	NOUN
ap-1688	19	5	,	,	PUNCT
ap-1688	19	6	we	we	PRON
ap-1688	19	7	will	will	AUX
ap-1688	19	8	discuss	discuss	VERB
ap-1688	19	9	the	the	DET
ap-1688	19	10	rate	rate	NOUN
ap-1688	19	11	of	of	ADP
ap-1688	19	12	energy	energy	NOUN
ap-1688	19	13	decay	decay	NOUN
ap-1688	19	14	and	and	CCONJ
ap-1688	19	15	we	we	PRON
ap-1688	19	16	will	will	AUX
ap-1688	19	17	present	present	VERB
ap-1688	19	18	a	a	DET
ap-1688	19	19	short	short	ADJ
ap-1688	19	20	survey	survey	NOUN
ap-1688	19	21	of	of	ADP
ap-1688	19	22	the	the	DET
ap-1688	19	23	results	result	NOUN
ap-1688	19	24	.	.	PUNCT
ap-1688	20	1	the	the	DET
ap-1688	20	2	third	third	ADJ
ap-1688	20	3	section	section	NOUN
ap-1688	20	4	will	will	AUX
ap-1688	20	5	be	be	AUX
ap-1688	20	6	devoted	devote	VERB
ap-1688	20	7	to	to	ADP
ap-1688	20	8	the	the	DET
ap-1688	20	9	phenomenon	phenomenon	NOUN
ap-1688	20	10	of	of	ADP
ap-1688	20	11	large	large	ADJ
ap-1688	20	12	time	time	NOUN
ap-1688	20	13	energy	energy	NOUN
ap-1688	20	14	concentration	concentration	NOUN
ap-1688	20	15	in	in	ADP
ap-1688	20	16	solutions	solution	NOUN
ap-1688	20	17	.	.	PUNCT
ap-1688	21	1	it	it	PRON
ap-1688	21	2	turns	turn	VERB
ap-1688	21	3	out	out	ADP
ap-1688	21	4	that	that	SCONJ
ap-1688	21	5	in	in	ADP
ap-1688	21	6	every	every	DET
ap-1688	21	7	(	(	PUNCT
ap-1688	21	8	turbulent	turbulent	ADJ
ap-1688	21	9	)	)	PUNCT
ap-1688	21	10	solution	solution	NOUN
ap-1688	21	11	the	the	DET
ap-1688	21	12	energy	energy	NOUN
ap-1688	21	13	concentrates	concentrate	VERB
ap-1688	21	14	for	for	ADP
ap-1688	21	15	large	large	ADJ
ap-1688	21	16	times	time	NOUN
ap-1688	21	17	in	in	ADP
ap-1688	21	18	frequencies	frequency	NOUN
ap-1688	21	19	forming	form	VERB
ap-1688	21	20	an	an	DET
ap-1688	21	21	annulus	annulus	NOUN
ap-1688	21	22	or	or	CCONJ
ap-1688	21	23	a	a	DET
ap-1688	21	24	ball	ball	NOUN
ap-1688	21	25	in	in	ADP
ap-1688	21	26	the	the	DET
ap-1688	21	27	frequency	frequency	NOUN
ap-1688	21	28	space	space	NOUN
ap-1688	21	29	.	.	PUNCT
ap-1688	22	1	this	this	DET
ap-1688	22	2	phenomenon	phenomenon	NOUN
ap-1688	22	3	seems	seem	VERB
ap-1688	22	4	to	to	PART
ap-1688	22	5	be	be	AUX
ap-1688	22	6	connected	connect	VERB
ap-1688	22	7	with	with	ADP
ap-1688	22	8	the	the	DET
ap-1688	22	9	rate	rate	NOUN
ap-1688	22	10	of	of	ADP
ap-1688	22	11	energy	energy	NOUN
ap-1688	22	12	decay	decay	NOUN
ap-1688	22	13	discussed	discuss	VERB
ap-1688	22	14	in	in	ADP
ap-1688	22	15	the	the	DET
ap-1688	22	16	second	second	ADJ
ap-1688	22	17	section	section	NOUN
ap-1688	22	18	,	,	PUNCT
ap-1688	22	19	and	and	CCONJ
ap-1688	22	20	we	we	PRON
ap-1688	22	21	will	will	AUX
ap-1688	22	22	present	present	VERB
ap-1688	22	23	several	several	ADJ
ap-1688	22	24	results	result	NOUN
ap-1688	22	25	concerning	concern	VERB
ap-1688	22	26	the	the	DET
ap-1688	22	27	existence	existence	NOUN
ap-1688	22	28	and	and	CCONJ
ap-1688	22	29	the	the	DET
ap-1688	22	30	rate	rate	NOUN
ap-1688	22	31	of	of	ADP
ap-1688	22	32	the	the	DET
ap-1688	22	33	energy	energy	NOUN
ap-1688	22	34	concentration	concentration	NOUN
ap-1688	22	35	the	the	DET
ap-1688	22	36	main	main	ADJ
ap-1688	22	37	results	result	NOUN
ap-1688	22	38	of	of	ADP
ap-1688	22	39	this	this	DET
ap-1688	22	40	paper	paper	NOUN
ap-1688	22	41	are	be	AUX
ap-1688	22	42	presented	present	VERB
ap-1688	22	43	in	in	ADP
ap-1688	22	44	the	the	DET
ap-1688	22	45	fourth	fourth	ADJ
ap-1688	22	46	section	section	NOUN
ap-1688	22	47	,	,	PUNCT
ap-1688	22	48	where	where	SCONJ
ap-1688	22	49	we	we	PRON
ap-1688	22	50	will	will	AUX
ap-1688	22	51	turn	turn	VERB
ap-1688	22	52	our	our	PRON
ap-1688	22	53	attention	attention	NOUN
ap-1688	22	54	to	to	ADP
ap-1688	22	55	the	the	DET
ap-1688	22	56	existence	existence	NOUN
ap-1688	22	57	of	of	ADP
ap-1688	22	58	solutions	solution	NOUN
ap-1688	22	59	in	in	ADP
ap-1688	22	60	besov	besov	NOUN
ap-1688	22	61	spaces	space	NOUN
ap-1688	22	62	.	.	PUNCT
ap-1688	23	1	these	these	DET
ap-1688	23	2	spaces	space	NOUN
ap-1688	23	3	are	be	AUX
ap-1688	23	4	defined	define	VERB
ap-1688	23	5	by	by	ADP
ap-1688	23	6	the	the	DET
ap-1688	23	7	use	use	NOUN
ap-1688	23	8	of	of	ADP
ap-1688	23	9	the	the	DET
ap-1688	23	10	fourier	fourier	NOUN
ap-1688	23	11	transform	transform	NOUN
ap-1688	23	12	,	,	PUNCT
ap-1688	23	13	and	and	CCONJ
ap-1688	23	14	enable	enable	VERB
ap-1688	23	15	a	a	DET
ap-1688	23	16	study	study	NOUN
ap-1688	23	17	of	of	ADP
ap-1688	23	18	the	the	DET
ap-1688	23	19	location	location	NOUN
ap-1688	23	20	of	of	ADP
ap-1688	23	21	the	the	DET
ap-1688	23	22	energy	energy	NOUN
ap-1688	23	23	in	in	ADP
ap-1688	23	24	the	the	DET
ap-1688	23	25	whole	whole	ADJ
ap-1688	23	26	spectrum	spectrum	NOUN
ap-1688	23	27	of	of	ADP
ap-1688	23	28	frequencies	frequency	NOUN
ap-1688	23	29	.	.	PUNCT
ap-1688	24	1	this	this	PRON
ap-1688	24	2	seems	seem	VERB
ap-1688	24	3	to	to	PART
ap-1688	24	4	be	be	AUX
ap-1688	24	5	a	a	DET
ap-1688	24	6	suitable	suitable	ADJ
ap-1688	24	7	way	way	NOUN
ap-1688	24	8	to	to	PART
ap-1688	24	9	study	study	VERB
ap-1688	24	10	the	the	DET
ap-1688	24	11	problems	problem	NOUN
ap-1688	24	12	presented	present	VERB
ap-1688	24	13	in	in	ADP
ap-1688	24	14	the	the	DET
ap-1688	24	15	third	third	ADJ
ap-1688	24	16	section	section	NOUN
ap-1688	24	17	.	.	PUNCT
ap-1688	25	1	we	we	PRON
ap-1688	25	2	improve	improve	VERB
ap-1688	25	3	a	a	DET
ap-1688	25	4	result	result	NOUN
ap-1688	25	5	presented	present	VERB
ap-1688	25	6	by	by	ADP
ap-1688	25	7	miyakawa	miyakawa	NOUN
ap-1688	25	8	in	in	ADP
ap-1688	25	9	[	[	X
ap-1688	25	10	8	8	NUM
ap-1688	25	11	]	]	PUNCT
ap-1688	25	12	,	,	PUNCT
ap-1688	25	13	and	and	CCONJ
ap-1688	25	14	show	show	VERB
ap-1688	25	15	here	here	ADV
ap-1688	25	16	that	that	SCONJ
ap-1688	25	17	there	there	PRON
ap-1688	25	18	exist	exist	VERB
ap-1688	25	19	some	some	DET
ap-1688	25	20	besov	besov	NOUN
ap-1688	25	21	spaces	space	NOUN
ap-1688	25	22	in	in	ADP
ap-1688	25	23	which	which	PRON
ap-1688	25	24	some	some	DET
ap-1688	25	25	solutions	solution	NOUN
ap-1688	25	26	do	do	AUX
ap-1688	25	27	not	not	PART
ap-1688	25	28	decrease	decrease	VERB
ap-1688	25	29	asymptotically	asymptotically	ADV
ap-1688	25	30	to	to	ADP
ap-1688	25	31	zero	zero	NUM
ap-1688	25	32	,	,	PUNCT
ap-1688	25	33	unlike	unlike	ADP
ap-1688	25	34	the	the	DET
ap-1688	25	35	decrease	decrease	NOUN
ap-1688	25	36	to	to	ADP
ap-1688	25	37	zero	zero	NUM
ap-1688	25	38	in	in	ADP
ap-1688	25	39	the	the	DET
ap-1688	25	40	energy	energy	NOUN
ap-1688	25	41	norm	norm	NOUN
ap-1688	25	42	mentioned	mention	VERB
ap-1688	25	43	above	above	ADV
ap-1688	25	44	.	.	PUNCT
ap-1688	26	1	for	for	ADP
ap-1688	26	2	the	the	DET
ap-1688	26	3	purposes	purpose	NOUN
ap-1688	26	4	of	of	ADP
ap-1688	26	5	clarity	clarity	NOUN
ap-1688	26	6	,	,	PUNCT
ap-1688	26	7	all	all	DET
ap-1688	26	8	the	the	DET
ap-1688	26	9	notation	notation	NOUN
ap-1688	26	10	used	use	VERB
ap-1688	26	11	in	in	ADP
ap-1688	26	12	this	this	DET
ap-1688	26	13	paper	paper	NOUN
ap-1688	26	14	,	,	PUNCT
ap-1688	26	15	and	and	CCONJ
ap-1688	26	16	also	also	ADV
ap-1688	26	17	definitions	definition	NOUN
ap-1688	26	18	of	of	ADP
ap-1688	26	19	some	some	DET
ap-1688	26	20	basic	basic	ADJ
ap-1688	26	21	mathematical	mathematical	ADJ
ap-1688	26	22	terms	term	NOUN
ap-1688	26	23	,	,	PUNCT
ap-1688	26	24	can	can	AUX
ap-1688	26	25	be	be	AUX
ap-1688	26	26	found	find	VERB
ap-1688	26	27	in	in	ADP
ap-1688	26	28	the	the	DET
ap-1688	26	29	appendix	appendix	NOUN
ap-1688	26	30	.	.	PUNCT
ap-1688	27	1	2	2	NUM
ap-1688	27	2	rate	rate	NOUN
ap-1688	27	3	of	of	ADP
ap-1688	27	4	energy	energy	NOUN
ap-1688	27	5	decay	decay	NOUN
ap-1688	27	6	as	as	SCONJ
ap-1688	27	7	was	be	AUX
ap-1688	27	8	mentioned	mention	VERB
ap-1688	27	9	in	in	ADP
ap-1688	27	10	the	the	DET
ap-1688	27	11	introduction	introduction	NOUN
ap-1688	27	12	,	,	PUNCT
ap-1688	27	13	the	the	DET
ap-1688	27	14	energy	energy	NOUN
ap-1688	27	15	of	of	ADP
ap-1688	27	16	every	every	DET
ap-1688	27	17	turbulent	turbulent	ADJ
ap-1688	27	18	solution	solution	NOUN
ap-1688	27	19	u	u	NOUN
ap-1688	27	20	decreases	decrease	VERB
ap-1688	27	21	asymptotically	asymptotically	ADV
ap-1688	27	22	to	to	ADP
ap-1688	27	23	zero	zero	NUM
ap-1688	27	24	,	,	PUNCT
ap-1688	27	25	i.e.	i.e.	X
ap-1688	27	26	limt→∞‖u(t)‖2	limt→∞‖u(t)‖2	NOUN
ap-1688	27	27	=	=	SYM
ap-1688	27	28	0	0	X
ap-1688	27	29	.	.	PUNCT
ap-1688	28	1	(	(	PUNCT
ap-1688	28	2	a	a	DET
ap-1688	28	3	precise	precise	ADJ
ap-1688	28	4	simple	simple	ADJ
ap-1688	28	5	proof	proof	NOUN
ap-1688	28	6	can	can	AUX
ap-1688	28	7	be	be	AUX
ap-1688	28	8	found	find	VERB
ap-1688	28	9	in	in	ADP
ap-1688	28	10	[	[	X
ap-1688	28	11	20	20	NUM
ap-1688	28	12	]	]	NUM
ap-1688	28	13	)	)	PUNCT
ap-1688	28	14	.	.	PUNCT
ap-1688	29	1	a	a	DET
ap-1688	29	2	further	further	ADJ
ap-1688	29	3	logical	logical	ADJ
ap-1688	29	4	step	step	NOUN
ap-1688	29	5	is	be	AUX
ap-1688	29	6	to	to	PART
ap-1688	29	7	study	study	VERB
ap-1688	29	8	the	the	DET
ap-1688	29	9	rate	rate	NOUN
ap-1688	29	10	of	of	ADP
ap-1688	29	11	energy	energy	NOUN
ap-1688	29	12	decay	decay	NOUN
ap-1688	29	13	,	,	PUNCT
ap-1688	29	14	and	and	CCONJ
ap-1688	29	15	to	to	PART
ap-1688	29	16	present	present	VERB
ap-1688	29	17	some	some	DET
ap-1688	29	18	classes	class	NOUN
ap-1688	29	19	of	of	ADP
ap-1688	29	20	initial	initial	ADJ
ap-1688	29	21	conditions	condition	NOUN
ap-1688	29	22	providing	provide	VERB
ap-1688	29	23	various	various	ADJ
ap-1688	29	24	rates	rate	NOUN
ap-1688	29	25	of	of	ADP
ap-1688	29	26	decay	decay	NOUN
ap-1688	29	27	.	.	PUNCT
ap-1688	30	1	many	many	ADJ
ap-1688	30	2	results	result	NOUN
ap-1688	30	3	concerning	concern	VERB
ap-1688	30	4	this	this	DET
ap-1688	30	5	problem	problem	NOUN
ap-1688	30	6	were	be	AUX
ap-1688	30	7	proved	prove	VERB
ap-1688	30	8	by	by	ADP
ap-1688	30	9	schonbek	schonbek	ADJ
ap-1688	30	10	(	(	PUNCT
ap-1688	30	11	see	see	VERB
ap-1688	30	12	,	,	PUNCT
ap-1688	30	13	for	for	ADP
ap-1688	30	14	example	example	NOUN
ap-1688	30	15	[	[	X
ap-1688	30	16	11	11	NUM
ap-1688	30	17	]	]	PUNCT
ap-1688	30	18	,	,	PUNCT
ap-1688	30	19	[	[	X
ap-1688	30	20	12	12	NUM
ap-1688	30	21	]	]	PUNCT
ap-1688	30	22	and	and	CCONJ
ap-1688	30	23	[	[	X
ap-1688	30	24	13	13	NUM
ap-1688	30	25	]	]	NUM
ap-1688	30	26	)	)	PUNCT
ap-1688	30	27	.	.	PUNCT
ap-1688	31	1	we	we	PRON
ap-1688	31	2	mention	mention	VERB
ap-1688	31	3	here	here	ADV
ap-1688	31	4	99	99	NUM
ap-1688	31	5	acta	acta	PROPN
ap-1688	31	6	polytechnica	polytechnica	PROPN
ap-1688	31	7	vol	vol	NOUN
ap-1688	31	8	.	.	PROPN
ap-1688	32	1	52	52	NUM
ap-1688	32	2	no	no	NOUN
ap-1688	32	3	.	.	PUNCT
ap-1688	33	1	6/2012	6/2012	NUM
ap-1688	33	2	as	as	ADP
ap-1688	33	3	an	an	DET
ap-1688	33	4	example	example	NOUN
ap-1688	33	5	a	a	DET
ap-1688	33	6	result	result	NOUN
ap-1688	33	7	proved	prove	VERB
ap-1688	33	8	in	in	ADP
ap-1688	33	9	[	[	X
ap-1688	33	10	12	12	NUM
ap-1688	33	11	]	]	X
ap-1688	33	12	:	:	PUNCT
ap-1688	33	13	if	if	SCONJ
ap-1688	33	14	the	the	DET
ap-1688	33	15	initial	initial	ADJ
ap-1688	33	16	condition	condition	NOUN
ap-1688	33	17	u0	u0	NOUN
ap-1688	33	18	belongs	belong	VERB
ap-1688	33	19	to	to	ADP
ap-1688	33	20	the	the	DET
ap-1688	33	21	space	space	NOUN
ap-1688	33	22	l1	l1	PROPN
ap-1688	33	23	∩l2	∩l2	PROPN
ap-1688	33	24	σ	σ	PROPN
ap-1688	33	25	,	,	PUNCT
ap-1688	33	26	then	then	ADV
ap-1688	33	27	there	there	PRON
ap-1688	33	28	exists	exist	VERB
ap-1688	33	29	a	a	DET
ap-1688	33	30	global	global	ADJ
ap-1688	33	31	weak	weak	ADJ
ap-1688	33	32	solution	solution	NOUN
ap-1688	33	33	of	of	ADP
ap-1688	33	34	(	(	PUNCT
ap-1688	33	35	1)–(3	1)–(3	NUM
ap-1688	33	36	)	)	PUNCT
ap-1688	33	37	and	and	CCONJ
ap-1688	33	38	c	c	X
ap-1688	33	39	>	>	X
ap-1688	33	40	0	0	NUM
ap-1688	34	1	such	such	ADJ
ap-1688	34	2	that	that	PRON
ap-1688	34	3	‖u(t)‖2	‖u(t)‖2	ADJ
ap-1688	34	4	≤	≤	NUM
ap-1688	34	5	c(t+	c(t+	NOUN
ap-1688	34	6	1)−1/4	1)−1/4	NUM
ap-1688	34	7	for	for	ADP
ap-1688	34	8	every	every	DET
ap-1688	34	9	t	t	PROPN
ap-1688	34	10	≥	≥	NOUN
ap-1688	34	11	0	0	NUM
ap-1688	34	12	.	.	PUNCT
ap-1688	35	1	a	a	DET
ap-1688	35	2	key	key	ADJ
ap-1688	35	3	paper	paper	NOUN
ap-1688	35	4	was	be	AUX
ap-1688	35	5	published	publish	VERB
ap-1688	35	6	by	by	ADP
ap-1688	35	7	wiegner	wiegner	NOUN
ap-1688	35	8	in	in	ADP
ap-1688	35	9	[	[	X
ap-1688	35	10	20	20	NUM
ap-1688	35	11	]	]	PUNCT
ap-1688	35	12	.	.	PUNCT
ap-1688	36	1	he	he	PRON
ap-1688	36	2	showed	show	VERB
ap-1688	36	3	that	that	SCONJ
ap-1688	36	4	,	,	PUNCT
ap-1688	36	5	roughly	roughly	ADV
ap-1688	36	6	speaking	speak	VERB
ap-1688	36	7	,	,	PUNCT
ap-1688	36	8	most	most	ADJ
ap-1688	36	9	of	of	ADP
ap-1688	36	10	the	the	DET
ap-1688	36	11	solutions	solution	NOUN
ap-1688	36	12	of	of	ADP
ap-1688	36	13	the	the	DET
ap-1688	36	14	navier	navier	NOUN
ap-1688	36	15	-	-	PUNCT
ap-1688	36	16	stokes	stoke	NOUN
ap-1688	36	17	equations	equation	NOUN
ap-1688	36	18	decrease	decrease	VERB
ap-1688	36	19	at	at	ADP
ap-1688	36	20	the	the	DET
ap-1688	36	21	same	same	ADJ
ap-1688	36	22	rate	rate	NOUN
ap-1688	36	23	as	as	ADP
ap-1688	36	24	the	the	DET
ap-1688	36	25	solutions	solution	NOUN
ap-1688	36	26	of	of	ADP
ap-1688	36	27	the	the	DET
ap-1688	36	28	so	so	ADV
ap-1688	36	29	called	call	VERB
ap-1688	36	30	stokes	stokes	PROPN
ap-1688	36	31	equations	equation	NOUN
ap-1688	36	32	(	(	PUNCT
ap-1688	36	33	the	the	DET
ap-1688	36	34	navier	navier	NOUN
ap-1688	36	35	-	-	PUNCT
ap-1688	36	36	stokes	stoke	NOUN
ap-1688	36	37	equations	equation	NOUN
ap-1688	36	38	deprived	deprive	VERB
ap-1688	36	39	of	of	ADP
ap-1688	36	40	the	the	DET
ap-1688	36	41	nonlinear	nonlinear	ADJ
ap-1688	36	42	term	term	NOUN
ap-1688	36	43	)	)	PUNCT
ap-1688	36	44	with	with	ADP
ap-1688	36	45	the	the	DET
ap-1688	36	46	same	same	ADJ
ap-1688	36	47	initial	initial	ADJ
ap-1688	36	48	conditions	condition	NOUN
ap-1688	36	49	.	.	PUNCT
ap-1688	37	1	more	more	ADV
ap-1688	37	2	precisely	precisely	ADV
ap-1688	37	3	,	,	PUNCT
ap-1688	37	4	a	a	DET
ap-1688	37	5	turbulent	turbulent	ADJ
ap-1688	37	6	solution	solution	NOUN
ap-1688	37	7	with	with	ADP
ap-1688	37	8	the	the	DET
ap-1688	37	9	initial	initial	ADJ
ap-1688	37	10	condition	condition	NOUN
ap-1688	37	11	u0	u0	NOUN
ap-1688	37	12	decreases	decrease	VERB
ap-1688	37	13	at	at	ADP
ap-1688	37	14	the	the	DET
ap-1688	37	15	rate	rate	NOUN
ap-1688	37	16	(	(	PUNCT
ap-1688	37	17	1	1	NUM
ap-1688	37	18	+	+	NUM
ap-1688	37	19	t)−α	t)−α	NUM
ap-1688	37	20	for	for	ADP
ap-1688	37	21	some	some	DET
ap-1688	37	22	α	α	NOUN
ap-1688	37	23	∈	∈	PROPN
ap-1688	37	24	(	(	PUNCT
ap-1688	37	25	0	0	NUM
ap-1688	37	26	,	,	PUNCT
ap-1688	37	27	5/4	5/4	NUM
ap-1688	37	28	]	]	PUNCT
ap-1688	37	29	if	if	SCONJ
ap-1688	37	30	also	also	ADV
ap-1688	37	31	et∆u0	et∆u0	VERB
ap-1688	37	32	decreases	decrease	NOUN
ap-1688	37	33	at	at	ADP
ap-1688	37	34	the	the	DET
ap-1688	37	35	same	same	ADJ
ap-1688	37	36	rate	rate	NOUN
ap-1688	37	37	.	.	PUNCT
ap-1688	38	1	solutions	solution	NOUN
ap-1688	38	2	with	with	ADP
ap-1688	38	3	an	an	DET
ap-1688	38	4	even	even	ADV
ap-1688	38	5	higher	high	ADJ
ap-1688	38	6	rate	rate	NOUN
ap-1688	38	7	of	of	ADP
ap-1688	38	8	decay	decay	NOUN
ap-1688	38	9	were	be	AUX
ap-1688	38	10	studied	study	VERB
ap-1688	38	11	by	by	ADP
ap-1688	38	12	miyakawa	miyakawa	NOUN
ap-1688	38	13	and	and	CCONJ
ap-1688	38	14	schonbek	schonbek	VERB
ap-1688	38	15	in	in	ADP
ap-1688	38	16	[	[	X
ap-1688	38	17	9	9	NUM
ap-1688	38	18	]	]	PUNCT
ap-1688	38	19	.	.	PUNCT
ap-1688	39	1	they	they	PRON
ap-1688	39	2	proved	prove	VERB
ap-1688	39	3	the	the	DET
ap-1688	39	4	following	follow	VERB
ap-1688	39	5	result	result	NOUN
ap-1688	39	6	:	:	PUNCT
ap-1688	39	7	theorem	theorem	NOUN
ap-1688	39	8	1	1	X
ap-1688	39	9	.	.	PUNCT
ap-1688	40	1	let	let	VERB
ap-1688	40	2	u0	u0	PROPN
ap-1688	40	3	∈	∈	PROPN
ap-1688	40	4	l2	l2	NOUN
ap-1688	40	5	σ	σ	NOUN
ap-1688	40	6	and	and	CCONJ
ap-1688	40	7	∫	∫	PROPN
ap-1688	40	8	|u0(x)|(1	|u0(x)|(1	PROPN
ap-1688	41	1	+	+	CCONJ
ap-1688	41	2	|x|)dx	|x|)dx	PROPN
ap-1688	41	3	<	<	X
ap-1688	41	4	∞.	∞.	PROPN
ap-1688	41	5	let	let	VERB
ap-1688	41	6	u	u	PRON
ap-1688	41	7	be	be	AUX
ap-1688	41	8	a	a	DET
ap-1688	41	9	turbulent	turbulent	ADJ
ap-1688	41	10	solution	solution	NOUN
ap-1688	41	11	to	to	ADP
ap-1688	41	12	the	the	DET
ap-1688	41	13	nse	nse	NOUN
ap-1688	41	14	with	with	ADP
ap-1688	41	15	the	the	DET
ap-1688	41	16	initial	initial	ADJ
ap-1688	41	17	condition	condition	NOUN
ap-1688	41	18	u0	u0	ADJ
ap-1688	41	19	such	such	ADJ
ap-1688	41	20	that	that	PRON
ap-1688	41	21	‖u(t)‖2	‖u(t)‖2	ADJ
ap-1688	41	22	≤	≤	PROPN
ap-1688	41	23	c(1	c(1	PROPN
ap-1688	41	24	+	+	NOUN
ap-1688	41	25	t)−5/4	t)−5/4	PROPN
ap-1688	41	26	.	.	PUNCT
ap-1688	42	1	we	we	PRON
ap-1688	42	2	set	set	VERB
ap-1688	42	3	bh	bh	NOUN
ap-1688	42	4	,	,	PUNCT
ap-1688	42	5	k	k	PROPN
ap-1688	43	1	=	=	SYM
ap-1688	44	1	∫	∫	PROPN
ap-1688	45	1	xhu0k(x)dx	xhu0k(x)dx	PROPN
ap-1688	45	2	and	and	CCONJ
ap-1688	45	3	λh	λh	ADP
ap-1688	45	4	,	,	PUNCT
ap-1688	45	5	k	k	PROPN
ap-1688	45	6	=	=	SYM
ap-1688	45	7	∫	∫	PROPN
ap-1688	45	8	∞	∞	PROPN
ap-1688	45	9	0	0	NUM
ap-1688	46	1	∫	∫	PROPN
ap-1688	46	2	(	(	PUNCT
ap-1688	46	3	uhuk)(x	uhuk)(x	PROPN
ap-1688	46	4	,	,	PUNCT
ap-1688	46	5	t)dxdt	t)dxdt	PROPN
ap-1688	46	6	(	(	PUNCT
ap-1688	46	7	h	h	NOUN
ap-1688	46	8	,	,	PUNCT
ap-1688	46	9	k	k	NOUN
ap-1688	46	10	=	=	SYM
ap-1688	46	11	1	1	NUM
ap-1688	46	12	,	,	PUNCT
ap-1688	46	13	2	2	NUM
ap-1688	46	14	,	,	PUNCT
ap-1688	46	15	3	3	NUM
ap-1688	46	16	)	)	PUNCT
ap-1688	46	17	.	.	PUNCT
ap-1688	47	1	then	then	ADV
ap-1688	47	2	if	if	SCONJ
ap-1688	47	3	(	(	PUNCT
ap-1688	47	4	bh	bh	NOUN
ap-1688	47	5	,	,	PUNCT
ap-1688	47	6	k	k	PROPN
ap-1688	47	7	)	)	PUNCT
ap-1688	47	8	≡	≡	PROPN
ap-1688	47	9	0	0	PUNCT
ap-1688	48	1	and	and	CCONJ
ap-1688	48	2	if	if	SCONJ
ap-1688	48	3	there	there	PRON
ap-1688	48	4	exists	exist	VERB
ap-1688	48	5	c	c	NOUN
ap-1688	48	6	∈	∈	PROPN
ap-1688	48	7	r	r	NOUN
ap-1688	48	8	such	such	ADJ
ap-1688	48	9	that	that	DET
ap-1688	48	10	λh	λh	ADP
ap-1688	48	11	,	,	PUNCT
ap-1688	48	12	k	k	NOUN
ap-1688	48	13	=	=	PUNCT
ap-1688	48	14	cδh	cδh	PROPN
ap-1688	48	15	,	,	PUNCT
ap-1688	48	16	k	k	NOUN
ap-1688	48	17	,	,	PUNCT
ap-1688	48	18	then	then	ADV
ap-1688	48	19	lim	lim	PROPN
ap-1688	48	20	t→∞	t→∞	ADP
ap-1688	48	21	t5/4‖u(t)‖2	t5/4‖u(t)‖2	PRON
ap-1688	48	22	=	=	NOUN
ap-1688	48	23	0	0	X
ap-1688	48	24	.	.	PUNCT
ap-1688	49	1	conversely	conversely	ADV
ap-1688	49	2	,	,	PUNCT
ap-1688	49	3	if	if	SCONJ
ap-1688	49	4	(	(	PUNCT
ap-1688	49	5	bh	bh	NOUN
ap-1688	49	6	,	,	PUNCT
ap-1688	49	7	k	k	PROPN
ap-1688	49	8	)	)	PUNCT
ap-1688	49	9	6=	6=	ADP
ap-1688	49	10	0	0	NUM
ap-1688	49	11	or	or	CCONJ
ap-1688	49	12	(	(	PUNCT
ap-1688	49	13	λh	λh	ADP
ap-1688	49	14	,	,	PUNCT
ap-1688	49	15	k	k	NOUN
ap-1688	49	16	)	)	PUNCT
ap-1688	49	17	is	be	AUX
ap-1688	49	18	not	not	PART
ap-1688	49	19	scalar	scalar	ADJ
ap-1688	49	20	,	,	PUNCT
ap-1688	49	21	then	then	ADV
ap-1688	49	22	lim	lim	PROPN
ap-1688	49	23	inf	inf	PROPN
ap-1688	49	24	t→∞	t→∞	ADP
ap-1688	49	25	t5/4‖u(t)‖2	t5/4‖u(t)‖2	PRON
ap-1688	49	26	>	>	X
ap-1688	49	27	0	0	X
ap-1688	49	28	.	.	PUNCT
ap-1688	49	29	theorem	theorem	VERB
ap-1688	49	30	1	1	NUM
ap-1688	49	31	presents	present	NOUN
ap-1688	49	32	conditions	condition	NOUN
ap-1688	49	33	under	under	ADP
ap-1688	49	34	which	which	PRON
ap-1688	49	35	a	a	DET
ap-1688	49	36	solution	solution	NOUN
ap-1688	49	37	decreases	decrease	VERB
ap-1688	49	38	at	at	ADP
ap-1688	49	39	a	a	DET
ap-1688	49	40	higher	high	ADJ
ap-1688	49	41	rate	rate	NOUN
ap-1688	49	42	than	than	ADP
ap-1688	49	43	(	(	PUNCT
ap-1688	49	44	1	1	NUM
ap-1688	49	45	+	+	NUM
ap-1688	49	46	t)−5/4	t)−5/4	PROPN
ap-1688	49	47	.	.	PUNCT
ap-1688	50	1	however	however	ADV
ap-1688	50	2	,	,	PUNCT
ap-1688	50	3	while	while	SCONJ
ap-1688	50	4	it	it	PRON
ap-1688	50	5	is	be	AUX
ap-1688	50	6	simple	simple	ADJ
ap-1688	50	7	to	to	PART
ap-1688	50	8	fulfill	fulfill	VERB
ap-1688	50	9	the	the	DET
ap-1688	50	10	condition	condition	NOUN
ap-1688	50	11	(	(	PUNCT
ap-1688	50	12	bh	bh	NOUN
ap-1688	50	13	,	,	PUNCT
ap-1688	50	14	k	k	PROPN
ap-1688	50	15	)	)	PUNCT
ap-1688	50	16	≡	≡	PROPN
ap-1688	50	17	0	0	PUNCT
ap-1688	51	1	just	just	ADV
ap-1688	51	2	by	by	ADP
ap-1688	51	3	a	a	DET
ap-1688	51	4	suitable	suitable	ADJ
ap-1688	51	5	choice	choice	NOUN
ap-1688	51	6	of	of	ADP
ap-1688	51	7	the	the	DET
ap-1688	51	8	initial	initial	ADJ
ap-1688	51	9	condition	condition	NOUN
ap-1688	51	10	,	,	PUNCT
ap-1688	51	11	it	it	PRON
ap-1688	51	12	is	be	AUX
ap-1688	51	13	difficult	difficult	ADJ
ap-1688	51	14	to	to	PART
ap-1688	51	15	observe	observe	VERB
ap-1688	51	16	the	the	DET
ap-1688	51	17	condition	condition	NOUN
ap-1688	51	18	λh	λh	ADP
ap-1688	51	19	,	,	PUNCT
ap-1688	51	20	k	k	NOUN
ap-1688	51	21	=	=	PUNCT
ap-1688	51	22	cδh	cδh	PROPN
ap-1688	51	23	,	,	PUNCT
ap-1688	51	24	k	k	PROPN
ap-1688	51	25	,	,	PUNCT
ap-1688	51	26	since	since	SCONJ
ap-1688	51	27	it	it	PRON
ap-1688	51	28	includes	include	VERB
ap-1688	51	29	the	the	DET
ap-1688	51	30	solution	solution	NOUN
ap-1688	51	31	itself	itself	PRON
ap-1688	51	32	.	.	PUNCT
ap-1688	52	1	thus	thus	ADV
ap-1688	52	2	,	,	PUNCT
ap-1688	52	3	theorem	theorem	VERB
ap-1688	52	4	1	1	NUM
ap-1688	52	5	neither	neither	CCONJ
ap-1688	52	6	ensures	ensure	VERB
ap-1688	52	7	the	the	DET
ap-1688	52	8	existence	existence	NOUN
ap-1688	52	9	of	of	ADP
ap-1688	52	10	solutions	solution	NOUN
ap-1688	52	11	decreasing	decrease	VERB
ap-1688	52	12	at	at	ADP
ap-1688	52	13	a	a	DET
ap-1688	52	14	rate	rate	NOUN
ap-1688	52	15	quicker	quick	ADJ
ap-1688	52	16	than	than	ADP
ap-1688	52	17	(	(	PUNCT
ap-1688	52	18	1	1	NUM
ap-1688	52	19	+	+	SYM
ap-1688	52	20	t)−5/4	t)−5/4	NUM
ap-1688	52	21	nor	nor	CCONJ
ap-1688	52	22	gives	give	VERB
ap-1688	52	23	a	a	DET
ap-1688	52	24	method	method	NOUN
ap-1688	52	25	for	for	ADP
ap-1688	52	26	possibly	possibly	ADV
ap-1688	52	27	constructing	construct	VERB
ap-1688	52	28	such	such	ADJ
ap-1688	52	29	solutions	solution	NOUN
ap-1688	52	30	.	.	PUNCT
ap-1688	53	1	this	this	DET
ap-1688	53	2	problem	problem	NOUN
ap-1688	53	3	is	be	AUX
ap-1688	53	4	solved	solve	VERB
ap-1688	53	5	by	by	ADP
ap-1688	53	6	brandolese	brandolese	PROPN
ap-1688	53	7	in	in	ADP
ap-1688	53	8	[	[	X
ap-1688	53	9	1	1	NUM
ap-1688	53	10	]	]	PUNCT
ap-1688	53	11	.	.	PUNCT
ap-1688	54	1	he	he	PRON
ap-1688	54	2	used	use	VERB
ap-1688	54	3	the	the	DET
ap-1688	54	4	following	follow	VERB
ap-1688	54	5	concept	concept	NOUN
ap-1688	54	6	of	of	ADP
ap-1688	54	7	a	a	DET
ap-1688	54	8	symmetric	symmetric	ADJ
ap-1688	54	9	solution	solution	NOUN
ap-1688	54	10	:	:	PUNCT
ap-1688	54	11	definition	definition	NOUN
ap-1688	54	12	1	1	NUM
ap-1688	54	13	.	.	PUNCT
ap-1688	55	1	a	a	DET
ap-1688	55	2	vector	vector	NOUN
ap-1688	55	3	field	field	NOUN
ap-1688	55	4	u0	u0	NOUN
ap-1688	55	5	=	=	SYM
ap-1688	55	6	(	(	PUNCT
ap-1688	55	7	u01	u01	PROPN
ap-1688	55	8	,	,	PUNCT
ap-1688	55	9	u02	u02	PROPN
ap-1688	55	10	,	,	PUNCT
ap-1688	55	11	u03	u03	NOUN
ap-1688	55	12	)	)	PUNCT
ap-1688	55	13	from	from	ADP
ap-1688	55	14	r3	r3	PROPN
ap-1688	55	15	to	to	ADP
ap-1688	55	16	r3	r3	PROPN
ap-1688	55	17	is	be	AUX
ap-1688	55	18	said	say	VERB
ap-1688	55	19	to	to	PART
ap-1688	55	20	be	be	AUX
ap-1688	55	21	symmetric	symmetric	ADJ
ap-1688	55	22	if	if	SCONJ
ap-1688	55	23	the	the	DET
ap-1688	55	24	following	follow	VERB
ap-1688	55	25	conditions	condition	NOUN
ap-1688	55	26	are	be	AUX
ap-1688	55	27	satisfied	satisfied	ADJ
ap-1688	55	28	for	for	ADP
ap-1688	55	29	all	all	DET
ap-1688	55	30	j	j	PROPN
ap-1688	55	31	,	,	PUNCT
ap-1688	55	32	k	k	PROPN
ap-1688	55	33	=	=	SYM
ap-1688	55	34	1	1	NUM
ap-1688	55	35	,	,	PUNCT
ap-1688	55	36	2	2	NUM
ap-1688	55	37	,	,	PUNCT
ap-1688	55	38	3	3	NUM
ap-1688	55	39	.	.	NOUN
ap-1688	55	40	1	1	NUM
ap-1688	55	41	.	.	X
ap-1688	56	1	u0j	u0j	NOUN
ap-1688	56	2	is	be	AUX
ap-1688	56	3	odd	odd	ADJ
ap-1688	56	4	with	with	ADP
ap-1688	56	5	respect	respect	NOUN
ap-1688	56	6	to	to	ADP
ap-1688	56	7	xj	xj	PROPN
ap-1688	56	8	and	and	CCONJ
ap-1688	56	9	even	even	ADV
ap-1688	56	10	with	with	ADP
ap-1688	56	11	respect	respect	NOUN
ap-1688	56	12	to	to	ADP
ap-1688	56	13	xk	xk	PROPN
ap-1688	56	14	,	,	PUNCT
ap-1688	56	15	j	j	PROPN
ap-1688	56	16	6=	6=	PROPN
ap-1688	56	17	k.	k.	PROPN
ap-1688	56	18	2	2	NUM
ap-1688	56	19	.	.	X
ap-1688	56	20	u01(x	u01(x	NOUN
ap-1688	56	21	)	)	PUNCT
ap-1688	56	22	=	=	SYM
ap-1688	56	23	u02(σx	u02(σx	X
ap-1688	56	24	)	)	PUNCT
ap-1688	56	25	=	=	SYM
ap-1688	56	26	u03(σ2x	u03(σ2x	NOUN
ap-1688	56	27	)	)	PUNCT
ap-1688	56	28	,	,	PUNCT
ap-1688	56	29	where	where	SCONJ
ap-1688	56	30	σ	σ	PROPN
ap-1688	56	31	is	be	AUX
ap-1688	56	32	the	the	DET
ap-1688	56	33	cycle	cycle	NOUN
ap-1688	56	34	σ(x1	σ(x1	NOUN
ap-1688	56	35	,	,	PUNCT
ap-1688	56	36	x2	x2	PRON
ap-1688	56	37	,	,	PUNCT
ap-1688	56	38	x3	x3	ADJ
ap-1688	56	39	)	)	PUNCT
ap-1688	57	1	=	=	SYM
ap-1688	57	2	(	(	PUNCT
ap-1688	57	3	x3	x3	ADJ
ap-1688	57	4	,	,	PUNCT
ap-1688	57	5	x1	x1	PROPN
ap-1688	57	6	,	,	PUNCT
ap-1688	57	7	x2	x2	PROPN
ap-1688	57	8	)	)	PUNCT
ap-1688	57	9	.	.	PUNCT
ap-1688	58	1	a	a	DET
ap-1688	58	2	simple	simple	ADJ
ap-1688	58	3	example	example	NOUN
ap-1688	58	4	of	of	ADP
ap-1688	58	5	a	a	DET
ap-1688	58	6	symmetric	symmetric	ADJ
ap-1688	58	7	and	and	CCONJ
ap-1688	58	8	solenoidal	solenoidal	ADJ
ap-1688	58	9	vector	vector	NOUN
ap-1688	58	10	field	field	NOUN
ap-1688	58	11	is	be	AUX
ap-1688	58	12	given	give	VERB
ap-1688	58	13	by	by	ADP
ap-1688	58	14	u0(x1	u0(x1	ADP
ap-1688	58	15	,	,	PUNCT
ap-1688	58	16	x2	x2	PROPN
ap-1688	58	17	,	,	PUNCT
ap-1688	58	18	x3	x3	ADJ
ap-1688	58	19	)	)	PUNCT
ap-1688	58	20	=	=	PUNCT
ap-1688	59	1			NUM
ap-1688	59	2	x1(x2	x1(x2	ADP
ap-1688	59	3	3	3	NUM
ap-1688	59	4	−	−	NOUN
ap-1688	60	1	x2	x2	NUM
ap-1688	60	2	2)e−|x|	2)e−|x|	NOUN
ap-1688	60	3	2	2	NUM
ap-1688	60	4	x2(x2	x2(x2	NUM
ap-1688	60	5	1	1	NUM
ap-1688	60	6	−	−	NOUN
ap-1688	61	1	x2	x2	NOUN
ap-1688	61	2	3)e−|x|	3)e−|x|	NOUN
ap-1688	61	3	2	2	NUM
ap-1688	61	4	x3(x2	x3(x2	NOUN
ap-1688	61	5	2	2	NUM
ap-1688	61	6	−	−	NOUN
ap-1688	61	7	x2	x2	SYM
ap-1688	61	8	1)e−|x|	1)e−|x|	PROPN
ap-1688	61	9	2	2	NUM
ap-1688	61	10			NOUN
ap-1688	61	11	.	.	PUNCT
ap-1688	62	1	(	(	PUNCT
ap-1688	62	2	4	4	X
ap-1688	62	3	)	)	PUNCT
ap-1688	62	4	brandolese	brandolese	NOUN
ap-1688	62	5	proved	prove	VERB
ap-1688	62	6	in	in	ADP
ap-1688	62	7	[	[	X
ap-1688	62	8	1	1	X
ap-1688	62	9	]	]	PUNCT
ap-1688	62	10	that	that	SCONJ
ap-1688	62	11	if	if	SCONJ
ap-1688	62	12	the	the	DET
ap-1688	62	13	initial	initial	ADJ
ap-1688	62	14	condition	condition	NOUN
ap-1688	62	15	u0	u0	NOUN
ap-1688	62	16	is	be	AUX
ap-1688	62	17	a	a	DET
ap-1688	62	18	solenoidal	solenoidal	ADJ
ap-1688	62	19	symmetric	symmetric	ADJ
ap-1688	62	20	vector	vector	NOUN
ap-1688	62	21	field	field	NOUN
ap-1688	62	22	then	then	ADV
ap-1688	62	23	there	there	PRON
ap-1688	62	24	exists	exist	VERB
ap-1688	62	25	a	a	DET
ap-1688	62	26	solution	solution	NOUN
ap-1688	62	27	which	which	PRON
ap-1688	62	28	is	be	AUX
ap-1688	62	29	symmetric	symmetric	ADJ
ap-1688	62	30	for	for	ADP
ap-1688	62	31	every	every	DET
ap-1688	62	32	time	time	NOUN
ap-1688	62	33	t	t	PROPN
ap-1688	62	34	≥	≥	NOUN
ap-1688	62	35	0	0	NUM
ap-1688	62	36	.	.	PUNCT
ap-1688	63	1	now	now	ADV
ap-1688	63	2	the	the	DET
ap-1688	63	3	existence	existence	NOUN
ap-1688	63	4	of	of	ADP
ap-1688	63	5	a	a	DET
ap-1688	63	6	solution	solution	NOUN
ap-1688	63	7	decreasing	decrease	VERB
ap-1688	63	8	at	at	ADP
ap-1688	63	9	the	the	DET
ap-1688	63	10	rate	rate	NOUN
ap-1688	63	11	o((1+t)−5/4	o((1+t)−5/4	PROPN
ap-1688	63	12	)	)	PUNCT
ap-1688	63	13	,	,	PUNCT
ap-1688	63	14	t→∞	t→∞	NUM
ap-1688	63	15	is	be	AUX
ap-1688	63	16	ensured	ensure	VERB
ap-1688	63	17	:	:	PUNCT
ap-1688	63	18	it	it	PRON
ap-1688	63	19	suffices	suffice	VERB
ap-1688	63	20	to	to	PART
ap-1688	63	21	take	take	VERB
ap-1688	63	22	the	the	DET
ap-1688	63	23	initial	initial	ADJ
ap-1688	63	24	condition	condition	NOUN
ap-1688	63	25	(	(	PUNCT
ap-1688	63	26	4	4	NUM
ap-1688	63	27	)	)	PUNCT
ap-1688	63	28	.	.	PUNCT
ap-1688	64	1	indeed	indeed	ADV
ap-1688	64	2	,	,	PUNCT
ap-1688	64	3	the	the	DET
ap-1688	64	4	associated	associated	ADJ
ap-1688	64	5	solution	solution	NOUN
ap-1688	64	6	u	u	NOUN
ap-1688	64	7	=	=	SYM
ap-1688	64	8	u(t	u(t	NOUN
ap-1688	64	9	)	)	PUNCT
ap-1688	64	10	is	be	AUX
ap-1688	64	11	then	then	ADV
ap-1688	64	12	symmetric	symmetric	ADJ
ap-1688	64	13	for	for	ADP
ap-1688	64	14	every	every	DET
ap-1688	64	15	time	time	NOUN
ap-1688	64	16	t	t	PROPN
ap-1688	64	17	≥	≥	NOUN
ap-1688	64	18	0	0	PUNCT
ap-1688	65	1	and	and	CCONJ
ap-1688	65	2	it	it	PRON
ap-1688	65	3	is	be	AUX
ap-1688	65	4	also	also	ADV
ap-1688	65	5	possible	possible	ADJ
ap-1688	65	6	to	to	PART
ap-1688	65	7	verify	verify	VERB
ap-1688	65	8	the	the	DET
ap-1688	65	9	veracity	veracity	NOUN
ap-1688	65	10	of	of	ADP
ap-1688	65	11	the	the	DET
ap-1688	65	12	condition	condition	NOUN
ap-1688	65	13	λh	λh	ADP
ap-1688	65	14	,	,	PUNCT
ap-1688	65	15	k	k	NOUN
ap-1688	65	16	=	=	PUNCT
ap-1688	65	17	cδh	cδh	PROPN
ap-1688	65	18	,	,	PUNCT
ap-1688	65	19	k.	k.	PROPN
ap-1688	65	20	moreover	moreover	ADV
ap-1688	65	21	,	,	PUNCT
ap-1688	65	22	the	the	DET
ap-1688	65	23	initial	initial	ADJ
ap-1688	65	24	condition	condition	NOUN
ap-1688	65	25	(	(	PUNCT
ap-1688	65	26	4	4	NUM
ap-1688	65	27	)	)	PUNCT
ap-1688	65	28	also	also	ADV
ap-1688	65	29	satisfies	satisfy	VERB
ap-1688	65	30	(	(	PUNCT
ap-1688	65	31	bh	bh	NOUN
ap-1688	65	32	,	,	PUNCT
ap-1688	65	33	k	k	PROPN
ap-1688	65	34	)	)	PUNCT
ap-1688	65	35	≡	≡	PROPN
ap-1688	65	36	0	0	NUM
ap-1688	65	37	and	and	CCONJ
ap-1688	65	38	the	the	DET
ap-1688	65	39	above	above	ADJ
ap-1688	65	40	conclusion	conclusion	NOUN
ap-1688	65	41	follows	follow	VERB
ap-1688	65	42	from	from	ADP
ap-1688	65	43	theorem	theorem	NOUN
ap-1688	65	44	1	1	NUM
ap-1688	65	45	.	.	PUNCT
ap-1688	65	46	using	use	VERB
ap-1688	65	47	symmetric	symmetric	ADJ
ap-1688	65	48	initial	initial	ADJ
ap-1688	65	49	conditions	condition	NOUN
ap-1688	65	50	,	,	PUNCT
ap-1688	65	51	brandolese	brandolese	PROPN
ap-1688	65	52	obtained	obtain	VERB
ap-1688	65	53	solutions	solution	NOUN
ap-1688	65	54	decreasing	decrease	VERB
ap-1688	65	55	at	at	ADP
ap-1688	65	56	the	the	DET
ap-1688	65	57	rate	rate	NOUN
ap-1688	65	58	(	(	PUNCT
ap-1688	65	59	1	1	NUM
ap-1688	65	60	+	+	CCONJ
ap-1688	65	61	t)−9/4	t)−9/4	PROPN
ap-1688	65	62	.	.	PUNCT
ap-1688	66	1	obtaining	obtain	VERB
ap-1688	66	2	solutions	solution	NOUN
ap-1688	66	3	which	which	PRON
ap-1688	66	4	decrease	decrease	VERB
ap-1688	66	5	at	at	ADP
ap-1688	66	6	an	an	DET
ap-1688	66	7	even	even	ADV
ap-1688	66	8	higher	high	ADJ
ap-1688	66	9	rate	rate	NOUN
ap-1688	66	10	seems	seem	VERB
ap-1688	66	11	to	to	PART
ap-1688	66	12	be	be	AUX
ap-1688	66	13	difficult	difficult	ADJ
ap-1688	66	14	,	,	PUNCT
ap-1688	66	15	with	with	ADP
ap-1688	66	16	only	only	ADV
ap-1688	66	17	one	one	NUM
ap-1688	66	18	exception	exception	NOUN
ap-1688	66	19	:	:	PUNCT
ap-1688	66	20	the	the	DET
ap-1688	66	21	existence	existence	NOUN
ap-1688	66	22	of	of	ADP
ap-1688	66	23	exponentially	exponentially	ADV
ap-1688	66	24	decreasing	decrease	VERB
ap-1688	66	25	solutions	solution	NOUN
ap-1688	66	26	is	be	AUX
ap-1688	66	27	very	very	ADV
ap-1688	66	28	well	well	ADV
ap-1688	66	29	known	know	VERB
ap-1688	66	30	,	,	PUNCT
ap-1688	66	31	as	as	SCONJ
ap-1688	66	32	was	be	AUX
ap-1688	66	33	described	describe	VERB
ap-1688	66	34	in	in	ADP
ap-1688	66	35	[	[	X
ap-1688	66	36	10	10	NUM
ap-1688	66	37	]	]	PUNCT
ap-1688	66	38	and	and	CCONJ
ap-1688	66	39	[	[	X
ap-1688	66	40	17	17	NUM
ap-1688	66	41	]	]	PUNCT
ap-1688	66	42	.	.	PUNCT
ap-1688	67	1	these	these	DET
ap-1688	67	2	solutions	solution	NOUN
ap-1688	67	3	are	be	AUX
ap-1688	67	4	very	very	ADV
ap-1688	67	5	rare	rare	ADJ
ap-1688	67	6	since	since	SCONJ
ap-1688	67	7	it	it	PRON
ap-1688	67	8	is	be	AUX
ap-1688	67	9	possible	possible	ADJ
ap-1688	67	10	to	to	PART
ap-1688	67	11	prove	prove	VERB
ap-1688	67	12	that	that	SCONJ
ap-1688	67	13	their	their	PRON
ap-1688	67	14	initial	initial	ADJ
ap-1688	67	15	conditions	condition	NOUN
ap-1688	67	16	lie	lie	VERB
ap-1688	67	17	only	only	ADV
ap-1688	67	18	on	on	ADP
ap-1688	67	19	a	a	DET
ap-1688	67	20	thin	thin	ADJ
ap-1688	67	21	manifold	manifold	NOUN
ap-1688	67	22	in	in	ADP
ap-1688	67	23	the	the	DET
ap-1688	67	24	phase	phase	NOUN
ap-1688	67	25	space	space	NOUN
ap-1688	67	26	.	.	PUNCT
ap-1688	68	1	the	the	DET
ap-1688	68	2	direct	direct	ADJ
ap-1688	68	3	construction	construction	NOUN
ap-1688	68	4	of	of	ADP
ap-1688	68	5	exponentially	exponentially	ADV
ap-1688	68	6	decreasing	decrease	VERB
ap-1688	68	7	solutions	solution	NOUN
ap-1688	68	8	in	in	ADP
ap-1688	68	9	three	three	NUM
ap-1688	68	10	dimensional	dimensional	ADJ
ap-1688	68	11	spaces	space	NOUN
ap-1688	68	12	is	be	AUX
ap-1688	68	13	still	still	ADV
ap-1688	68	14	an	an	DET
ap-1688	68	15	open	open	ADJ
ap-1688	68	16	problem	problem	NOUN
ap-1688	68	17	.	.	PUNCT
ap-1688	69	1	3	3	NUM
ap-1688	69	2	concentration	concentration	NOUN
ap-1688	69	3	of	of	ADP
ap-1688	69	4	energy	energy	NOUN
ap-1688	69	5	in	in	ADP
ap-1688	69	6	this	this	DET
ap-1688	69	7	section	section	NOUN
ap-1688	69	8	,	,	PUNCT
ap-1688	69	9	we	we	PRON
ap-1688	69	10	present	present	VERB
ap-1688	69	11	several	several	ADJ
ap-1688	69	12	results	result	NOUN
ap-1688	69	13	concerning	concern	VERB
ap-1688	69	14	the	the	DET
ap-1688	69	15	large	large	ADJ
ap-1688	69	16	time	time	NOUN
ap-1688	69	17	energy	energy	NOUN
ap-1688	69	18	concentration	concentration	NOUN
ap-1688	69	19	which	which	PRON
ap-1688	69	20	occurs	occur	VERB
ap-1688	69	21	in	in	ADP
ap-1688	69	22	every	every	DET
ap-1688	69	23	(	(	PUNCT
ap-1688	69	24	turbulent	turbulent	ADJ
ap-1688	69	25	)	)	PUNCT
ap-1688	69	26	solution	solution	NOUN
ap-1688	69	27	:	:	PUNCT
ap-1688	69	28	the	the	DET
ap-1688	69	29	energy	energy	NOUN
ap-1688	69	30	of	of	ADP
ap-1688	69	31	the	the	DET
ap-1688	69	32	solution	solution	NOUN
ap-1688	69	33	concentrates	concentrate	VERB
ap-1688	69	34	in	in	ADP
ap-1688	69	35	frequencies	frequency	NOUN
ap-1688	69	36	localized	localize	VERB
ap-1688	69	37	in	in	ADP
ap-1688	69	38	an	an	DET
ap-1688	69	39	annulus	annulus	NOUN
ap-1688	69	40	(	(	PUNCT
ap-1688	69	41	for	for	ADP
ap-1688	69	42	the	the	DET
ap-1688	69	43	case	case	NOUN
ap-1688	69	44	of	of	ADP
ap-1688	69	45	solutions	solution	NOUN
ap-1688	69	46	decreasing	decrease	VERB
ap-1688	69	47	exponentially	exponentially	ADV
ap-1688	69	48	)	)	PUNCT
ap-1688	69	49	or	or	CCONJ
ap-1688	69	50	a	a	DET
ap-1688	69	51	ball	ball	NOUN
ap-1688	69	52	(	(	PUNCT
ap-1688	69	53	for	for	ADP
ap-1688	69	54	the	the	DET
ap-1688	69	55	case	case	NOUN
ap-1688	69	56	of	of	ADP
ap-1688	69	57	solutions	solution	NOUN
ap-1688	69	58	not	not	PART
ap-1688	69	59	decreasing	decrease	VERB
ap-1688	69	60	exponentially	exponentially	ADV
ap-1688	69	61	)	)	PUNCT
ap-1688	69	62	in	in	ADP
ap-1688	69	63	the	the	DET
ap-1688	69	64	frequency	frequency	NOUN
ap-1688	69	65	space	space	NOUN
ap-1688	69	66	.	.	PUNCT
ap-1688	70	1	the	the	DET
ap-1688	70	2	diameter	diameter	NOUN
ap-1688	70	3	of	of	ADP
ap-1688	70	4	the	the	DET
ap-1688	70	5	annulus	annulus	NOUN
ap-1688	70	6	determines	determine	VERB
ap-1688	70	7	the	the	DET
ap-1688	70	8	rate	rate	NOUN
ap-1688	70	9	of	of	ADP
ap-1688	70	10	the	the	DET
ap-1688	70	11	exponential	exponential	ADJ
ap-1688	70	12	decay	decay	NOUN
ap-1688	70	13	of	of	ADP
ap-1688	70	14	the	the	DET
ap-1688	70	15	solution	solution	NOUN
ap-1688	70	16	,	,	PUNCT
ap-1688	70	17	and	and	CCONJ
ap-1688	70	18	it	it	PRON
ap-1688	70	19	can	can	AUX
ap-1688	70	20	be	be	AUX
ap-1688	70	21	arbitrarily	arbitrarily	ADV
ap-1688	70	22	narrow	narrow	ADJ
ap-1688	70	23	.	.	PUNCT
ap-1688	71	1	the	the	DET
ap-1688	71	2	ball	ball	NOUN
ap-1688	71	3	is	be	AUX
ap-1688	71	4	centered	center	VERB
ap-1688	71	5	in	in	ADP
ap-1688	71	6	the	the	DET
ap-1688	71	7	origin	origin	NOUN
ap-1688	71	8	of	of	ADP
ap-1688	71	9	the	the	DET
ap-1688	71	10	coordinates	coordinate	NOUN
ap-1688	71	11	,	,	PUNCT
ap-1688	71	12	and	and	CCONJ
ap-1688	71	13	can	can	AUX
ap-1688	71	14	have	have	VERB
ap-1688	71	15	an	an	DET
ap-1688	71	16	arbitrarily	arbitrarily	ADV
ap-1688	71	17	small	small	ADJ
ap-1688	71	18	diameter	diameter	NOUN
ap-1688	71	19	(	(	PUNCT
ap-1688	71	20	for	for	ADP
ap-1688	71	21	the	the	DET
ap-1688	71	22	results	result	NOUN
ap-1688	71	23	presented	present	VERB
ap-1688	71	24	in	in	ADP
ap-1688	71	25	this	this	DET
ap-1688	71	26	section	section	NOUN
ap-1688	71	27	,	,	PUNCT
ap-1688	71	28	see	see	VERB
ap-1688	71	29	[	[	X
ap-1688	71	30	15	15	NUM
ap-1688	71	31	]	]	PUNCT
ap-1688	71	32	,	,	PUNCT
ap-1688	72	1	[	[	X
ap-1688	72	2	16	16	NUM
ap-1688	72	3	]	]	PUNCT
ap-1688	72	4	,	,	PUNCT
ap-1688	72	5	[	[	X
ap-1688	72	6	17	17	NUM
ap-1688	72	7	]	]	PUNCT
ap-1688	72	8	and	and	CCONJ
ap-1688	72	9	[	[	X
ap-1688	72	10	18	18	NUM
ap-1688	72	11	]	]	NUM
ap-1688	72	12	)	)	PUNCT
ap-1688	72	13	.	.	PUNCT
ap-1688	73	1	the	the	DET
ap-1688	73	2	following	follow	VERB
ap-1688	73	3	theorem	theorem	NOUN
ap-1688	73	4	and	and	CCONJ
ap-1688	73	5	the	the	DET
ap-1688	73	6	ensuing	ensue	VERB
ap-1688	73	7	remarks	remark	NOUN
ap-1688	73	8	provide	provide	VERB
ap-1688	73	9	precise	precise	ADJ
ap-1688	73	10	information	information	NOUN
ap-1688	73	11	.	.	PUNCT
ap-1688	74	1	theorem	theorem	NOUN
ap-1688	74	2	2	2	NUM
ap-1688	74	3	.	.	PUNCT
ap-1688	75	1	let	let	VERB
ap-1688	75	2	u	u	PRON
ap-1688	75	3	be	be	AUX
ap-1688	75	4	a	a	DET
ap-1688	75	5	nonzero	nonzero	ADJ
ap-1688	75	6	turbulent	turbulent	ADJ
ap-1688	75	7	solution	solution	NOUN
ap-1688	75	8	of	of	ADP
ap-1688	75	9	(	(	PUNCT
ap-1688	75	10	1)–(3	1)–(3	NUM
ap-1688	75	11	)	)	PUNCT
ap-1688	75	12	.	.	PUNCT
ap-1688	76	1	then	then	ADV
ap-1688	76	2	there	there	PRON
ap-1688	76	3	exists	exist	VERB
ap-1688	76	4	a	a	DET
ap-1688	76	5	∈	∈	NOUN
ap-1688	77	1	[	[	X
ap-1688	77	2	0,∞	0,∞	NOUN
ap-1688	77	3	)	)	PUNCT
ap-1688	78	1	such	such	ADJ
ap-1688	78	2	that	that	SCONJ
ap-1688	78	3	lim	lim	PROPN
ap-1688	78	4	t→∞	t→∞	PRON
ap-1688	78	5	‖(ea+ε	‖(ea+ε	PROPN
ap-1688	78	6	−	−	PROPN
ap-1688	78	7	ea−ε)u(t)‖2	ea−ε)u(t)‖2	PROPN
ap-1688	78	8	‖u(t)‖2	‖u(t)‖2	PUNCT
ap-1688	78	9	=	=	SYM
ap-1688	78	10	1	1	NUM
ap-1688	78	11	(	(	PUNCT
ap-1688	78	12	5	5	NUM
ap-1688	78	13	)	)	PUNCT
ap-1688	78	14	for	for	ADP
ap-1688	78	15	every	every	DET
ap-1688	78	16	ε	ε	PROPN
ap-1688	78	17	>	>	X
ap-1688	78	18	0	0	PROPN
ap-1688	78	19	,	,	PUNCT
ap-1688	78	20	where	where	SCONJ
ap-1688	78	21	we	we	PRON
ap-1688	78	22	put	put	VERB
ap-1688	78	23	ea−ε	ea−ε	NOUN
ap-1688	78	24	=	=	SYM
ap-1688	78	25	0	0	PUNCT
ap-1688	79	1	if	if	SCONJ
ap-1688	79	2	a	a	DET
ap-1688	79	3	−	−	PROPN
ap-1688	79	4	ε	ε	PROPN
ap-1688	79	5	<	<	X
ap-1688	79	6	0	0	NUM
ap-1688	79	7	.	.	PUNCT
ap-1688	80	1	the	the	DET
ap-1688	80	2	number	number	NOUN
ap-1688	80	3	a	a	PRON
ap-1688	80	4	can	can	AUX
ap-1688	80	5	be	be	AUX
ap-1688	80	6	explicitly	explicitly	ADV
ap-1688	80	7	computed	compute	VERB
ap-1688	80	8	as	as	ADP
ap-1688	80	9	a	a	DET
ap-1688	80	10	=	=	NOUN
ap-1688	80	11	limt→∞‖a1/2u(t)‖22/‖u(t)‖22	limt→∞‖a1/2u(t)‖22/‖u(t)‖22	NOUN
ap-1688	80	12	.	.	PUNCT
ap-1688	81	1	100	100	NUM
ap-1688	81	2	acta	acta	PROPN
ap-1688	81	3	polytechnica	polytechnica	PROPN
ap-1688	81	4	vol	vol	NOUN
ap-1688	81	5	.	.	PROPN
ap-1688	82	1	52	52	NUM
ap-1688	82	2	no	no	NOUN
ap-1688	82	3	.	.	PUNCT
ap-1688	83	1	6/2012	6/2012	NUM
ap-1688	83	2	further	far	ADV
ap-1688	83	3	,	,	PUNCT
ap-1688	83	4	a	a	DET
ap-1688	83	5	=	=	X
ap-1688	83	6	sup{λ	sup{λ	NOUN
ap-1688	83	7	≥	≥	NOUN
ap-1688	83	8	0	0	NUM
ap-1688	83	9	;	;	PUNCT
ap-1688	83	10	lim	lim	PROPN
ap-1688	83	11	t→∞	t→∞	X
ap-1688	83	12	‖u(t)‖2eλt	‖u(t)‖2eλt	PROPN
ap-1688	83	13	=	=	PUNCT
ap-1688	83	14	0	0	NUM
ap-1688	83	15	}	}	PUNCT
ap-1688	83	16	,	,	PUNCT
ap-1688	83	17	which	which	PRON
ap-1688	83	18	implies	imply	VERB
ap-1688	83	19	that	that	SCONJ
ap-1688	83	20	the	the	DET
ap-1688	83	21	energy	energy	NOUN
ap-1688	83	22	of	of	ADP
ap-1688	83	23	u	u	PRON
ap-1688	83	24	decreases	decrease	VERB
ap-1688	83	25	exponentially	exponentially	ADV
ap-1688	83	26	for	for	ADP
ap-1688	83	27	t→∞	t→∞	NUM
ap-1688	83	28	if	if	SCONJ
ap-1688	83	29	and	and	CCONJ
ap-1688	83	30	only	only	ADV
ap-1688	83	31	if	if	SCONJ
ap-1688	83	32	a	a	DET
ap-1688	83	33	>	>	X
ap-1688	83	34	0	0	NUM
ap-1688	83	35	.	.	PUNCT
ap-1688	84	1	finally	finally	ADV
ap-1688	84	2	,	,	PUNCT
ap-1688	84	3	if	if	SCONJ
ap-1688	84	4	a	a	DET
ap-1688	84	5	>	>	X
ap-1688	84	6	0	0	PUNCT
ap-1688	84	7	and	and	CCONJ
ap-1688	84	8	ε	ε	PROPN
ap-1688	84	9	>	>	X
ap-1688	84	10	0	0	PROPN
ap-1688	84	11	,	,	PUNCT
ap-1688	84	12	then	then	ADV
ap-1688	84	13	lim	lim	PROPN
ap-1688	84	14	t→∞	t→∞	PRON
ap-1688	84	15	e(a−ε)t‖u(t)‖2	e(a−ε)t‖u(t)‖2	NOUN
ap-1688	84	16	=	=	SYM
ap-1688	84	17	0	0	NUM
ap-1688	84	18	and	and	CCONJ
ap-1688	84	19	lim	lim	PROPN
ap-1688	84	20	t→∞	t→∞	PRON
ap-1688	84	21	e(a+ε)t‖u(t)‖2	e(a+ε)t‖u(t)‖2	NOUN
ap-1688	85	1	=	=	VERB
ap-1688	85	2	∞.	∞.	PROPN
ap-1688	85	3	remark	remark	NOUN
ap-1688	85	4	1	1	NUM
ap-1688	85	5	.	.	PUNCT
ap-1688	86	1	it	it	PRON
ap-1688	86	2	is	be	AUX
ap-1688	86	3	possible	possible	ADJ
ap-1688	86	4	to	to	PART
ap-1688	86	5	show	show	VERB
ap-1688	86	6	(	(	PUNCT
ap-1688	86	7	see	see	VERB
ap-1688	86	8	[	[	X
ap-1688	86	9	5	5	NUM
ap-1688	86	10	]	]	PUNCT
ap-1688	86	11	)	)	PUNCT
ap-1688	86	12	that	that	SCONJ
ap-1688	86	13	for	for	ADP
ap-1688	86	14	every	every	DET
ap-1688	86	15	λ	λ	PROPN
ap-1688	86	16	>	>	X
ap-1688	87	1	0	0	PUNCT
ap-1688	87	2	f	f	X
ap-1688	87	3	(	(	PUNCT
ap-1688	87	4	eλu(t))(ξ	eλu(t))(ξ	NOUN
ap-1688	87	5	)	)	PUNCT
ap-1688	87	6	=	=	PUNCT
ap-1688	87	7	χb√λ(0)f	χb√λ(0)f	NUM
ap-1688	87	8	(	(	PUNCT
ap-1688	87	9	u(t))(ξ	u(t))(ξ	PROPN
ap-1688	87	10	)	)	PUNCT
ap-1688	87	11	,	,	PUNCT
ap-1688	87	12	where	where	SCONJ
ap-1688	87	13	f	f	PROPN
ap-1688	87	14	denotes	denote	VERB
ap-1688	87	15	the	the	DET
ap-1688	87	16	fourier	fourier	NOUN
ap-1688	87	17	transform	transform	NOUN
ap-1688	87	18	and	and	CCONJ
ap-1688	87	19	χb√λ(0	χb√λ(0	NOUN
ap-1688	87	20	)	)	PUNCT
ap-1688	87	21	is	be	AUX
ap-1688	87	22	the	the	DET
ap-1688	87	23	characteristic	characteristic	ADJ
ap-1688	87	24	function	function	NOUN
ap-1688	87	25	of	of	ADP
ap-1688	87	26	b√λ(0	b√λ(0	NOUN
ap-1688	87	27	)	)	PUNCT
ap-1688	88	1	=	=	PRON
ap-1688	88	2	{	{	PUNCT
ap-1688	88	3	x	x	SYM
ap-1688	88	4	∈	∈	PROPN
ap-1688	88	5	r3	r3	PROPN
ap-1688	88	6	;	;	PUNCT
ap-1688	88	7	‖x‖	‖x‖	PROPN
ap-1688	88	8	≤	≤	NOUN
ap-1688	88	9	√	√	NUM
ap-1688	88	10	λ	λ	SYM
ap-1688	88	11	}	}	PUNCT
ap-1688	88	12	.	.	PUNCT
ap-1688	89	1	consequently	consequently	ADV
ap-1688	89	2	,	,	PUNCT
ap-1688	89	3	the	the	DET
ap-1688	89	4	equality	equality	NOUN
ap-1688	89	5	(	(	PUNCT
ap-1688	89	6	5	5	NUM
ap-1688	89	7	)	)	PUNCT
ap-1688	89	8	can	can	AUX
ap-1688	89	9	be	be	AUX
ap-1688	89	10	written	write	VERB
ap-1688	89	11	in	in	ADP
ap-1688	89	12	the	the	DET
ap-1688	89	13	form	form	NOUN
ap-1688	89	14	lim	lim	PROPN
ap-1688	89	15	t→∞	t→∞	DET
ap-1688	89	16	∫	∫	PROPN
ap-1688	89	17	b√a+ε(0)\b√a−ε(0	b√a+ε(0)\b√a−ε(0	PROPN
ap-1688	89	18	)	)	PUNCT
ap-1688	89	19	|f	|f	PROPN
ap-1688	90	1	(	(	PUNCT
ap-1688	90	2	u(t))(ξ)|2	u(t))(ξ)|2	PROPN
ap-1688	90	3	dξ∫	dξ∫	PROPN
ap-1688	90	4	r3	r3	PROPN
ap-1688	90	5	|f	|f	PROPN
ap-1688	90	6	(	(	PUNCT
ap-1688	90	7	u(t))(ξ)|2	u(t))(ξ)|2	PROPN
ap-1688	90	8	dξ	dξ	PROPN
ap-1688	90	9	=	=	PUNCT
ap-1688	90	10	1	1	NUM
ap-1688	90	11	if	if	SCONJ
ap-1688	90	12	a	a	DET
ap-1688	90	13	>	>	X
ap-1688	90	14	0	0	NUM
ap-1688	90	15	and	and	CCONJ
ap-1688	90	16	ε	ε	PROPN
ap-1688	90	17	∈	∈	PROPN
ap-1688	90	18	(	(	PUNCT
ap-1688	90	19	0	0	NUM
ap-1688	90	20	,	,	PUNCT
ap-1688	90	21	a	a	PRON
ap-1688	90	22	)	)	PUNCT
ap-1688	90	23	and	and	CCONJ
ap-1688	90	24	lim	lim	PROPN
ap-1688	90	25	t→∞	t→∞	DET
ap-1688	90	26	∫	∫	PROPN
ap-1688	90	27	b√ε(0	b√ε(0	PROPN
ap-1688	90	28	)	)	PUNCT
ap-1688	90	29	|f	|f	PROPN
ap-1688	91	1	(	(	PUNCT
ap-1688	91	2	u(t))(ξ)|2	u(t))(ξ)|2	PROPN
ap-1688	91	3	dξ∫	dξ∫	PROPN
ap-1688	91	4	r3	r3	PROPN
ap-1688	91	5	|f	|f	PROPN
ap-1688	91	6	(	(	PUNCT
ap-1688	91	7	u(t))(ξ)|2	u(t))(ξ)|2	PROPN
ap-1688	91	8	dξ	dξ	PROPN
ap-1688	91	9	=	=	PUNCT
ap-1688	91	10	1	1	NUM
ap-1688	91	11	if	if	SCONJ
ap-1688	91	12	a	a	DET
ap-1688	91	13	=	=	SYM
ap-1688	91	14	0	0	NUM
ap-1688	91	15	and	and	CCONJ
ap-1688	91	16	ε	ε	X
ap-1688	91	17	>	>	X
ap-1688	91	18	0	0	PROPN
ap-1688	91	19	.	.	PUNCT
ap-1688	92	1	the	the	DET
ap-1688	92	2	results	result	NOUN
ap-1688	92	3	from	from	ADP
ap-1688	92	4	theorem	theorem	ADJ
ap-1688	92	5	2	2	NUM
ap-1688	92	6	and	and	CCONJ
ap-1688	92	7	remark	remark	NOUN
ap-1688	92	8	1	1	NUM
ap-1688	92	9	can	can	AUX
ap-1688	92	10	be	be	AUX
ap-1688	92	11	interpreted	interpret	VERB
ap-1688	92	12	in	in	ADP
ap-1688	92	13	a	a	DET
ap-1688	92	14	way	way	NOUN
ap-1688	92	15	that	that	SCONJ
ap-1688	92	16	in	in	ADP
ap-1688	92	17	every	every	DET
ap-1688	92	18	turbulent	turbulent	ADJ
ap-1688	92	19	solution	solution	NOUN
ap-1688	92	20	the	the	DET
ap-1688	92	21	frequencies	frequency	NOUN
ap-1688	92	22	outside	outside	ADP
ap-1688	92	23	the	the	DET
ap-1688	92	24	annulus	annulus	NOUN
ap-1688	92	25	or	or	CCONJ
ap-1688	92	26	the	the	DET
ap-1688	92	27	ball	ball	NOUN
ap-1688	92	28	disappear	disappear	VERB
ap-1688	92	29	asymptotically	asymptotically	ADV
ap-1688	92	30	.	.	PUNCT
ap-1688	93	1	this	this	DET
ap-1688	93	2	result	result	NOUN
ap-1688	93	3	can	can	AUX
ap-1688	93	4	be	be	AUX
ap-1688	93	5	further	far	ADV
ap-1688	93	6	strengthened	strengthen	VERB
ap-1688	93	7	in	in	ADP
ap-1688	93	8	the	the	DET
ap-1688	93	9	following	following	ADJ
ap-1688	93	10	way	way	NOUN
ap-1688	93	11	:	:	PUNCT
ap-1688	93	12	theorem	theorem	NOUN
ap-1688	93	13	3	3	X
ap-1688	93	14	.	.	PUNCT
ap-1688	94	1	let	let	VERB
ap-1688	94	2	α	α	PRON
ap-1688	94	3	≥	≥	NOUN
ap-1688	94	4	0	0	NUM
ap-1688	94	5	.	.	PUNCT
ap-1688	95	1	then	then	ADV
ap-1688	95	2	lim	lim	PROPN
ap-1688	95	3	t→∞	t→∞	X
ap-1688	95	4	∫	∫	PROPN
ap-1688	95	5	kc	kc	PROPN
ap-1688	95	6	a	a	PROPN
ap-1688	95	7	,	,	PUNCT
ap-1688	95	8	ε	ε	PROPN
ap-1688	95	9	|ξ|4α|f	|ξ|4α|f	X
ap-1688	95	10	(	(	PUNCT
ap-1688	95	11	u(t))(ξ)|2	u(t))(ξ)|2	PROPN
ap-1688	95	12	dξ∫	dξ∫	PROPN
ap-1688	95	13	r3	r3	PROPN
ap-1688	95	14	|f	|f	PROPN
ap-1688	95	15	(	(	PUNCT
ap-1688	95	16	u(t))(ξ)|2	u(t))(ξ)|2	PROPN
ap-1688	95	17	dξ	dξ	PROPN
ap-1688	95	18	=	=	PUNCT
ap-1688	95	19	0	0	PROPN
ap-1688	95	20	,	,	PUNCT
ap-1688	95	21	where	where	SCONJ
ap-1688	95	22	kc	kc	PROPN
ap-1688	95	23	a	a	X
ap-1688	95	24	,	,	PUNCT
ap-1688	95	25	ε	ε	PROPN
ap-1688	95	26	=	=	SYM
ap-1688	95	27	r3	r3	PROPN
ap-1688	95	28	\ka	\ka	PROPN
ap-1688	95	29	,	,	PUNCT
ap-1688	95	30	ε	ε	PROPN
ap-1688	95	31	,	,	PUNCT
ap-1688	95	32	ka	ka	PROPN
ap-1688	95	33	,	,	PUNCT
ap-1688	95	34	ε	ε	PROPN
ap-1688	95	35	=	=	PUNCT
ap-1688	95	36	b√a+ε(0)\b√a−ε(0	b√a+ε(0)\b√a−ε(0	PROPN
ap-1688	95	37	)	)	PUNCT
ap-1688	95	38	if	if	SCONJ
ap-1688	95	39	a	a	DET
ap-1688	95	40	>	>	X
ap-1688	95	41	0	0	NUM
ap-1688	95	42	and	and	CCONJ
ap-1688	95	43	ka	ka	PROPN
ap-1688	95	44	,	,	PUNCT
ap-1688	95	45	ε	ε	PROPN
ap-1688	95	46	=	=	PUNCT
ap-1688	95	47	b√ε(0	b√ε(0	PROPN
ap-1688	95	48	)	)	PUNCT
ap-1688	95	49	if	if	SCONJ
ap-1688	95	50	a	a	PRON
ap-1688	95	51	=	=	NOUN
ap-1688	95	52	0	0	NUM
ap-1688	95	53	.	.	NOUN
ap-1688	95	54	up	up	ADP
ap-1688	95	55	to	to	ADP
ap-1688	95	56	now	now	ADV
ap-1688	95	57	we	we	PRON
ap-1688	95	58	have	have	AUX
ap-1688	95	59	discussed	discuss	VERB
ap-1688	95	60	the	the	DET
ap-1688	95	61	phenomenon	phenomenon	NOUN
ap-1688	95	62	of	of	ADP
ap-1688	95	63	the	the	DET
ap-1688	95	64	large	large	ADJ
ap-1688	95	65	time	time	NOUN
ap-1688	95	66	energy	energy	NOUN
ap-1688	95	67	concentration	concentration	NOUN
ap-1688	95	68	in	in	ADP
ap-1688	95	69	the	the	DET
ap-1688	95	70	frequency	frequency	NOUN
ap-1688	95	71	space	space	NOUN
ap-1688	95	72	which	which	PRON
ap-1688	95	73	occurs	occur	VERB
ap-1688	95	74	in	in	ADP
ap-1688	95	75	any	any	DET
ap-1688	95	76	turbulent	turbulent	ADJ
ap-1688	95	77	solution	solution	NOUN
ap-1688	95	78	.	.	PUNCT
ap-1688	96	1	in	in	ADP
ap-1688	96	2	theorem	theorem	NOUN
ap-1688	96	3	4	4	NUM
ap-1688	96	4	,	,	PUNCT
ap-1688	96	5	we	we	PRON
ap-1688	96	6	will	will	AUX
ap-1688	96	7	present	present	VERB
ap-1688	96	8	an	an	DET
ap-1688	96	9	example	example	NOUN
ap-1688	96	10	of	of	ADP
ap-1688	96	11	a	a	DET
ap-1688	96	12	concrete	concrete	ADJ
ap-1688	96	13	class	class	NOUN
ap-1688	96	14	of	of	ADP
ap-1688	96	15	initial	initial	ADJ
ap-1688	96	16	conditions	condition	NOUN
ap-1688	96	17	such	such	ADJ
ap-1688	96	18	that	that	SCONJ
ap-1688	96	19	if	if	SCONJ
ap-1688	96	20	u0	u0	ADJ
ap-1688	96	21	belongs	belong	VERB
ap-1688	96	22	to	to	ADP
ap-1688	96	23	this	this	DET
ap-1688	96	24	class	class	NOUN
ap-1688	96	25	and	and	CCONJ
ap-1688	96	26	u	u	NOUN
ap-1688	96	27	is	be	AUX
ap-1688	96	28	a	a	DET
ap-1688	96	29	turbulent	turbulent	ADJ
ap-1688	96	30	solution	solution	NOUN
ap-1688	96	31	with	with	ADP
ap-1688	96	32	u(0	u(0	NOUN
ap-1688	96	33	)	)	PUNCT
ap-1688	96	34	=	=	PUNCT
ap-1688	97	1	u0	u0	ADJ
ap-1688	97	2	,	,	PUNCT
ap-1688	97	3	then	then	ADV
ap-1688	97	4	the	the	DET
ap-1688	97	5	energy	energy	NOUN
ap-1688	97	6	of	of	ADP
ap-1688	97	7	the	the	DET
ap-1688	97	8	solution	solution	NOUN
ap-1688	97	9	concentrates	concentrate	VERB
ap-1688	97	10	asymptotically	asymptotically	ADV
ap-1688	97	11	in	in	ADP
ap-1688	97	12	frequencies	frequency	NOUN
ap-1688	97	13	from	from	ADP
ap-1688	97	14	an	an	DET
ap-1688	97	15	arbitrarily	arbitrarily	ADV
ap-1688	97	16	small	small	ADJ
ap-1688	97	17	ball	ball	NOUN
ap-1688	97	18	in	in	ADP
ap-1688	97	19	the	the	DET
ap-1688	97	20	frequency	frequency	NOUN
ap-1688	97	21	space	space	NOUN
ap-1688	97	22	centered	center	VERB
ap-1688	97	23	in	in	ADP
ap-1688	97	24	the	the	DET
ap-1688	97	25	origin	origin	NOUN
ap-1688	97	26	of	of	ADP
ap-1688	97	27	the	the	DET
ap-1688	97	28	coordinates	coordinate	NOUN
ap-1688	97	29	.	.	PUNCT
ap-1688	98	1	we	we	PRON
ap-1688	98	2	also	also	ADV
ap-1688	98	3	present	present	VERB
ap-1688	98	4	two	two	NUM
ap-1688	98	5	estimates	estimate	NOUN
ap-1688	98	6	of	of	ADP
ap-1688	98	7	the	the	DET
ap-1688	98	8	rate	rate	NOUN
ap-1688	98	9	of	of	ADP
ap-1688	98	10	energy	energy	NOUN
ap-1688	98	11	concentration	concentration	NOUN
ap-1688	98	12	(	(	PUNCT
ap-1688	98	13	see	see	VERB
ap-1688	98	14	[	[	X
ap-1688	98	15	18	18	NUM
ap-1688	98	16	]	]	NUM
ap-1688	98	17	)	)	PUNCT
ap-1688	98	18	.	.	PUNCT
ap-1688	99	1	for	for	ADP
ap-1688	99	2	a	a	DET
ap-1688	99	3	description	description	NOUN
ap-1688	99	4	of	of	ADP
ap-1688	99	5	the	the	DET
ap-1688	99	6	class	class	NOUN
ap-1688	99	7	mentioned	mention	VERB
ap-1688	99	8	in	in	ADP
ap-1688	99	9	the	the	DET
ap-1688	99	10	previous	previous	ADJ
ap-1688	99	11	paragraph	paragraph	NOUN
ap-1688	99	12	we	we	PRON
ap-1688	99	13	need	need	VERB
ap-1688	99	14	the	the	DET
ap-1688	99	15	following	follow	VERB
ap-1688	99	16	definition	definition	NOUN
ap-1688	99	17	.	.	PUNCT
ap-1688	100	1	definition	definition	NOUN
ap-1688	100	2	2	2	NUM
ap-1688	100	3	.	.	PUNCT
ap-1688	101	1	let	let	VERB
ap-1688	101	2	α	α	PRON
ap-1688	101	3	,	,	PUNCT
ap-1688	101	4	δ	δ	PROPN
ap-1688	101	5	>	>	X
ap-1688	101	6	0	0	PROPN
ap-1688	101	7	,	,	PUNCT
ap-1688	101	8	and	and	CCONJ
ap-1688	101	9	m	m	AUX
ap-1688	101	10	be	be	AUX
ap-1688	101	11	a	a	DET
ap-1688	101	12	real	real	ADJ
ap-1688	101	13	number	number	NOUN
ap-1688	101	14	.	.	PUNCT
ap-1688	102	1	we	we	PRON
ap-1688	102	2	define	define	VERB
ap-1688	102	3	kδ	kδ	PROPN
ap-1688	102	4	m	m	PROPN
ap-1688	102	5	,	,	PUNCT
ap-1688	102	6	α	α	X
ap-1688	102	7	=	=	PUNCT
ap-1688	102	8	{	{	PUNCT
ap-1688	102	9	v	v	NUM
ap-1688	102	10	∈	∈	PROPN
ap-1688	102	11	l2	l2	NOUN
ap-1688	102	12	σ	σ	PROPN
ap-1688	102	13	;	;	PUNCT
ap-1688	102	14	|f	|f	PROPN
ap-1688	102	15	(	(	PUNCT
ap-1688	102	16	v)(ξ)|	v)(ξ)|	PROPN
ap-1688	102	17	≥	≥	NUM
ap-1688	102	18	α|ξ|m,∀|ξ|	α|ξ|m,∀|ξ|	NOUN
ap-1688	102	19	≤	≤	NUM
ap-1688	102	20	δ	δ	PROPN
ap-1688	102	21	}	}	PUNCT
ap-1688	102	22	.	.	PUNCT
ap-1688	103	1	theorem	theorem	ADJ
ap-1688	103	2	4	4	NUM
ap-1688	103	3	.	.	PUNCT
ap-1688	104	1	let	let	VERB
ap-1688	104	2	α	α	PRON
ap-1688	104	3	,	,	PUNCT
ap-1688	104	4	δ	δ	PROPN
ap-1688	104	5	>	>	X
ap-1688	104	6	0	0	PROPN
ap-1688	104	7	,	,	PUNCT
ap-1688	104	8	m	m	VERB
ap-1688	104	9	>	>	X
ap-1688	104	10	−3/2	−3/2	ADJ
ap-1688	104	11	,	,	PUNCT
ap-1688	104	12	p	p	PROPN
ap-1688	104	13	∈	∈	PROPN
ap-1688	105	1	[	[	X
ap-1688	105	2	1	1	NUM
ap-1688	105	3	,	,	PUNCT
ap-1688	105	4	2	2	NUM
ap-1688	105	5	]	]	PUNCT
ap-1688	105	6	and	and	CCONJ
ap-1688	105	7	3	3	NUM
ap-1688	105	8	p	p	NOUN
ap-1688	105	9	−	−	PROPN
ap-1688	105	10	3	3	NUM
ap-1688	105	11	2	2	NUM
ap-1688	105	12	≤	≤	NOUN
ap-1688	105	13	m+	m+	NUM
ap-1688	105	14	3/2	3/2	NUM
ap-1688	105	15	<	<	X
ap-1688	105	16	min	min	NOUN
ap-1688	105	17	(	(	PUNCT
ap-1688	105	18	6	6	NUM
ap-1688	105	19	p	p	NOUN
ap-1688	105	20	−	−	NUM
ap-1688	105	21	5	5	NUM
ap-1688	105	22	2	2	NUM
ap-1688	105	23	,	,	PUNCT
ap-1688	105	24	5	5	NUM
ap-1688	105	25	2	2	NUM
ap-1688	105	26	)	)	PUNCT
ap-1688	105	27	.	.	PUNCT
ap-1688	106	1	let	let	VERB
ap-1688	106	2	further	far	ADV
ap-1688	106	3	u0	u0	VERB
ap-1688	106	4	∈	∈	PROPN
ap-1688	106	5	l2	l2	NOUN
ap-1688	106	6	σ	σ	PROPN
ap-1688	106	7	∩kδ	∩kδ	PROPN
ap-1688	106	8	m	m	PROPN
ap-1688	106	9	,	,	PUNCT
ap-1688	106	10	α	α	PROPN
ap-1688	106	11	∩	∩	ADJ
ap-1688	106	12	lp	lp	NOUN
ap-1688	106	13	and	and	CCONJ
ap-1688	106	14	u	u	PRON
ap-1688	106	15	be	be	VERB
ap-1688	106	16	a	a	DET
ap-1688	106	17	turbulent	turbulent	ADJ
ap-1688	106	18	solution	solution	NOUN
ap-1688	106	19	of	of	ADP
ap-1688	106	20	(	(	PUNCT
ap-1688	106	21	1)–(3	1)–(3	NUM
ap-1688	106	22	)	)	PUNCT
ap-1688	106	23	with	with	ADP
ap-1688	106	24	the	the	DET
ap-1688	106	25	initial	initial	ADJ
ap-1688	106	26	condition	condition	NOUN
ap-1688	106	27	u0	u0	ADJ
ap-1688	106	28	.	.	PUNCT
ap-1688	107	1	if	if	SCONJ
ap-1688	107	2	q	q	PROPN
ap-1688	107	3	≥	≥	NUM
ap-1688	107	4	1/2	1/2	NUM
ap-1688	107	5	,	,	PUNCT
ap-1688	107	6	then	then	ADV
ap-1688	107	7	there	there	PRON
ap-1688	107	8	exists	exist	VERB
ap-1688	107	9	c	c	NOUN
ap-1688	107	10	>	>	X
ap-1688	107	11	0	0	PUNCT
ap-1688	108	1	dependent	dependent	ADJ
ap-1688	108	2	only	only	ADV
ap-1688	108	3	on	on	ADP
ap-1688	108	4	‖u0‖2	‖u0‖2	PROPN
ap-1688	108	5	,	,	PUNCT
ap-1688	108	6	‖u0‖p	‖u0‖p	PROPN
ap-1688	108	7	,	,	PUNCT
ap-1688	108	8	δ	δ	PROPN
ap-1688	108	9	,	,	PUNCT
ap-1688	108	10	m	m	PROPN
ap-1688	108	11	,	,	PUNCT
ap-1688	108	12	α	α	PROPN
ap-1688	108	13	and	and	CCONJ
ap-1688	108	14	q	q	NOUN
ap-1688	108	15	such	such	ADJ
ap-1688	108	16	that	that	SCONJ
ap-1688	108	17	1−	1−	NUM
ap-1688	108	18	‖eλu(t)‖2	‖eλu(t)‖2	NUM
ap-1688	108	19	‖u(t)‖2	‖u(t)‖2	VERB
ap-1688	108	20	≤	≤	NUM
ap-1688	109	1	c	c	NOUN
ap-1688	109	2	λ	λ	X
ap-1688	109	3	t−1+(m−3	t−1+(m−3	ADV
ap-1688	109	4	/	/	SYM
ap-1688	109	5	p+3)/(2q	p+3)/(2q	NOUN
ap-1688	109	6	)	)	PUNCT
ap-1688	109	7	(	(	PUNCT
ap-1688	109	8	6	6	NUM
ap-1688	109	9	)	)	PUNCT
ap-1688	109	10	for	for	ADP
ap-1688	109	11	every	every	DET
ap-1688	109	12	λ	λ	PROPN
ap-1688	109	13	>	>	X
ap-1688	109	14	0	0	PUNCT
ap-1688	110	1	and	and	CCONJ
ap-1688	110	2	every	every	DET
ap-1688	110	3	t	t	NOUN
ap-1688	110	4	≥	≥	NOUN
ap-1688	110	5	1	1	NUM
ap-1688	110	6	.	.	PUNCT
ap-1688	111	1	let	let	VERB
ap-1688	111	2	λ0	λ0	NOUN
ap-1688	111	3	>	>	X
ap-1688	111	4	0	0	X
ap-1688	111	5	.	.	PUNCT
ap-1688	112	1	then	then	ADV
ap-1688	112	2	there	there	PRON
ap-1688	112	3	exists	exist	VERB
ap-1688	112	4	c	c	NOUN
ap-1688	112	5	>	>	X
ap-1688	112	6	0	0	PUNCT
ap-1688	113	1	dependent	dependent	ADJ
ap-1688	113	2	only	only	ADV
ap-1688	113	3	on	on	ADP
ap-1688	113	4	‖u0‖2	‖u0‖2	PROPN
ap-1688	113	5	,	,	PUNCT
ap-1688	113	6	‖u0‖p	‖u0‖p	PROPN
ap-1688	113	7	,	,	PUNCT
ap-1688	113	8	δ	δ	PROPN
ap-1688	113	9	,	,	PUNCT
ap-1688	113	10	m	m	PROPN
ap-1688	113	11	,	,	PUNCT
ap-1688	113	12	α	α	NOUN
ap-1688	113	13	and	and	CCONJ
ap-1688	113	14	λ0	λ0	NOUN
ap-1688	113	15	such	such	ADJ
ap-1688	113	16	that	that	SCONJ
ap-1688	113	17	1−	1−	NUM
ap-1688	113	18	‖eλu(t)‖2	‖eλu(t)‖2	PROPN
ap-1688	113	19	‖u(t)‖2	‖u(t)‖2	VERB
ap-1688	113	20	≤	≤	NUM
ap-1688	114	1	c	c	NOUN
ap-1688	114	2	λ2	λ2	NOUN
ap-1688	114	3	t	t	PROPN
ap-1688	114	4	−3	−3	PROPN
ap-1688	114	5	/	/	SYM
ap-1688	114	6	p−1	p−1	PROPN
ap-1688	114	7	(	(	PUNCT
ap-1688	114	8	7	7	NUM
ap-1688	114	9	)	)	PUNCT
ap-1688	114	10	for	for	ADP
ap-1688	114	11	all	all	DET
ap-1688	114	12	λ	λ	PROPN
ap-1688	114	13	≥	≥	NOUN
ap-1688	114	14	λ0	λ0	NOUN
ap-1688	114	15	and	and	CCONJ
ap-1688	114	16	t	t	NOUN
ap-1688	114	17	≥	≥	NUM
ap-1688	114	18	1	1	NUM
ap-1688	114	19	.	.	PUNCT
ap-1688	114	20	inequalities	inequality	NOUN
ap-1688	114	21	(	(	PUNCT
ap-1688	114	22	6	6	NUM
ap-1688	114	23	)	)	PUNCT
ap-1688	114	24	and	and	CCONJ
ap-1688	114	25	(	(	PUNCT
ap-1688	114	26	7	7	X
ap-1688	114	27	)	)	PUNCT
ap-1688	114	28	provide	provide	VERB
ap-1688	114	29	information	information	NOUN
ap-1688	114	30	about	about	ADP
ap-1688	114	31	the	the	DET
ap-1688	114	32	concentration	concentration	NOUN
ap-1688	114	33	of	of	ADP
ap-1688	114	34	the	the	DET
ap-1688	114	35	energy	energy	NOUN
ap-1688	114	36	in	in	ADP
ap-1688	114	37	low	low	ADJ
ap-1688	114	38	frequencies	frequency	NOUN
ap-1688	114	39	and	and	CCONJ
ap-1688	114	40	about	about	ADP
ap-1688	114	41	the	the	DET
ap-1688	114	42	rate	rate	NOUN
ap-1688	114	43	of	of	ADP
ap-1688	114	44	this	this	DET
ap-1688	114	45	concentration	concentration	NOUN
ap-1688	114	46	.	.	PUNCT
ap-1688	115	1	since	since	SCONJ
ap-1688	115	2	the	the	DET
ap-1688	115	3	energy	energy	NOUN
ap-1688	115	4	of	of	ADP
ap-1688	115	5	the	the	DET
ap-1688	115	6	solutions	solution	NOUN
ap-1688	115	7	described	describe	VERB
ap-1688	115	8	in	in	ADP
ap-1688	115	9	theorem	theorem	ADJ
ap-1688	115	10	5	5	NUM
ap-1688	115	11	concentrates	concentrate	NOUN
ap-1688	115	12	at	at	ADP
ap-1688	115	13	low	low	ADJ
ap-1688	115	14	frequencies	frequency	NOUN
ap-1688	115	15	,	,	PUNCT
ap-1688	115	16	the	the	DET
ap-1688	115	17	number	number	NOUN
ap-1688	115	18	a	a	NOUN
ap-1688	115	19	from	from	ADP
ap-1688	115	20	theorem	theorem	ADJ
ap-1688	115	21	2	2	NUM
ap-1688	115	22	is	be	AUX
ap-1688	115	23	equal	equal	ADJ
ap-1688	115	24	to	to	ADP
ap-1688	115	25	zero	zero	NUM
ap-1688	115	26	and	and	CCONJ
ap-1688	115	27	the	the	DET
ap-1688	115	28	energy	energy	NOUN
ap-1688	115	29	of	of	ADP
ap-1688	115	30	these	these	DET
ap-1688	115	31	solutions	solution	NOUN
ap-1688	115	32	does	do	AUX
ap-1688	115	33	not	not	PART
ap-1688	115	34	decrease	decrease	VERB
ap-1688	115	35	exponentially	exponentially	ADV
ap-1688	115	36	.	.	PUNCT
ap-1688	116	1	let	let	VERB
ap-1688	116	2	us	we	PRON
ap-1688	116	3	mention	mention	VERB
ap-1688	116	4	here	here	ADV
ap-1688	116	5	one	one	NUM
ap-1688	116	6	open	open	ADJ
ap-1688	116	7	problem	problem	NOUN
ap-1688	116	8	:	:	PUNCT
ap-1688	116	9	to	to	PART
ap-1688	116	10	find	find	VERB
ap-1688	116	11	a	a	DET
ap-1688	116	12	turbulent	turbulent	ADJ
ap-1688	116	13	solution	solution	NOUN
ap-1688	116	14	u	u	NOUN
ap-1688	116	15	with	with	ADP
ap-1688	116	16	an	an	DET
ap-1688	116	17	initial	initial	ADJ
ap-1688	116	18	condition	condition	NOUN
ap-1688	116	19	u0	u0	ADJ
ap-1688	116	20	such	such	ADJ
ap-1688	116	21	that	that	SCONJ
ap-1688	116	22	the	the	DET
ap-1688	116	23	number	number	NOUN
ap-1688	116	24	a	a	NOUN
ap-1688	116	25	from	from	ADP
ap-1688	116	26	theorem	theorem	ADJ
ap-1688	116	27	2	2	NUM
ap-1688	116	28	is	be	AUX
ap-1688	116	29	positive	positive	ADJ
ap-1688	116	30	.	.	PUNCT
ap-1688	117	1	in	in	ADP
ap-1688	117	2	other	other	ADJ
ap-1688	117	3	words	word	NOUN
ap-1688	117	4	,	,	PUNCT
ap-1688	117	5	to	to	PART
ap-1688	117	6	find	find	VERB
ap-1688	117	7	a	a	DET
ap-1688	117	8	solution	solution	NOUN
ap-1688	117	9	with	with	ADP
ap-1688	117	10	the	the	DET
ap-1688	117	11	energy	energy	NOUN
ap-1688	117	12	decreasing	decrease	VERB
ap-1688	117	13	at	at	ADP
ap-1688	117	14	the	the	DET
ap-1688	117	15	exponential	exponential	ADJ
ap-1688	117	16	rate	rate	NOUN
ap-1688	117	17	e−at	e−at	PROPN
ap-1688	117	18	,	,	PUNCT
ap-1688	117	19	t→∞	t→∞	PRON
ap-1688	117	20	and	and	CCONJ
ap-1688	117	21	concentrating	concentrate	VERB
ap-1688	117	22	in	in	ADP
ap-1688	117	23	frequencies	frequency	NOUN
ap-1688	117	24	from	from	ADP
ap-1688	117	25	an	an	DET
ap-1688	117	26	arbitrarily	arbitrarily	ADV
ap-1688	117	27	narrow	narrow	ADJ
ap-1688	117	28	annulus	annulus	NOUN
ap-1688	117	29	with	with	ADP
ap-1688	117	30	the	the	DET
ap-1688	117	31	middle	middle	PROPN
ap-1688	117	32	diameter	diameter	NOUN
ap-1688	117	33	a	a	DET
ap-1688	117	34	>	>	X
ap-1688	117	35	0	0	NUM
ap-1688	117	36	.	.	NOUN
ap-1688	117	37	4	4	NUM
ap-1688	117	38	solutions	solution	NOUN
ap-1688	117	39	in	in	ADP
ap-1688	117	40	besov	besov	NOUN
ap-1688	117	41	spaces	space	NOUN
ap-1688	117	42	we	we	PRON
ap-1688	117	43	start	start	VERB
ap-1688	117	44	this	this	DET
ap-1688	117	45	section	section	NOUN
ap-1688	117	46	with	with	ADP
ap-1688	117	47	a	a	DET
ap-1688	117	48	definition	definition	NOUN
ap-1688	117	49	of	of	ADP
ap-1688	117	50	homogeneous	homogeneous	ADJ
ap-1688	117	51	besov	besov	NOUN
ap-1688	117	52	spaces	space	NOUN
ap-1688	117	53	(	(	PUNCT
ap-1688	117	54	see	see	VERB
ap-1688	117	55	also	also	ADV
ap-1688	117	56	[	[	X
ap-1688	117	57	2	2	NUM
ap-1688	117	58	]	]	PUNCT
ap-1688	117	59	)	)	PUNCT
ap-1688	117	60	.	.	PUNCT
ap-1688	118	1	let	let	VERB
ap-1688	118	2	c	c	PRON
ap-1688	118	3	be	be	AUX
ap-1688	118	4	the	the	DET
ap-1688	118	5	annulus	annulus	NOUN
ap-1688	118	6	{	{	PUNCT
ap-1688	118	7	ξ	ξ	PROPN
ap-1688	118	8	∈	∈	PROPN
ap-1688	118	9	r3	r3	PROPN
ap-1688	118	10	;	;	PUNCT
ap-1688	118	11	3/4	3/4	NUM
ap-1688	118	12	≤	≤	NUM
ap-1688	118	13	|ξ|	|ξ|	PROPN
ap-1688	118	14	≤	≤	NOUN
ap-1688	118	15	8/3	8/3	NUM
ap-1688	118	16	}	}	PUNCT
ap-1688	118	17	.	.	PUNCT
ap-1688	119	1	there	there	PRON
ap-1688	119	2	exist	exist	VERB
ap-1688	119	3	the	the	DET
ap-1688	119	4	smooth	smooth	ADJ
ap-1688	119	5	radial	radial	ADJ
ap-1688	119	6	function	function	NOUN
ap-1688	119	7	χ	χ	NOUN
ap-1688	119	8	and	and	CCONJ
ap-1688	119	9	ϕ	ϕ	NOUN
ap-1688	119	10	with	with	ADP
ap-1688	119	11	the	the	DET
ap-1688	119	12	support	support	NOUN
ap-1688	119	13	b(0	b(0	NOUN
ap-1688	119	14	,	,	PUNCT
ap-1688	119	15	4/3	4/3	NUM
ap-1688	119	16	)	)	PUNCT
ap-1688	119	17	and	and	CCONJ
ap-1688	119	18	c	c	NOUN
ap-1688	119	19	,	,	PUNCT
ap-1688	119	20	resp	resp	NOUN
ap-1688	119	21	.	.	PUNCT
ap-1688	119	22	,	,	PUNCT
ap-1688	119	23	with	with	ADP
ap-1688	119	24	values	value	NOUN
ap-1688	119	25	in	in	ADP
ap-1688	119	26	[	[	X
ap-1688	119	27	0	0	NUM
ap-1688	119	28	,	,	PUNCT
ap-1688	119	29	1	1	NUM
ap-1688	119	30	]	]	PUNCT
ap-1688	119	31	,	,	PUNCT
ap-1688	119	32	and	and	CCONJ
ap-1688	119	33	such	such	ADJ
ap-1688	119	34	that	that	SCONJ
ap-1688	119	35	χ(ξ	χ(ξ	NOUN
ap-1688	119	36	)	)	PUNCT
ap-1688	119	37	+	+	CCONJ
ap-1688	119	38	∑	∑	ADP
ap-1688	119	39	j≥0	j≥0	PROPN
ap-1688	119	40	ϕ(2−jξ	ϕ(2−jξ	PROPN
ap-1688	119	41	)	)	PUNCT
ap-1688	119	42	=	=	SYM
ap-1688	120	1	1	1	NUM
ap-1688	120	2	,	,	PUNCT
ap-1688	120	3	∀ξ	∀ξ	X
ap-1688	120	4	∈	∈	PROPN
ap-1688	120	5	r3	r3	PROPN
ap-1688	120	6	∑	∑	PROPN
ap-1688	120	7	j∈z	j∈z	PROPN
ap-1688	120	8	ϕ(2−jξ	ϕ(2−jξ	PROPN
ap-1688	120	9	)	)	PUNCT
ap-1688	121	1	=	=	SYM
ap-1688	121	2	1	1	NUM
ap-1688	121	3	,	,	PUNCT
ap-1688	121	4	∀ξ	∀ξ	X
ap-1688	121	5	∈	∈	PROPN
ap-1688	121	6	r3	r3	X
ap-1688	121	7	\	\	PROPN
ap-1688	121	8	{	{	PUNCT
ap-1688	121	9	0	0	NUM
ap-1688	121	10	}	}	PUNCT
ap-1688	121	11	,	,	PUNCT
ap-1688	121	12	101	101	NUM
ap-1688	121	13	acta	acta	PROPN
ap-1688	121	14	polytechnica	polytechnica	PROPN
ap-1688	121	15	vol	vol	NOUN
ap-1688	121	16	.	.	PROPN
ap-1688	122	1	52	52	NUM
ap-1688	122	2	no	no	NOUN
ap-1688	122	3	.	.	PUNCT
ap-1688	123	1	6/2012	6/2012	NUM
ap-1688	123	2	suppϕ(2−j	suppϕ(2−j	ADP
ap-1688	123	3	·	·	PUNCT
ap-1688	123	4	)	)	PUNCT
ap-1688	123	5	∩	∩	NOUN
ap-1688	123	6	suppϕ(2−j	suppϕ(2−j	ADP
ap-1688	123	7	′	′	NUM
ap-1688	123	8	·	·	PUNCT
ap-1688	123	9	)	)	PUNCT
ap-1688	124	1	=	=	PUNCT
ap-1688	124	2	∅	∅	NOUN
ap-1688	124	3	if	if	SCONJ
ap-1688	124	4	|j	|j	PROPN
ap-1688	124	5	−	−	PROPN
ap-1688	124	6	j′|	j′|	ADP
ap-1688	124	7	≥	≥	NOUN
ap-1688	124	8	2	2	NUM
ap-1688	124	9	,	,	PUNCT
ap-1688	124	10	suppχ	suppχ	NOUN
ap-1688	124	11	∩	∩	NOUN
ap-1688	124	12	suppϕ(2−j	suppϕ(2−j	ADP
ap-1688	124	13	·	·	PUNCT
ap-1688	124	14	)	)	PUNCT
ap-1688	124	15	=	=	PUNCT
ap-1688	124	16	∅	∅	NOUN
ap-1688	124	17	if	if	SCONJ
ap-1688	124	18	j	j	PROPN
ap-1688	124	19	≥	≥	PROPN
ap-1688	124	20	1	1	NUM
ap-1688	124	21	.	.	PUNCT
ap-1688	125	1	let	let	VERB
ap-1688	125	2	h	h	NOUN
ap-1688	125	3	=	=	NOUN
ap-1688	125	4	f−1ϕ.	f−1ϕ.	PROPN
ap-1688	125	5	if	if	SCONJ
ap-1688	125	6	u	u	PROPN
ap-1688	125	7	∈	∈	PROPN
ap-1688	125	8	s′	s′	X
ap-1688	125	9	(	(	PUNCT
ap-1688	125	10	the	the	DET
ap-1688	125	11	space	space	NOUN
ap-1688	125	12	of	of	ADP
ap-1688	125	13	tempered	temper	VERB
ap-1688	125	14	distributions	distribution	NOUN
ap-1688	125	15	)	)	PUNCT
ap-1688	125	16	,	,	PUNCT
ap-1688	125	17	then	then	ADV
ap-1688	125	18	the	the	DET
ap-1688	125	19	homogeneous	homogeneous	ADJ
ap-1688	125	20	dyadic	dyadic	ADJ
ap-1688	125	21	blocks	block	NOUN
ap-1688	125	22	are	be	AUX
ap-1688	125	23	defined	define	VERB
ap-1688	125	24	for	for	ADP
ap-1688	125	25	j	j	PROPN
ap-1688	125	26	∈	∈	PROPN
ap-1688	125	27	z	z	PROPN
ap-1688	125	28	as	as	ADP
ap-1688	125	29	∆ju	∆ju	NOUN
ap-1688	125	30	=	=	SYM
ap-1688	125	31	23j	23j	NUM
ap-1688	125	32	∫	∫	PROPN
ap-1688	125	33	r3	r3	PROPN
ap-1688	125	34	h(2jy)u(x−	h(2jy)u(x−	PROPN
ap-1688	125	35	y)dy	y)dy	PROPN
ap-1688	125	36	.	.	PUNCT
ap-1688	126	1	the	the	DET
ap-1688	126	2	space	space	NOUN
ap-1688	126	3	of	of	ADP
ap-1688	126	4	the	the	DET
ap-1688	126	5	homogeneous	homogeneous	ADJ
ap-1688	126	6	distributions	distribution	NOUN
ap-1688	126	7	s′h	s′h	ADJ
ap-1688	126	8	is	be	AUX
ap-1688	126	9	defined	define	VERB
ap-1688	126	10	in	in	ADP
ap-1688	126	11	the	the	DET
ap-1688	126	12	following	following	ADJ
ap-1688	126	13	way	way	NOUN
ap-1688	126	14	:	:	PUNCT
ap-1688	126	15	u	u	PROPN
ap-1688	126	16	∈	∈	PROPN
ap-1688	126	17	s′	s′	PUNCT
ap-1688	126	18	belongs	belong	VERB
ap-1688	126	19	to	to	ADP
ap-1688	126	20	s′h	s′h	NOUN
ap-1688	126	21	if	if	SCONJ
ap-1688	126	22	and	and	CCONJ
ap-1688	126	23	only	only	ADV
ap-1688	126	24	if	if	SCONJ
ap-1688	126	25	u	u	PROPN
ap-1688	126	26	=	=	PUNCT
ap-1688	126	27	∑	∑	PROPN
ap-1688	126	28	j∈z	j∈z	PROPN
ap-1688	126	29	∆ju	∆ju	NOUN
ap-1688	126	30	.	.	PUNCT
ap-1688	127	1	we	we	PRON
ap-1688	127	2	can	can	AUX
ap-1688	127	3	now	now	ADV
ap-1688	127	4	define	define	VERB
ap-1688	127	5	the	the	DET
ap-1688	127	6	homogeneous	homogeneous	ADJ
ap-1688	127	7	besov	besov	NOUN
ap-1688	127	8	space	space	NOUN
ap-1688	127	9	bsp,∞	bsp,∞	PROPN
ap-1688	127	10	,	,	PUNCT
ap-1688	127	11	s	s	NOUN
ap-1688	127	12	∈	∈	PROPN
ap-1688	127	13	r	r	NOUN
ap-1688	127	14	,	,	PUNCT
ap-1688	127	15	p	p	NOUN
ap-1688	127	16	∈	∈	PROPN
ap-1688	128	1	[	[	X
ap-1688	128	2	1,∞	1,∞	NUM
ap-1688	128	3	]	]	PUNCT
ap-1688	128	4	.	.	PUNCT
ap-1688	129	1	this	this	DET
ap-1688	129	2	space	space	NOUN
ap-1688	129	3	consists	consist	VERB
ap-1688	129	4	of	of	ADP
ap-1688	129	5	those	those	DET
ap-1688	129	6	distributions	distribution	NOUN
ap-1688	129	7	from	from	ADP
ap-1688	129	8	s	s	NOUN
ap-1688	129	9	′	′	NUM
ap-1688	129	10	h	h	NOUN
ap-1688	129	11	such	such	ADJ
ap-1688	129	12	that	that	SCONJ
ap-1688	129	13	‖u‖bsp,∞	‖u‖bsp,∞	NOUN
ap-1688	129	14	=	=	SYM
ap-1688	129	15	sup	sup	NOUN
ap-1688	129	16	j∈z	j∈z	NOUN
ap-1688	130	1	2js‖∆ju‖p	2js‖∆ju‖p	PROPN
ap-1688	130	2	<	<	X
ap-1688	130	3	∞.	∞.	PROPN
ap-1688	130	4	suppose	suppose	VERB
ap-1688	130	5	now	now	ADV
ap-1688	130	6	that	that	SCONJ
ap-1688	130	7	u	u	PRON
ap-1688	130	8	is	be	AUX
ap-1688	130	9	a	a	DET
ap-1688	130	10	turbulent	turbulent	ADJ
ap-1688	130	11	solution	solution	NOUN
ap-1688	130	12	of	of	ADP
ap-1688	130	13	(	(	PUNCT
ap-1688	130	14	1	1	NUM
ap-1688	130	15	)	)	PUNCT
ap-1688	130	16	(	(	PUNCT
ap-1688	130	17	3	3	X
ap-1688	130	18	)	)	PUNCT
ap-1688	130	19	with	with	ADP
ap-1688	130	20	an	an	DET
ap-1688	130	21	initial	initial	ADJ
ap-1688	130	22	condition	condition	NOUN
ap-1688	130	23	u0	u0	ADJ
ap-1688	130	24	.	.	PUNCT
ap-1688	131	1	then	then	ADV
ap-1688	131	2	(	(	PUNCT
ap-1688	131	3	see	see	VERB
ap-1688	131	4	[	[	X
ap-1688	131	5	19	19	NUM
ap-1688	131	6	]	]	SYM
ap-1688	131	7	)	)	PUNCT
ap-1688	131	8	u(t	u(t	NOUN
ap-1688	131	9	)	)	PUNCT
ap-1688	131	10	=	=	PRON
ap-1688	131	11	et∆u0	et∆u0	NOUN
ap-1688	132	1	+	+	CCONJ
ap-1688	133	1	∫	∫	PROPN
ap-1688	133	2	t	t	PROPN
ap-1688	133	3	0	0	NUM
ap-1688	133	4	e∆(t−s)pσ∇(u⊗	e∆(t−s)pσ∇(u⊗	NUM
ap-1688	133	5	u(s))ds	u(s))ds	PROPN
ap-1688	133	6	.	.	PUNCT
ap-1688	134	1	if	if	SCONJ
ap-1688	134	2	we	we	PRON
ap-1688	134	3	denote	denote	VERB
ap-1688	134	4	the	the	DET
ap-1688	134	5	integral	integral	ADJ
ap-1688	134	6	from	from	ADP
ap-1688	134	7	the	the	DET
ap-1688	134	8	previous	previous	ADJ
ap-1688	134	9	equality	equality	NOUN
ap-1688	134	10	as	as	ADP
ap-1688	134	11	w(t	w(t	PROPN
ap-1688	134	12	)	)	PUNCT
ap-1688	134	13	and	and	CCONJ
ap-1688	134	14	use	use	VERB
ap-1688	134	15	the	the	DET
ap-1688	134	16	fact	fact	NOUN
ap-1688	134	17	that	that	SCONJ
ap-1688	134	18	the	the	DET
ap-1688	134	19	operator	operator	NOUN
ap-1688	134	20	pσ∇	pσ∇	NOUN
ap-1688	134	21	is	be	AUX
ap-1688	134	22	homogeneous	homogeneous	ADJ
ap-1688	134	23	of	of	ADP
ap-1688	134	24	degree	degree	NOUN
ap-1688	134	25	1	1	NUM
ap-1688	134	26	,	,	PUNCT
ap-1688	134	27	we	we	PRON
ap-1688	134	28	can	can	AUX
ap-1688	134	29	derive	derive	VERB
ap-1688	134	30	‖∆jw(t)‖1	‖∆jw(t)‖1	NOUN
ap-1688	134	31	≤	≤	NUM
ap-1688	134	32	∫	∫	PROPN
ap-1688	134	33	t	t	PROPN
ap-1688	134	34	0	0	NUM
ap-1688	134	35	ce−c(t−s)2	ce−c(t−s)2	PROPN
ap-1688	134	36	2j	2j	NUM
ap-1688	134	37	2j‖∆j(u⊗	2j‖∆j(u⊗	NUM
ap-1688	134	38	u(s))‖1ds	u(s))‖1ds	NOUN
ap-1688	134	39	.	.	PUNCT
ap-1688	135	1	so	so	ADV
ap-1688	135	2	,	,	PUNCT
ap-1688	135	3	we	we	PRON
ap-1688	135	4	have	have	VERB
ap-1688	135	5	for	for	ADP
ap-1688	135	6	every	every	DET
ap-1688	135	7	t	t	NOUN
ap-1688	135	8	>	>	X
ap-1688	135	9	0	0	PUNCT
ap-1688	136	1	‖w(t)‖b−1	‖w(t)‖b−1	NOUN
ap-1688	136	2	1,∞	1,∞	NUM
ap-1688	136	3	=	=	SYM
ap-1688	136	4	sup	sup	NOUN
ap-1688	136	5	j∈z	j∈z	NOUN
ap-1688	136	6	2−j‖∆jw(t)‖1	2−j‖∆jw(t)‖1	NUM
ap-1688	136	7	≤	≤	NUM
ap-1688	136	8	c	c	NOUN
ap-1688	136	9	sup	sup	NOUN
ap-1688	137	1	j∈z	j∈z	PROPN
ap-1688	137	2	∫	∫	PROPN
ap-1688	137	3	t	t	PROPN
ap-1688	137	4	0	0	NUM
ap-1688	138	1	e−c(t−s)2	e−c(t−s)2	NOUN
ap-1688	138	2	2j	2j	NUM
ap-1688	138	3	‖∆j(u⊗	‖∆j(u⊗	VERB
ap-1688	138	4	u(s))‖1ds	u(s))‖1ds	SYM
ap-1688	138	5	≤	≤	NUM
ap-1688	139	1	c	c	NOUN
ap-1688	139	2	sup	sup	NOUN
ap-1688	139	3	j∈z	j∈z	PROPN
ap-1688	139	4	∫	∫	PROPN
ap-1688	140	1	t	t	PROPN
ap-1688	140	2	0	0	NUM
ap-1688	140	3	e−c(t−s)2	e−c(t−s)2	NOUN
ap-1688	140	4	2j	2j	NUM
ap-1688	140	5	‖u(s)‖22ds	‖u(s)‖22ds	NOUN
ap-1688	140	6	≤	≤	NUM
ap-1688	140	7	c	c	NOUN
ap-1688	140	8	∫	∫	PROPN
ap-1688	140	9	t	t	PROPN
ap-1688	140	10	0	0	NUM
ap-1688	141	1	‖u(s)‖22ds	‖u(s)‖22ds	NOUN
ap-1688	141	2	<	<	X
ap-1688	141	3	∞.	∞.	PROPN
ap-1688	141	4	it	it	PRON
ap-1688	141	5	follows	follow	VERB
ap-1688	141	6	that	that	SCONJ
ap-1688	141	7	w(t	w(t	PROPN
ap-1688	141	8	)	)	PUNCT
ap-1688	141	9	∈	∈	PROPN
ap-1688	142	1	b−1	b−1	PROPN
ap-1688	142	2	1,∞	1,∞	NUM
ap-1688	142	3	and	and	CCONJ
ap-1688	142	4	so	so	ADV
ap-1688	143	1	w(t	w(t	PROPN
ap-1688	143	2	)	)	PUNCT
ap-1688	143	3	∈	∈	PROPN
ap-1688	143	4	b	b	PROPN
ap-1688	143	5	−5/2	−5/2	PROPN
ap-1688	143	6	2,∞	2,∞	NUM
ap-1688	143	7	,	,	PUNCT
ap-1688	143	8	since	since	SCONJ
ap-1688	143	9	b−1	b−1	PROPN
ap-1688	143	10	1,∞	1,∞	PROPN
ap-1688	143	11	is	be	AUX
ap-1688	143	12	continuously	continuously	ADV
ap-1688	143	13	embedded	embed	VERB
ap-1688	143	14	into	into	ADP
ap-1688	143	15	b−5/2	b−5/2	PROPN
ap-1688	143	16	2,∞	2,∞	NUM
ap-1688	143	17	(	(	PUNCT
ap-1688	143	18	as	as	SCONJ
ap-1688	143	19	follows	follow	VERB
ap-1688	143	20	from	from	ADP
ap-1688	143	21	the	the	DET
ap-1688	143	22	bernstein	bernstein	PROPN
ap-1688	143	23	inequalities	inequalities	PROPN
ap-1688	143	24	,	,	PUNCT
ap-1688	143	25	see	see	VERB
ap-1688	143	26	[	[	X
ap-1688	143	27	3	3	NUM
ap-1688	143	28	]	]	NUM
ap-1688	143	29	)	)	PUNCT
ap-1688	143	30	.	.	PUNCT
ap-1688	144	1	thus	thus	ADV
ap-1688	144	2	,	,	PUNCT
ap-1688	144	3	if	if	SCONJ
ap-1688	144	4	the	the	DET
ap-1688	144	5	initial	initial	ADJ
ap-1688	144	6	condition	condition	NOUN
ap-1688	144	7	u0	u0	NOUN
ap-1688	144	8	is	be	AUX
ap-1688	144	9	from	from	ADP
ap-1688	144	10	the	the	DET
ap-1688	144	11	space	space	NOUN
ap-1688	144	12	b−5/2	b−5/2	PROPN
ap-1688	144	13	2,∞	2,∞	PROPN
ap-1688	144	14	,	,	PUNCT
ap-1688	144	15	then	then	ADV
ap-1688	144	16	e∆tu0	e∆tu0	PROPN
ap-1688	144	17	is	be	AUX
ap-1688	144	18	also	also	ADV
ap-1688	144	19	from	from	ADP
ap-1688	144	20	the	the	DET
ap-1688	144	21	same	same	ADJ
ap-1688	144	22	space	space	NOUN
ap-1688	144	23	.	.	PUNCT
ap-1688	145	1	this	this	PRON
ap-1688	145	2	means	mean	VERB
ap-1688	145	3	that	that	SCONJ
ap-1688	145	4	u(t	u(t	NOUN
ap-1688	145	5	)	)	PUNCT
ap-1688	145	6	∈	∈	PROPN
ap-1688	145	7	b−5/2	b−5/2	NOUN
ap-1688	145	8	2,∞	2,∞	NUM
ap-1688	145	9	for	for	ADP
ap-1688	145	10	every	every	DET
ap-1688	145	11	t	t	NOUN
ap-1688	145	12	>	>	X
ap-1688	145	13	0	0	PROPN
ap-1688	145	14	,	,	PUNCT
ap-1688	145	15	||et∆u0||2	||et∆u0||2	PROPN
ap-1688	145	16	decreases	decrease	VERB
ap-1688	145	17	at	at	ADP
ap-1688	145	18	the	the	DET
ap-1688	145	19	rate	rate	NOUN
ap-1688	145	20	(	(	PUNCT
ap-1688	145	21	1	1	NUM
ap-1688	145	22	+	+	NUM
ap-1688	145	23	t)−5/4	t)−5/4	PROPN
ap-1688	145	24	(	(	PUNCT
ap-1688	145	25	see	see	VERB
ap-1688	145	26	[	[	X
ap-1688	145	27	3	3	NUM
ap-1688	145	28	]	]	PUNCT
ap-1688	145	29	)	)	PUNCT
ap-1688	145	30	and	and	CCONJ
ap-1688	145	31	using	use	VERB
ap-1688	145	32	the	the	DET
ap-1688	145	33	result	result	NOUN
ap-1688	145	34	from	from	ADP
ap-1688	145	35	[	[	X
ap-1688	145	36	20	20	NUM
ap-1688	145	37	]	]	PUNCT
ap-1688	145	38	mentioned	mention	VERB
ap-1688	145	39	in	in	ADP
ap-1688	145	40	the	the	DET
ap-1688	145	41	second	second	ADJ
ap-1688	145	42	section	section	NOUN
ap-1688	145	43	we	we	PRON
ap-1688	145	44	also	also	ADV
ap-1688	145	45	have	have	VERB
ap-1688	145	46	‖u(t)‖2	‖u(t)‖2	VERB
ap-1688	145	47	≤	≤	NUM
ap-1688	145	48	c2(1	c2(1	NOUN
ap-1688	145	49	+	+	CCONJ
ap-1688	145	50	t)−5/4	t)−5/4	NUM
ap-1688	145	51	for	for	ADP
ap-1688	145	52	every	every	DET
ap-1688	145	53	t	t	PROPN
ap-1688	145	54	≥	≥	NOUN
ap-1688	145	55	0	0	NUM
ap-1688	145	56	.	.	PUNCT
ap-1688	145	57	suppose	suppose	VERB
ap-1688	145	58	now	now	ADV
ap-1688	145	59	that	that	SCONJ
ap-1688	145	60	the	the	DET
ap-1688	145	61	initial	initial	ADJ
ap-1688	145	62	condition	condition	NOUN
ap-1688	145	63	was	be	AUX
ap-1688	145	64	chosen	choose	VERB
ap-1688	145	65	in	in	ADP
ap-1688	145	66	such	such	DET
ap-1688	145	67	a	a	DET
ap-1688	145	68	way	way	NOUN
ap-1688	145	69	that	that	PRON
ap-1688	145	70	c1(1	c1(1	ADJ
ap-1688	145	71	+	+	SYM
ap-1688	145	72	t)−5/4	t)−5/4	PROPN
ap-1688	145	73	≤	≤	NOUN
ap-1688	145	74	‖u(t)‖2	‖u(t)‖2	VERB
ap-1688	145	75	(	(	PUNCT
ap-1688	145	76	8)	8)	NUM
ap-1688	145	77	for	for	ADP
ap-1688	145	78	some	some	DET
ap-1688	145	79	c1	c1	NOUN
ap-1688	145	80	>	>	X
ap-1688	145	81	0	0	PUNCT
ap-1688	146	1	and	and	CCONJ
ap-1688	146	2	every	every	DET
ap-1688	146	3	t	t	NOUN
ap-1688	146	4	≥	≥	NOUN
ap-1688	146	5	0	0	NUM
ap-1688	146	6	.	.	PUNCT
ap-1688	147	1	it	it	PRON
ap-1688	147	2	follows	follow	VERB
ap-1688	147	3	from	from	ADP
ap-1688	147	4	[	[	X
ap-1688	147	5	14	14	NUM
ap-1688	147	6	]	]	PUNCT
ap-1688	147	7	that	that	PRON
ap-1688	147	8	‖aαu(t)‖2	‖aαu(t)‖2	PROPN
ap-1688	147	9	≤	≤	ADV
ap-1688	147	10	c3(1	c3(1	PROPN
ap-1688	147	11	+	+	CCONJ
ap-1688	147	12	t)−α−5/4	t)−α−5/4	NOUN
ap-1688	147	13	for	for	ADP
ap-1688	147	14	every	every	DET
ap-1688	147	15	α	α	NOUN
ap-1688	147	16	>	>	X
ap-1688	147	17	0	0	PUNCT
ap-1688	148	1	and	and	CCONJ
ap-1688	148	2	every	every	DET
ap-1688	148	3	sufficiently	sufficiently	ADV
ap-1688	148	4	large	large	ADJ
ap-1688	148	5	t.	t.	NOUN
ap-1688	148	6	if	if	SCONJ
ap-1688	148	7	µ(t	µ(t	ADJ
ap-1688	148	8	)	)	PUNCT
ap-1688	148	9	=	=	SYM
ap-1688	148	10	c4(1	c4(1	PROPN
ap-1688	148	11	+	+	CCONJ
ap-1688	148	12	t)−1	t)−1	NOUN
ap-1688	148	13	,	,	PUNCT
ap-1688	148	14	we	we	PRON
ap-1688	148	15	get	get	VERB
ap-1688	148	16	c23(1	c23(1	ADJ
ap-1688	149	1	+	+	CCONJ
ap-1688	149	2	t)−2α−5/2c−2	t)−2α−5/2c−2	VERB
ap-1688	149	3	1	1	NUM
ap-1688	149	4	(	(	PUNCT
ap-1688	149	5	1	1	NUM
ap-1688	149	6	+	+	CCONJ
ap-1688	149	7	t)5/2	t)5/2	NOUN
ap-1688	149	8	≥	≥	NOUN
ap-1688	149	9	‖a	‖a	NOUN
ap-1688	149	10	αu(t)‖22	αu(t)‖22	PROPN
ap-1688	149	11	‖u(t)‖22	‖u(t)‖22	NOUN
ap-1688	149	12	≥	≥	NOUN
ap-1688	149	13	c2α4	c2α4	X
ap-1688	149	14	(	(	PUNCT
ap-1688	149	15	1	1	NUM
ap-1688	149	16	+	+	NUM
ap-1688	149	17	t)−2α	t)−2α	PROPN
ap-1688	150	1	(	(	PUNCT
ap-1688	150	2	1−	1−	NUM
ap-1688	150	3	‖eµ(t)u(t)‖22	‖eµ(t)u(t)‖22	NOUN
ap-1688	150	4	‖u(t)‖22	‖u(t)‖22	NOUN
ap-1688	150	5	)	)	PUNCT
ap-1688	150	6	.	.	PUNCT
ap-1688	151	1	so	so	ADV
ap-1688	151	2	,	,	PUNCT
ap-1688	151	3	if	if	SCONJ
ap-1688	151	4	c4	c4	NOUN
ap-1688	151	5	is	be	AUX
ap-1688	151	6	sufficiently	sufficiently	ADV
ap-1688	151	7	large	large	ADJ
ap-1688	151	8	then	then	ADV
ap-1688	151	9	1−	1−	NUM
ap-1688	151	10	‖eµ(t)u(t)‖22	‖eµ(t)u(t)‖22	NOUN
ap-1688	151	11	‖u(t)‖22	‖u(t)‖22	NOUN
ap-1688	151	12	≤	≤	NOUN
ap-1688	151	13	c−2α	c−2α	ADJ
ap-1688	151	14	4	4	NUM
ap-1688	151	15	c23c	c23c	NOUN
ap-1688	151	16	−2	−2	PROPN
ap-1688	151	17	1	1	NUM
ap-1688	151	18	<	<	SYM
ap-1688	151	19	1	1	NUM
ap-1688	151	20	and	and	CCONJ
ap-1688	151	21	‖eµ(t)u(t)‖2	‖eµ(t)u(t)‖2	NUM
ap-1688	151	22	≥	≥	NOUN
ap-1688	151	23	c‖u(t)‖2	c‖u(t)‖2	ADV
ap-1688	151	24	(	(	PUNCT
ap-1688	151	25	9	9	NUM
ap-1688	151	26	)	)	PUNCT
ap-1688	151	27	for	for	ADP
ap-1688	151	28	every	every	DET
ap-1688	151	29	sufficiently	sufficiently	ADV
ap-1688	151	30	large	large	ADJ
ap-1688	151	31	t	t	NOUN
ap-1688	151	32	and	and	CCONJ
ap-1688	151	33	some	some	DET
ap-1688	151	34	c	c	PROPN
ap-1688	151	35	>	>	X
ap-1688	151	36	0	0	X
ap-1688	151	37	.	.	PUNCT
ap-1688	152	1	we	we	PRON
ap-1688	152	2	will	will	AUX
ap-1688	152	3	now	now	ADV
ap-1688	152	4	prove	prove	VERB
ap-1688	152	5	the	the	DET
ap-1688	152	6	existence	existence	NOUN
ap-1688	152	7	of	of	ADP
ap-1688	152	8	a	a	DET
ap-1688	152	9	constant	constant	ADJ
ap-1688	152	10	c	c	NOUN
ap-1688	152	11	such	such	ADJ
ap-1688	152	12	that	that	SCONJ
ap-1688	152	13	lim	lim	PROPN
ap-1688	152	14	inft→∞‖u(t)‖b−5/2	inft→∞‖u(t)‖b−5/2	PROPN
ap-1688	152	15	2,∞	2,∞	NUM
ap-1688	152	16	≥	≥	NOUN
ap-1688	152	17	c	c	NOUN
ap-1688	152	18	>	>	X
ap-1688	152	19	0	0	X
ap-1688	152	20	.	.	PUNCT
ap-1688	153	1	we	we	PRON
ap-1688	153	2	proceed	proceed	VERB
ap-1688	153	3	by	by	ADP
ap-1688	153	4	contradiction	contradiction	NOUN
ap-1688	153	5	.	.	PUNCT
ap-1688	154	1	suppose	suppose	VERB
ap-1688	154	2	that	that	SCONJ
ap-1688	154	3	there	there	PRON
ap-1688	154	4	exists	exist	VERB
ap-1688	154	5	a	a	DET
ap-1688	154	6	sequence	sequence	NOUN
ap-1688	154	7	{	{	PUNCT
ap-1688	154	8	tn}∞n=1	tn}∞n=1	NUM
ap-1688	154	9	,	,	PUNCT
ap-1688	154	10	limn→∞	limn→∞	PROPN
ap-1688	154	11	tn	tn	NOUN
ap-1688	154	12	=	=	SYM
ap-1688	154	13	∞	∞	NUM
ap-1688	154	14	such	such	ADJ
ap-1688	154	15	that	that	SCONJ
ap-1688	154	16	limn→∞	limn→∞	PROPN
ap-1688	154	17	c(n	c(n	NOUN
ap-1688	154	18	)	)	PUNCT
ap-1688	155	1	=	=	SYM
ap-1688	155	2	0	0	NUM
ap-1688	155	3	,	,	PUNCT
ap-1688	155	4	where	where	SCONJ
ap-1688	155	5	c(n	c(n	VERB
ap-1688	155	6	)	)	PUNCT
ap-1688	155	7	=	=	SYM
ap-1688	155	8	‖u(tn)‖b−5/2	‖u(tn)‖b−5/2	NOUN
ap-1688	155	9	2,∞	2,∞	NUM
ap-1688	155	10	=	=	PUNCT
ap-1688	156	1	supj∈z	supj∈z	PROPN
ap-1688	156	2	2−5j/2‖∆ju(tn)‖2	2−5j/2‖∆ju(tn)‖2	NUM
ap-1688	156	3	.	.	PUNCT
ap-1688	157	1	then	then	ADV
ap-1688	157	2	‖∆ju(tn)‖22	‖∆ju(tn)‖22	PROPN
ap-1688	157	3	≤	≤	NOUN
ap-1688	157	4	25jc(n)2	25jc(n)2	ADV
ap-1688	157	5	.	.	PUNCT
ap-1688	158	1	choose	choose	VERB
ap-1688	158	2	j0	j0	PROPN
ap-1688	158	3	so	so	SCONJ
ap-1688	158	4	that	that	SCONJ
ap-1688	158	5	2j0	2j0	NUM
ap-1688	158	6	∼	∼	NOUN
ap-1688	158	7	(	(	PUNCT
ap-1688	158	8	1	1	NUM
ap-1688	158	9	+	+	NUM
ap-1688	158	10	tn)−1/2	tn)−1/2	PRON
ap-1688	158	11	and	and	CCONJ
ap-1688	158	12	sum	sum	VERB
ap-1688	158	13	up	up	ADP
ap-1688	158	14	the	the	DET
ap-1688	158	15	last	last	ADJ
ap-1688	158	16	inequality	inequality	NOUN
ap-1688	158	17	over	over	ADP
ap-1688	158	18	j	j	PROPN
ap-1688	158	19	from	from	ADP
ap-1688	158	20	−∞	−∞	X
ap-1688	158	21	to	to	ADP
ap-1688	158	22	j0	j0	PROPN
ap-1688	158	23	.	.	PUNCT
ap-1688	159	1	we	we	PRON
ap-1688	159	2	get∑	get∑	PRON
ap-1688	159	3	j≤j0	j≤j0	PUNCT
ap-1688	160	1	‖∆ju(tn)‖22	‖∆ju(tn)‖22	PROPN
ap-1688	160	2	≤	≤	PROPN
ap-1688	160	3	∑	∑	PUNCT
ap-1688	160	4	j≤j0	j≤j0	PROPN
ap-1688	160	5	25jc(n)2	25jc(n)2	ADV
ap-1688	160	6	.	.	PUNCT
ap-1688	161	1	due	due	ADP
ap-1688	161	2	to	to	ADP
ap-1688	161	3	the	the	DET
ap-1688	161	4	definition	definition	NOUN
ap-1688	161	5	of	of	ADP
ap-1688	161	6	µ	µ	NUM
ap-1688	161	7	,	,	PUNCT
ap-1688	161	8	(	(	PUNCT
ap-1688	161	9	9	9	NUM
ap-1688	161	10	)	)	PUNCT
ap-1688	161	11	and	and	CCONJ
ap-1688	161	12	the	the	DET
ap-1688	161	13	choice	choice	NOUN
ap-1688	161	14	of	of	ADP
ap-1688	161	15	j0	j0	PROPN
ap-1688	161	16	,	,	PUNCT
ap-1688	161	17	the	the	DET
ap-1688	161	18	left	left	ADJ
ap-1688	161	19	hand	hand	NOUN
ap-1688	161	20	side	side	NOUN
ap-1688	161	21	is	be	AUX
ap-1688	161	22	greater	great	ADJ
ap-1688	161	23	than	than	ADP
ap-1688	161	24	c‖u(tn)‖22	c‖u(tn)‖22	NOUN
ap-1688	161	25	for	for	SCONJ
ap-1688	161	26	some	some	DET
ap-1688	161	27	c	c	PROPN
ap-1688	161	28	>	>	X
ap-1688	161	29	0	0	PROPN
ap-1688	161	30	independent	independent	ADJ
ap-1688	161	31	of	of	ADP
ap-1688	161	32	n.	n.	PROPN
ap-1688	161	33	the	the	DET
ap-1688	161	34	right	right	ADJ
ap-1688	161	35	hand	hand	NOUN
ap-1688	161	36	side	side	NOUN
ap-1688	161	37	is	be	AUX
ap-1688	161	38	smaller	small	ADJ
ap-1688	161	39	than	than	ADP
ap-1688	161	40	2c(n)225j0	2c(n)225j0	NUM
ap-1688	161	41	∼	∼	NOUN
ap-1688	161	42	2c(n)2(1	2c(n)2(1	NOUN
ap-1688	161	43	+	+	NOUN
ap-1688	161	44	tn)−5/2	tn)−5/2	X
ap-1688	161	45	.	.	PUNCT
ap-1688	162	1	we	we	PRON
ap-1688	162	2	get	get	VERB
ap-1688	162	3	finally	finally	ADV
ap-1688	162	4	c‖u(tn)‖2	c‖u(tn)‖2	VERB
ap-1688	162	5	≤	≤	NUM
ap-1688	162	6	c(n)(1	c(n)(1	NOUN
ap-1688	162	7	+	+	CCONJ
ap-1688	162	8	tn)−5/4	tn)−5/4	NUM
ap-1688	162	9	for	for	ADP
ap-1688	162	10	every	every	DET
ap-1688	162	11	n	n	PRON
ap-1688	162	12	∈	∈	NOUN
ap-1688	162	13	n	n	NOUN
ap-1688	163	1	and	and	CCONJ
ap-1688	163	2	this	this	PRON
ap-1688	163	3	is	be	AUX
ap-1688	163	4	in	in	ADP
ap-1688	163	5	contradiction	contradiction	NOUN
ap-1688	163	6	with	with	ADP
ap-1688	163	7	(	(	PUNCT
ap-1688	163	8	8)	8)	NUM
ap-1688	163	9	.	.	PUNCT
ap-1688	164	1	we	we	PRON
ap-1688	164	2	sum	sum	VERB
ap-1688	164	3	up	up	ADP
ap-1688	164	4	the	the	DET
ap-1688	164	5	result	result	NOUN
ap-1688	164	6	from	from	ADP
ap-1688	164	7	this	this	DET
ap-1688	164	8	section	section	NOUN
ap-1688	164	9	in	in	ADP
ap-1688	164	10	the	the	DET
ap-1688	164	11	following	follow	VERB
ap-1688	164	12	theorem	theorem	PROPN
ap-1688	164	13	.	.	PUNCT
ap-1688	164	14	theorem	theorem	NOUN
ap-1688	164	15	5	5	NUM
ap-1688	164	16	.	.	PUNCT
ap-1688	165	1	let	let	VERB
ap-1688	165	2	u0	u0	PROPN
ap-1688	165	3	∈	∈	PROPN
ap-1688	165	4	b−5/2	b−5/2	PROPN
ap-1688	166	1	2,∞	2,∞	NUM
ap-1688	166	2	∩l2	∩l2	PROPN
ap-1688	166	3	σ	σ	PROPN
ap-1688	166	4	.	.	PUNCT
ap-1688	167	1	let	let	VERB
ap-1688	167	2	u	u	PRON
ap-1688	167	3	be	be	AUX
ap-1688	167	4	a	a	DET
ap-1688	167	5	turbulent	turbulent	ADJ
ap-1688	167	6	solution	solution	NOUN
ap-1688	167	7	of	of	ADP
ap-1688	167	8	(	(	PUNCT
ap-1688	167	9	1)–(3	1)–(3	NUM
ap-1688	167	10	)	)	PUNCT
ap-1688	167	11	with	with	ADP
ap-1688	167	12	the	the	DET
ap-1688	167	13	initial	initial	ADJ
ap-1688	167	14	condition	condition	NOUN
ap-1688	167	15	u0	u0	ADJ
ap-1688	167	16	and	and	CCONJ
ap-1688	167	17	such	such	ADJ
ap-1688	167	18	that	that	SCONJ
ap-1688	167	19	‖u(t)‖2	‖u(t)‖2	ADJ
ap-1688	167	20	≥	≥	PROPN
ap-1688	167	21	c(1	c(1	PROPN
ap-1688	167	22	+	+	PUNCT
ap-1688	167	23	t)−5/4	t)−5/4	PROPN
ap-1688	167	24	for	for	ADP
ap-1688	167	25	some	some	DET
ap-1688	167	26	c	c	PROPN
ap-1688	167	27	>	>	X
ap-1688	167	28	0	0	PUNCT
ap-1688	168	1	and	and	CCONJ
ap-1688	168	2	all	all	DET
ap-1688	168	3	t	t	PROPN
ap-1688	168	4	≥	≥	NOUN
ap-1688	168	5	0	0	NUM
ap-1688	168	6	.	.	PUNCT
ap-1688	169	1	then	then	ADV
ap-1688	169	2	there	there	PRON
ap-1688	169	3	exist	exist	VERB
ap-1688	169	4	constants	constant	NOUN
ap-1688	169	5	c′	c′	VERB
ap-1688	169	6	and	and	CCONJ
ap-1688	169	7	c′′	c′′	NOUN
ap-1688	169	8	such	such	ADJ
ap-1688	169	9	that	that	SCONJ
ap-1688	169	10	0	0	NUM
ap-1688	169	11	<	<	X
ap-1688	169	12	c′	c′	ADJ
ap-1688	169	13	≤	≤	NUM
ap-1688	169	14	‖u(t)‖	‖u(t)‖	NOUN
ap-1688	169	15	b	b	NOUN
ap-1688	169	16	−5/2	−5/2	PROPN
ap-1688	169	17	2,∞	2,∞	PROPN
ap-1688	169	18	≤	≤	NUM
ap-1688	169	19	c′′	c′′	NOUN
ap-1688	169	20	(	(	PUNCT
ap-1688	169	21	10	10	NUM
ap-1688	169	22	)	)	PUNCT
ap-1688	169	23	for	for	ADP
ap-1688	169	24	every	every	DET
ap-1688	169	25	t	t	PROPN
ap-1688	169	26	≥	≥	NOUN
ap-1688	169	27	0	0	NUM
ap-1688	169	28	.	.	PUNCT
ap-1688	170	1	we	we	PRON
ap-1688	170	2	will	will	AUX
ap-1688	170	3	now	now	ADV
ap-1688	170	4	show	show	VERB
ap-1688	170	5	that	that	SCONJ
ap-1688	170	6	theorem	theorem	NOUN
ap-1688	170	7	5	5	NUM
ap-1688	170	8	improves	improve	VERB
ap-1688	170	9	the	the	DET
ap-1688	170	10	result	result	NOUN
ap-1688	170	11	presented	present	VERB
ap-1688	170	12	by	by	ADP
ap-1688	170	13	miyakawa	miyakawa	NOUN
ap-1688	170	14	in	in	ADP
ap-1688	170	15	[	[	X
ap-1688	170	16	8	8	NUM
ap-1688	170	17	]	]	PUNCT
ap-1688	170	18	.	.	PUNCT
ap-1688	171	1	miyakawa	miyakawa	NOUN
ap-1688	171	2	studied	study	VERB
ap-1688	171	3	102	102	NUM
ap-1688	171	4	acta	acta	PROPN
ap-1688	171	5	polytechnica	polytechnica	PROPN
ap-1688	171	6	vol	vol	NOUN
ap-1688	171	7	.	.	PROPN
ap-1688	172	1	52	52	NUM
ap-1688	172	2	no	no	NOUN
ap-1688	172	3	.	.	PUNCT
ap-1688	173	1	6/2012	6/2012	NUM
ap-1688	173	2	turbulent	turbulent	ADJ
ap-1688	173	3	solutions	solution	NOUN
ap-1688	173	4	with	with	ADP
ap-1688	173	5	initial	initial	ADJ
ap-1688	173	6	conditions	condition	NOUN
ap-1688	173	7	u0	u0	PROPN
ap-1688	173	8	∈	∈	PROPN
ap-1688	173	9	l2	l2	NOUN
ap-1688	173	10	σ	σ	NOUN
ap-1688	173	11	such	such	ADJ
ap-1688	173	12	that	that	DET
ap-1688	173	13	∫	∫	PROPN
ap-1688	173	14	(	(	PUNCT
ap-1688	173	15	1	1	NUM
ap-1688	173	16	+	+	NUM
ap-1688	173	17	|x|)|u0(x)|dx	|x|)|u0(x)|dx	NUM
ap-1688	173	18	<	<	X
ap-1688	173	19	∞.	∞.	PROPN
ap-1688	173	20	(	(	PUNCT
ap-1688	173	21	11	11	NUM
ap-1688	173	22	)	)	PUNCT
ap-1688	173	23	he	he	PRON
ap-1688	173	24	proved	prove	VERB
ap-1688	173	25	that	that	SCONJ
ap-1688	173	26	0	0	PUNCT
ap-1688	173	27	<	<	X
ap-1688	173	28	c0	c0	X
ap-1688	173	29	≤	≤	ADJ
ap-1688	173	30	‖u(t)‖b−1	‖u(t)‖b−1	X
ap-1688	173	31	1,∞	1,∞	NUM
ap-1688	173	32	≤	≤	ADJ
ap-1688	173	33	c1	c1	NOUN
ap-1688	173	34	(	(	PUNCT
ap-1688	173	35	12	12	NUM
ap-1688	173	36	)	)	PUNCT
ap-1688	173	37	for	for	ADP
ap-1688	173	38	large	large	ADJ
ap-1688	173	39	t	t	PROPN
ap-1688	173	40	>	>	X
ap-1688	173	41	0	0	PUNCT
ap-1688	173	42	and	and	CCONJ
ap-1688	173	43	some	some	DET
ap-1688	173	44	constants	constant	NOUN
ap-1688	173	45	c0	c0	NOUN
ap-1688	173	46	and	and	CCONJ
ap-1688	173	47	c1	c1	PROPN
ap-1688	173	48	if	if	SCONJ
ap-1688	173	49	and	and	CCONJ
ap-1688	173	50	only	only	ADV
ap-1688	173	51	if(∫	if(∫	VERB
ap-1688	173	52	xju0m(x)dx	xju0m(x)dx	NOUN
ap-1688	173	53	,	,	PUNCT
ap-1688	173	54	∫	∫	PROPN
ap-1688	173	55	∞	∞	PROPN
ap-1688	173	56	0	0	NUM
ap-1688	173	57	∫	∫	PROPN
ap-1688	173	58	(	(	PUNCT
ap-1688	173	59	ukul)(x	ukul)(x	PROPN
ap-1688	173	60	,	,	PUNCT
ap-1688	173	61	s)dxds	s)dxds	NOUN
ap-1688	173	62	)	)	PUNCT
ap-1688	173	63	6=	6=	ADP
ap-1688	173	64	(	(	PUNCT
ap-1688	173	65	0	0	NUM
ap-1688	173	66	,	,	PUNCT
ap-1688	173	67	cδkl	cδkl	ADJ
ap-1688	173	68	)	)	PUNCT
ap-1688	173	69	.	.	PUNCT
ap-1688	174	1	(	(	PUNCT
ap-1688	174	2	13	13	NUM
ap-1688	174	3	)	)	PUNCT
ap-1688	174	4	moreover	moreover	ADV
ap-1688	174	5	,	,	PUNCT
ap-1688	174	6	it	it	PRON
ap-1688	174	7	was	be	AUX
ap-1688	174	8	proved	prove	VERB
ap-1688	174	9	by	by	ADP
ap-1688	174	10	miyakawa	miyakawa	NOUN
ap-1688	174	11	and	and	CCONJ
ap-1688	174	12	schonbek	schonbek	VERB
ap-1688	174	13	in	in	ADP
ap-1688	174	14	[	[	X
ap-1688	174	15	9	9	NUM
ap-1688	174	16	]	]	PUNCT
ap-1688	174	17	that	that	SCONJ
ap-1688	174	18	(	(	PUNCT
ap-1688	174	19	13	13	NUM
ap-1688	174	20	)	)	PUNCT
ap-1688	174	21	holds	hold	VERB
ap-1688	174	22	if	if	SCONJ
ap-1688	174	23	and	and	CCONJ
ap-1688	174	24	only	only	ADV
ap-1688	174	25	if	if	SCONJ
ap-1688	174	26	there	there	PRON
ap-1688	174	27	exist	exist	VERB
ap-1688	174	28	constant	constant	ADJ
ap-1688	174	29	c0	c0	NOUN
ap-1688	174	30	and	and	CCONJ
ap-1688	174	31	c1	c1	PROPN
ap-1688	174	32	such	such	ADJ
ap-1688	174	33	that	that	SCONJ
ap-1688	174	34	0	0	NUM
ap-1688	174	35	<	<	X
ap-1688	174	36	c0	c0	PROPN
ap-1688	174	37	≤	≤	X
ap-1688	174	38	t5/4‖u(t)‖2	t5/4‖u(t)‖2	PRON
ap-1688	174	39	≤	≤	PROPN
ap-1688	174	40	c1	c1	NOUN
ap-1688	174	41	(	(	PUNCT
ap-1688	174	42	14	14	NUM
ap-1688	174	43	)	)	PUNCT
ap-1688	174	44	for	for	ADP
ap-1688	174	45	large	large	ADJ
ap-1688	174	46	t	t	PROPN
ap-1688	174	47	>	>	X
ap-1688	174	48	0	0	X
ap-1688	174	49	.	.	PUNCT
ap-1688	175	1	since	since	SCONJ
ap-1688	175	2	the	the	DET
ap-1688	175	3	initial	initial	ADJ
ap-1688	175	4	conditions	condition	NOUN
ap-1688	175	5	satisfying	satisfy	VERB
ap-1688	175	6	(	(	PUNCT
ap-1688	175	7	11	11	NUM
ap-1688	175	8	)	)	PUNCT
ap-1688	175	9	belong	belong	VERB
ap-1688	175	10	to	to	ADP
ap-1688	175	11	the	the	DET
ap-1688	175	12	space	space	NOUN
ap-1688	175	13	b−5/2	b−5/2	PROPN
ap-1688	175	14	2,∞	2,∞	NUM
ap-1688	175	15	∩	∩	ADJ
ap-1688	175	16	l2	l2	NOUN
ap-1688	175	17	σ	σ	PROPN
ap-1688	175	18	,	,	PUNCT
ap-1688	175	19	it	it	PRON
ap-1688	175	20	is	be	AUX
ap-1688	175	21	clear	clear	ADJ
ap-1688	175	22	that	that	SCONJ
ap-1688	175	23	theorem	theorem	VERB
ap-1688	175	24	5	5	NUM
ap-1688	175	25	generalizes	generalize	VERB
ap-1688	175	26	the	the	DET
ap-1688	175	27	result	result	NOUN
ap-1688	175	28	by	by	ADP
ap-1688	175	29	miyakawa	miyakawa	NOUN
ap-1688	175	30	mentioned	mention	VERB
ap-1688	175	31	above	above	ADV
ap-1688	175	32	:	:	PUNCT
ap-1688	175	33	for	for	ADP
ap-1688	175	34	initial	initial	ADJ
ap-1688	175	35	conditions	condition	NOUN
ap-1688	175	36	satisfying	satisfy	VERB
ap-1688	175	37	(	(	PUNCT
ap-1688	175	38	11	11	NUM
ap-1688	175	39	)	)	PUNCT
ap-1688	175	40	and	and	CCONJ
ap-1688	175	41	under	under	ADP
ap-1688	175	42	condition	condition	NOUN
ap-1688	175	43	(	(	PUNCT
ap-1688	175	44	13	13	NUM
ap-1688	175	45	)	)	PUNCT
ap-1688	175	46	(	(	PUNCT
ap-1688	175	47	resp	resp	NOUN
ap-1688	175	48	.	.	PUNCT
ap-1688	176	1	(	(	PUNCT
ap-1688	176	2	14	14	NUM
ap-1688	176	3	)	)	PUNCT
ap-1688	176	4	)	)	PUNCT
ap-1688	177	1	both	both	DET
ap-1688	177	2	results	result	NOUN
ap-1688	177	3	give	give	VERB
ap-1688	177	4	lower	low	ADJ
ap-1688	177	5	estimates	estimate	NOUN
ap-1688	177	6	of	of	ADP
ap-1688	177	7	u(t	u(t	NOUN
ap-1688	177	8	)	)	PUNCT
ap-1688	177	9	,	,	PUNCT
ap-1688	177	10	but	but	CCONJ
ap-1688	177	11	while	while	SCONJ
ap-1688	177	12	miyakawa	miyakawa	PROPN
ap-1688	177	13	’s	’s	PART
ap-1688	177	14	estimate	estimate	NOUN
ap-1688	177	15	uses	use	VERB
ap-1688	177	16	the	the	DET
ap-1688	177	17	space	space	NOUN
ap-1688	177	18	b−1	b−1	PROPN
ap-1688	177	19	1,∞	1,∞	NUM
ap-1688	177	20	,	,	PUNCT
ap-1688	177	21	in	in	ADP
ap-1688	177	22	theorem	theorem	NOUN
ap-1688	177	23	5	5	NUM
ap-1688	177	24	we	we	PRON
ap-1688	177	25	use	use	VERB
ap-1688	177	26	the	the	DET
ap-1688	177	27	space	space	NOUN
ap-1688	177	28	b−5/2	b−5/2	PROPN
ap-1688	177	29	2,∞	2,∞	PROPN
ap-1688	177	30	.	.	PUNCT
ap-1688	178	1	since	since	SCONJ
ap-1688	178	2	b−1	b−1	PROPN
ap-1688	178	3	1,∞	1,∞	PROPN
ap-1688	178	4	is	be	AUX
ap-1688	178	5	continuously	continuously	ADV
ap-1688	178	6	embedded	embed	VERB
ap-1688	178	7	into	into	ADP
ap-1688	178	8	b−5/2	b−5/2	PROPN
ap-1688	178	9	2,∞	2,∞	NUM
ap-1688	178	10	,	,	PUNCT
ap-1688	178	11	the	the	DET
ap-1688	178	12	result	result	NOUN
ap-1688	178	13	from	from	ADP
ap-1688	178	14	theorem	theorem	ADJ
ap-1688	178	15	5	5	NUM
ap-1688	178	16	is	be	AUX
ap-1688	178	17	stronger	strong	ADJ
ap-1688	178	18	.	.	PUNCT
ap-1688	179	1	moreover	moreover	ADV
ap-1688	179	2	,	,	PUNCT
ap-1688	179	3	theorem	theorem	ADJ
ap-1688	179	4	5	5	NUM
ap-1688	179	5	also	also	ADV
ap-1688	179	6	describes	describe	VERB
ap-1688	179	7	lower	low	ADJ
ap-1688	179	8	estimates	estimate	NOUN
ap-1688	179	9	for	for	SCONJ
ap-1688	179	10	solutions	solution	NOUN
ap-1688	179	11	with	with	ADP
ap-1688	179	12	initial	initial	ADJ
ap-1688	179	13	conditions	condition	NOUN
ap-1688	179	14	not	not	PART
ap-1688	179	15	satisfying	satisfy	VERB
ap-1688	179	16	(	(	PUNCT
ap-1688	179	17	11	11	NUM
ap-1688	179	18	)	)	PUNCT
ap-1688	179	19	(	(	PUNCT
ap-1688	179	20	in	in	ADP
ap-1688	179	21	this	this	DET
ap-1688	179	22	paper	paper	NOUN
ap-1688	179	23	we	we	PRON
ap-1688	179	24	have	have	AUX
ap-1688	179	25	not	not	PART
ap-1688	179	26	dealt	deal	VERB
ap-1688	179	27	with	with	ADP
ap-1688	179	28	their	their	PRON
ap-1688	179	29	existence	existence	NOUN
ap-1688	179	30	)	)	PUNCT
ap-1688	179	31	.	.	PUNCT
ap-1688	180	1	5	5	NUM
ap-1688	180	2	appendix	appendix	VERB
ap-1688	180	3	the	the	DET
ap-1688	180	4	definitions	definition	NOUN
ap-1688	180	5	and	and	CCONJ
ap-1688	180	6	some	some	DET
ap-1688	180	7	basic	basic	ADJ
ap-1688	180	8	properties	property	NOUN
ap-1688	180	9	of	of	ADP
ap-1688	180	10	the	the	DET
ap-1688	180	11	following	follow	VERB
ap-1688	180	12	concepts	concept	NOUN
ap-1688	180	13	can	can	AUX
ap-1688	180	14	be	be	AUX
ap-1688	180	15	found	find	VERB
ap-1688	180	16	in	in	ADP
ap-1688	180	17	[	[	X
ap-1688	180	18	19	19	NUM
ap-1688	180	19	]	]	NUM
ap-1688	180	20	:	:	PUNCT
ap-1688	180	21	•	•	NUM
ap-1688	180	22	lp	lp	NOUN
ap-1688	180	23	,	,	PUNCT
ap-1688	180	24	p	p	PROPN
ap-1688	180	25	∈	∈	PROPN
ap-1688	181	1	[	[	X
ap-1688	181	2	1,∞	1,∞	NUM
ap-1688	181	3	]	]	X
ap-1688	181	4	,	,	PUNCT
ap-1688	181	5	the	the	DET
ap-1688	181	6	lebesgue	lebesgue	NOUN
ap-1688	181	7	space	space	NOUN
ap-1688	181	8	with	with	ADP
ap-1688	181	9	the	the	DET
ap-1688	181	10	norm	norm	NOUN
ap-1688	181	11	‖·‖p	‖·‖p	ADJ
ap-1688	181	12	;	;	PUNCT
ap-1688	181	13	•	•	X
ap-1688	181	14	w	w	PROPN
ap-1688	182	1	k	k	PROPN
ap-1688	182	2	,	,	PUNCT
ap-1688	182	3	p	p	X
ap-1688	182	4	,	,	PUNCT
ap-1688	182	5	k	k	PROPN
ap-1688	182	6	∈	∈	PROPN
ap-1688	182	7	n	n	PROPN
ap-1688	182	8	,	,	PUNCT
ap-1688	182	9	p	p	PROPN
ap-1688	182	10	∈	∈	PROPN
ap-1688	183	1	[	[	X
ap-1688	183	2	1,∞	1,∞	NUM
ap-1688	183	3	]	]	X
ap-1688	183	4	,	,	PUNCT
ap-1688	183	5	the	the	DET
ap-1688	183	6	sobolev	sobolev	NOUN
ap-1688	183	7	space	space	NOUN
ap-1688	183	8	with	with	ADP
ap-1688	183	9	the	the	DET
ap-1688	183	10	norm	norm	NOUN
ap-1688	183	11	‖·‖k	‖·‖k	NOUN
ap-1688	183	12	,	,	PUNCT
ap-1688	183	13	p	p	X
ap-1688	183	14	;	;	PUNCT
ap-1688	183	15	•	•	NUM
ap-1688	183	16	c∞0,σ	c∞0,σ	NOUN
ap-1688	183	17	=	=	PUNCT
ap-1688	183	18	{	{	PUNCT
ap-1688	183	19	ϕ	ϕ	NOUN
ap-1688	183	20	∈	∈	PROPN
ap-1688	183	21	(	(	PUNCT
ap-1688	183	22	c∞0	c∞0	X
ap-1688	183	23	)	)	PUNCT
ap-1688	184	1	3;∇·ϕ	3;∇·ϕ	NUM
ap-1688	184	2	=	=	SYM
ap-1688	184	3	0	0	NUM
ap-1688	184	4	}	}	PUNCT
ap-1688	184	5	,	,	PUNCT
ap-1688	184	6	the	the	DET
ap-1688	184	7	set	set	NOUN
ap-1688	184	8	of	of	ADP
ap-1688	184	9	smooth	smooth	ADJ
ap-1688	184	10	solenoidal	solenoidal	ADJ
ap-1688	184	11	vector	vector	NOUN
ap-1688	184	12	functions	function	NOUN
ap-1688	184	13	with	with	ADP
ap-1688	184	14	compact	compact	ADJ
ap-1688	184	15	support	support	NOUN
ap-1688	184	16	in	in	ADP
ap-1688	184	17	r3	r3	PROPN
ap-1688	184	18	;	;	PUNCT
ap-1688	184	19	•	•	NUM
ap-1688	184	20	l2	l2	PROPN
ap-1688	184	21	σ	σ	PROPN
ap-1688	184	22	,	,	PUNCT
ap-1688	184	23	resp	resp	NOUN
ap-1688	184	24	.	.	PUNCT
ap-1688	185	1	w	w	ADP
ap-1688	185	2	1,2	1,2	NUM
ap-1688	185	3	0,σ	0,σ	NOUN
ap-1688	185	4	,	,	PUNCT
ap-1688	185	5	the	the	DET
ap-1688	185	6	closure	closure	NOUN
ap-1688	185	7	of	of	ADP
ap-1688	185	8	c∞0,σ	c∞0,σ	VERB
ap-1688	185	9	in	in	ADP
ap-1688	185	10	(	(	PUNCT
ap-1688	185	11	l2)3	l2)3	NOUN
ap-1688	185	12	,	,	PUNCT
ap-1688	185	13	resp	resp	NOUN
ap-1688	185	14	.	.	PUNCT
ap-1688	186	1	(	(	PUNCT
ap-1688	186	2	w	w	PROPN
ap-1688	186	3	1,2)3	1,2)3	NUM
ap-1688	186	4	;	;	PUNCT
ap-1688	186	5	•	•	NUM
ap-1688	186	6	pσ	pσ	NOUN
ap-1688	186	7	,	,	PUNCT
ap-1688	186	8	the	the	DET
ap-1688	186	9	orthogonal	orthogonal	ADJ
ap-1688	186	10	projection	projection	NOUN
ap-1688	186	11	of	of	ADP
ap-1688	186	12	l2(ω)3	l2(ω)3	PROPN
ap-1688	186	13	onto	onto	ADP
ap-1688	186	14	l2	l2	PROPN
ap-1688	186	15	σ	σ	PROPN
ap-1688	186	16	;	;	PUNCT
ap-1688	186	17	•	•	ADP
ap-1688	186	18	a	a	NOUN
ap-1688	186	19	,	,	PUNCT
ap-1688	186	20	the	the	DET
ap-1688	186	21	stokes	stoke	NOUN
ap-1688	186	22	operator	operator	NOUN
ap-1688	186	23	in	in	ADP
ap-1688	186	24	l2	l2	NOUN
ap-1688	186	25	σ	σ	NOUN
ap-1688	186	26	defined	define	VERB
ap-1688	186	27	as	as	ADP
ap-1688	186	28	au	au	ADV
ap-1688	186	29	=	=	VERB
ap-1688	186	30	−pσ∆u	−pσ∆u	NOUN
ap-1688	186	31	for	for	ADP
ap-1688	186	32	every	every	DET
ap-1688	186	33	u	u	PROPN
ap-1688	186	34	∈	∈	PROPN
ap-1688	186	35	d(a	d(a	PROPN
ap-1688	186	36	)	)	PUNCT
ap-1688	186	37	=	=	SYM
ap-1688	186	38	w	w	ADP
ap-1688	186	39	1,2	1,2	NUM
ap-1688	186	40	0,σ	0,σ	NUM
ap-1688	186	41	∩	∩	NOUN
ap-1688	186	42	(	(	PUNCT
ap-1688	186	43	w	w	PROPN
ap-1688	186	44	2,2)3	2,2)3	NUM
ap-1688	186	45	;	;	PUNCT
ap-1688	186	46	a	a	PRON
ap-1688	186	47	is	be	AUX
ap-1688	186	48	a	a	DET
ap-1688	186	49	positive	positive	ADJ
ap-1688	186	50	self	self	NOUN
ap-1688	186	51	-	-	PUNCT
ap-1688	186	52	adjoint	adjoint	NOUN
ap-1688	186	53	operator	operator	NOUN
ap-1688	186	54	;	;	PUNCT
ap-1688	186	55	for	for	ADP
ap-1688	186	56	the	the	DET
ap-1688	186	57	case	case	NOUN
ap-1688	186	58	of	of	ADP
ap-1688	186	59	the	the	DET
ap-1688	186	60	whole	whole	ADJ
ap-1688	186	61	space	space	NOUN
ap-1688	186	62	au	au	ADV
ap-1688	186	63	=	=	PUNCT
ap-1688	186	64	−∆u	−∆u	NOUN
ap-1688	186	65	;	;	PUNCT
ap-1688	186	66	•	•	PRON
ap-1688	186	67	{	{	PUNCT
ap-1688	186	68	eλ;λ	eλ;λ	NOUN
ap-1688	186	69	≥	≥	NOUN
ap-1688	186	70	0	0	NUM
ap-1688	186	71	}	}	PUNCT
ap-1688	186	72	,	,	PUNCT
ap-1688	186	73	the	the	DET
ap-1688	186	74	resolution	resolution	NOUN
ap-1688	186	75	of	of	ADP
ap-1688	186	76	identity	identity	NOUN
ap-1688	186	77	of	of	ADP
ap-1688	186	78	a	a	PRON
ap-1688	186	79	;	;	PUNCT
ap-1688	186	80	•	•	ADP
ap-1688	186	81	aµ	aµ	PROPN
ap-1688	186	82	,	,	PUNCT
ap-1688	186	83	µ	µ	PROPN
ap-1688	186	84	∈	∈	PROPN
ap-1688	186	85	r	r	NOUN
ap-1688	186	86	,	,	PUNCT
ap-1688	186	87	the	the	DET
ap-1688	186	88	powers	power	NOUN
ap-1688	186	89	of	of	ADP
ap-1688	186	90	a	a	PRON
ap-1688	186	91	with	with	ADP
ap-1688	186	92	domains	domain	NOUN
ap-1688	186	93	d(aµ	d(aµ	PROPN
ap-1688	186	94	)	)	PUNCT
ap-1688	186	95	and	and	CCONJ
ap-1688	186	96	ranges	range	VERB
ap-1688	186	97	r(aµ	r(aµ	NOUN
ap-1688	186	98	)	)	PUNCT
ap-1688	186	99	;	;	PUNCT
ap-1688	186	100	•	•	X
ap-1688	186	101	{	{	PUNCT
ap-1688	186	102	et∆	et∆	PROPN
ap-1688	186	103	;	;	PUNCT
ap-1688	186	104	t	t	PROPN
ap-1688	186	105	≥	≥	NOUN
ap-1688	186	106	0	0	NUM
ap-1688	186	107	}	}	PUNCT
ap-1688	186	108	,	,	PUNCT
ap-1688	186	109	the	the	DET
ap-1688	186	110	semigroup	semigroup	NOUN
ap-1688	186	111	generated	generate	VERB
ap-1688	186	112	by	by	ADP
ap-1688	186	113	the	the	DET
ap-1688	186	114	laplace	laplace	NOUN
ap-1688	186	115	operator	operator	NOUN
ap-1688	186	116	−∆.	−∆.	NOUN
ap-1688	186	117	definition	definition	NOUN
ap-1688	186	118	3	3	X
ap-1688	186	119	.	.	PUNCT
ap-1688	187	1	if	if	SCONJ
ap-1688	187	2	u0	u0	PROPN
ap-1688	187	3	∈	∈	PROPN
ap-1688	187	4	l2	l2	NOUN
ap-1688	187	5	σ	σ	PROPN
ap-1688	187	6	,	,	PUNCT
ap-1688	187	7	a	a	DET
ap-1688	187	8	measurable	measurable	ADJ
ap-1688	187	9	function	function	NOUN
ap-1688	187	10	u	u	NOUN
ap-1688	187	11	defined	define	VERB
ap-1688	187	12	on	on	ADP
ap-1688	187	13	r3×(0,∞	r3×(0,∞	NOUN
ap-1688	187	14	)	)	PUNCT
ap-1688	187	15	is	be	AUX
ap-1688	187	16	called	call	VERB
ap-1688	187	17	a	a	DET
ap-1688	187	18	global	global	ADJ
ap-1688	187	19	weak	weak	ADJ
ap-1688	187	20	solution	solution	NOUN
ap-1688	187	21	of	of	ADP
ap-1688	187	22	(	(	PUNCT
ap-1688	187	23	1)–(3	1)–(3	NUM
ap-1688	187	24	)	)	PUNCT
ap-1688	187	25	if	if	SCONJ
ap-1688	187	26	u	u	PROPN
ap-1688	187	27	∈	∈	PROPN
ap-1688	187	28	l∞((0,∞);l2	l∞((0,∞);l2	PROPN
ap-1688	187	29	σ	σ	PROPN
ap-1688	187	30	)	)	PUNCT
ap-1688	187	31	∩	∩	ADJ
ap-1688	187	32	l2((0	l2((0	PROPN
ap-1688	187	33	,	,	PUNCT
ap-1688	187	34	t	t	PROPN
ap-1688	187	35	)	)	PUNCT
ap-1688	187	36	;	;	PUNCT
ap-1688	187	37	w	w	X
ap-1688	187	38	1,2	1,2	NUM
ap-1688	187	39	0,σ	0,σ	NUM
ap-1688	187	40	)	)	PUNCT
ap-1688	187	41	for	for	ADP
ap-1688	187	42	every	every	DET
ap-1688	187	43	t	t	NOUN
ap-1688	187	44	>	>	X
ap-1688	187	45	0	0	PUNCT
ap-1688	187	46	and	and	CCONJ
ap-1688	187	47	the	the	DET
ap-1688	187	48	integral	integral	ADJ
ap-1688	187	49	relation∫	relation∫	NOUN
ap-1688	187	50	∞	∞	PROPN
ap-1688	187	51	0	0	PUNCT
ap-1688	188	1	[	[	PUNCT
ap-1688	188	2	−	−	PROPN
ap-1688	188	3	(	(	PUNCT
ap-1688	188	4	u(t	u(t	PROPN
ap-1688	188	5	)	)	PUNCT
ap-1688	188	6	,	,	PUNCT
ap-1688	188	7	∂tφ(t	∂tφ(t	NUM
ap-1688	188	8	)	)	PUNCT
ap-1688	188	9	)	)	PUNCT
ap-1688	189	1	+	+	CCONJ
ap-1688	189	2	(	(	PUNCT
ap-1688	189	3	∇u(t),∇φ(t	∇u(t),∇φ(t	NUM
ap-1688	189	4	)	)	PUNCT
ap-1688	189	5	)	)	PUNCT
ap-1688	190	1	+	+	CCONJ
ap-1688	190	2	(	(	PUNCT
ap-1688	190	3	u(t	u(t	NOUN
ap-1688	190	4	)	)	PUNCT
ap-1688	190	5	·	·	PUNCT
ap-1688	190	6	∇u(t	∇u(t	PROPN
ap-1688	190	7	)	)	PUNCT
ap-1688	190	8	,	,	PUNCT
ap-1688	190	9	φ(t	φ(t	PROPN
ap-1688	190	10	)	)	PUNCT
ap-1688	190	11	)	)	PUNCT
ap-1688	190	12	]	]	PUNCT
ap-1688	190	13	dt	dt	X
ap-1688	190	14	=	=	PUNCT
ap-1688	190	15	(	(	PUNCT
ap-1688	190	16	u0	u0	ADJ
ap-1688	190	17	,	,	PUNCT
ap-1688	190	18	φ(0	φ(0	ADJ
ap-1688	190	19	)	)	PUNCT
ap-1688	190	20	)	)	PUNCT
ap-1688	190	21	holds	hold	VERB
ap-1688	190	22	for	for	ADP
ap-1688	190	23	all	all	DET
ap-1688	190	24	φ	φ	NOUN
ap-1688	190	25	∈	∈	PROPN
ap-1688	190	26	c∞0	c∞0	PROPN
ap-1688	190	27	(	(	PUNCT
ap-1688	190	28	[	[	X
ap-1688	190	29	0,∞);c∞0,σ	0,∞);c∞0,σ	NOUN
ap-1688	190	30	)	)	PUNCT
ap-1688	190	31	.	.	PUNCT
ap-1688	191	1	definition	definition	NOUN
ap-1688	191	2	4	4	NUM
ap-1688	191	3	.	.	PUNCT
ap-1688	192	1	a	a	DET
ap-1688	192	2	global	global	ADJ
ap-1688	192	3	weak	weak	ADJ
ap-1688	192	4	solution	solution	NOUN
ap-1688	192	5	u	u	NOUN
ap-1688	192	6	satisfies	satisfy	VERB
ap-1688	192	7	the	the	DET
ap-1688	192	8	strong	strong	ADJ
ap-1688	192	9	energy	energy	NOUN
ap-1688	192	10	inequality	inequality	NOUN
ap-1688	192	11	if	if	SCONJ
ap-1688	192	12	‖u(t)‖2	‖u(t)‖2	ADJ
ap-1688	193	1	+	+	X
ap-1688	193	2	2	2	NUM
ap-1688	193	3	∫	∫	NOUN
ap-1688	193	4	t	t	PROPN
ap-1688	193	5	s	s	PROPN
ap-1688	193	6	‖∇u(σ)‖2dσ	‖∇u(σ)‖2dσ	NOUN
ap-1688	193	7	≤	≤	NUM
ap-1688	193	8	‖u(s)‖2	‖u(s)‖2	NOUN
ap-1688	193	9	for	for	ADP
ap-1688	193	10	s	s	NOUN
ap-1688	193	11	=	=	SYM
ap-1688	193	12	0	0	NUM
ap-1688	194	1	and	and	CCONJ
ap-1688	194	2	almost	almost	ADV
ap-1688	194	3	all	all	PRON
ap-1688	194	4	s	s	VERB
ap-1688	194	5	>	>	X
ap-1688	194	6	0	0	NUM
ap-1688	194	7	,	,	PUNCT
ap-1688	194	8	and	and	CCONJ
ap-1688	194	9	all	all	DET
ap-1688	194	10	t	t	PROPN
ap-1688	194	11	≥	≥	NUM
ap-1688	194	12	s.	s.	PROPN
ap-1688	195	1	a	a	DET
ap-1688	195	2	global	global	ADJ
ap-1688	195	3	weak	weak	ADJ
ap-1688	195	4	solution	solution	NOUN
ap-1688	195	5	satisfying	satisfy	VERB
ap-1688	195	6	the	the	DET
ap-1688	195	7	strong	strong	ADJ
ap-1688	195	8	energy	energy	NOUN
ap-1688	195	9	inequality	inequality	NOUN
ap-1688	195	10	is	be	AUX
ap-1688	195	11	called	call	VERB
ap-1688	195	12	turbulent	turbulent	ADJ
ap-1688	195	13	.	.	PUNCT
ap-1688	196	1	definition	definition	NOUN
ap-1688	196	2	5	5	NUM
ap-1688	196	3	.	.	PUNCT
ap-1688	197	1	let	let	VERB
ap-1688	197	2	u0	u0	PROPN
ap-1688	197	3	∈	∈	PROPN
ap-1688	197	4	d(a	d(a	PROPN
ap-1688	197	5	)	)	PUNCT
ap-1688	197	6	.	.	PUNCT
ap-1688	198	1	a	a	DET
ap-1688	198	2	function	function	NOUN
ap-1688	198	3	u	u	PROPN
ap-1688	198	4	∈	∈	PROPN
ap-1688	198	5	c([0,∞);d(a	c([0,∞);d(a	NOUN
ap-1688	198	6	)	)	PUNCT
ap-1688	198	7	)	)	PUNCT
ap-1688	198	8	∩	∩	PROPN
ap-1688	198	9	c1((0,∞);l2	c1((0,∞);l2	PROPN
ap-1688	198	10	σ	σ	PROPN
ap-1688	198	11	)	)	PUNCT
ap-1688	198	12	is	be	AUX
ap-1688	198	13	called	call	VERB
ap-1688	198	14	a	a	DET
ap-1688	198	15	global	global	ADJ
ap-1688	198	16	strong	strong	ADJ
ap-1688	198	17	solution	solution	NOUN
ap-1688	198	18	of	of	ADP
ap-1688	198	19	(	(	PUNCT
ap-1688	198	20	1)–(3	1)–(3	NUM
ap-1688	198	21	)	)	PUNCT
ap-1688	198	22	if	if	SCONJ
ap-1688	198	23	u(0	u(0	NOUN
ap-1688	198	24	)	)	PUNCT
ap-1688	198	25	=	=	PUNCT
ap-1688	198	26	u0	u0	ADJ
ap-1688	198	27	and	and	CCONJ
ap-1688	198	28	du	du	ADJ
ap-1688	198	29	/	/	SYM
ap-1688	198	30	dt	dt	NOUN
ap-1688	199	1	+	+	CCONJ
ap-1688	200	1	au+	au+	PROPN
ap-1688	200	2	pσ(u	pσ(u	X
ap-1688	200	3	·	·	PUNCT
ap-1688	200	4	∇u	∇u	X
ap-1688	200	5	)	)	PUNCT
ap-1688	200	6	=	=	SYM
ap-1688	200	7	0	0	NUM
ap-1688	200	8	for	for	ADP
ap-1688	200	9	every	every	DET
ap-1688	200	10	t	t	NOUN
ap-1688	200	11	>	>	X
ap-1688	200	12	0	0	X
ap-1688	200	13	.	.	PUNCT
ap-1688	201	1	if	if	SCONJ
ap-1688	201	2	u0	u0	PROPN
ap-1688	201	3	∈	∈	PROPN
ap-1688	201	4	l2	l2	NOUN
ap-1688	201	5	σ	σ	NOUN
ap-1688	201	6	then	then	ADV
ap-1688	201	7	there	there	PRON
ap-1688	201	8	exists	exist	VERB
ap-1688	201	9	at	at	ADP
ap-1688	201	10	least	least	ADV
ap-1688	201	11	one	one	NUM
ap-1688	201	12	turbulent	turbulent	ADJ
ap-1688	201	13	solution	solution	NOUN
ap-1688	201	14	of	of	ADP
ap-1688	201	15	(	(	PUNCT
ap-1688	201	16	1)–(3	1)–(3	NUM
ap-1688	201	17	)	)	PUNCT
ap-1688	201	18	(	(	PUNCT
ap-1688	201	19	see	see	VERB
ap-1688	201	20	[	[	X
ap-1688	201	21	4	4	NUM
ap-1688	201	22	]	]	NUM
ap-1688	201	23	)	)	PUNCT
ap-1688	201	24	.	.	PUNCT
ap-1688	202	1	every	every	DET
ap-1688	202	2	turbulent	turbulent	ADJ
ap-1688	202	3	solution	solution	NOUN
ap-1688	202	4	becomes	become	VERB
ap-1688	202	5	strong	strong	ADJ
ap-1688	202	6	after	after	ADP
ap-1688	202	7	some	some	DET
ap-1688	202	8	transient	transient	ADJ
ap-1688	202	9	time	time	NOUN
ap-1688	202	10	(	(	PUNCT
ap-1688	202	11	see	see	VERB
ap-1688	202	12	[	[	X
ap-1688	202	13	19	19	NUM
ap-1688	202	14	]	]	PUNCT
ap-1688	202	15	,	,	PUNCT
ap-1688	202	16	chapter	chapter	NOUN
ap-1688	202	17	v.	v.	NOUN
ap-1688	202	18	)	)	PUNCT
ap-1688	202	19	.	.	PUNCT
ap-1688	203	1	this	this	PRON
ap-1688	203	2	means	mean	VERB
ap-1688	203	3	that	that	SCONJ
ap-1688	203	4	there	there	PRON
ap-1688	203	5	exists	exist	VERB
ap-1688	203	6	t0	t0	PROPN
ap-1688	203	7	≥	≥	NUM
ap-1688	203	8	0	0	NUM
ap-1688	204	1	such	such	ADJ
ap-1688	204	2	that	that	SCONJ
ap-1688	204	3	u	u	PROPN
ap-1688	204	4	∈	∈	PROPN
ap-1688	204	5	c((t0,∞);d(a	c((t0,∞);d(a	PROPN
ap-1688	204	6	)	)	PUNCT
ap-1688	204	7	)	)	PUNCT
ap-1688	204	8	∩	∩	PROPN
ap-1688	204	9	c1((t0,∞);l2	c1((t0,∞);l2	PROPN
ap-1688	204	10	σ	σ	PROPN
ap-1688	204	11	)	)	PUNCT
ap-1688	204	12	and	and	CCONJ
ap-1688	204	13	du	du	PROPN
ap-1688	204	14	/	/	SYM
ap-1688	204	15	dt+au+	dt+au+	PROPN
ap-1688	204	16	pσ(u	pσ(u	X
ap-1688	204	17	·	·	PUNCT
ap-1688	205	1	∇u	∇u	ADJ
ap-1688	205	2	)	)	PUNCT
ap-1688	205	3	=	=	SYM
ap-1688	205	4	0	0	NUM
ap-1688	205	5	for	for	ADP
ap-1688	205	6	every	every	DET
ap-1688	205	7	t	t	PROPN
ap-1688	205	8	>	>	X
ap-1688	205	9	t0	t0	PROPN
ap-1688	205	10	.	.	PUNCT
ap-1688	206	1	6	6	NUM
ap-1688	206	2	conclusion	conclusion	NOUN
ap-1688	206	3	in	in	ADP
ap-1688	206	4	this	this	DET
ap-1688	206	5	paper	paper	NOUN
ap-1688	206	6	we	we	PRON
ap-1688	206	7	have	have	AUX
ap-1688	206	8	presented	present	VERB
ap-1688	206	9	a	a	DET
ap-1688	206	10	survey	survey	NOUN
ap-1688	206	11	of	of	ADP
ap-1688	206	12	some	some	DET
ap-1688	206	13	results	result	NOUN
ap-1688	206	14	on	on	ADP
ap-1688	206	15	the	the	DET
ap-1688	206	16	large	large	ADJ
ap-1688	206	17	time	time	NOUN
ap-1688	206	18	decay	decay	NOUN
ap-1688	206	19	of	of	ADP
ap-1688	206	20	energy	energy	NOUN
ap-1688	206	21	in	in	ADP
ap-1688	206	22	turbulent	turbulent	ADJ
ap-1688	206	23	solutions	solution	NOUN
ap-1688	206	24	to	to	ADP
ap-1688	206	25	the	the	DET
ap-1688	206	26	navier	navier	NOUN
ap-1688	206	27	-	-	PUNCT
ap-1688	206	28	stokes	stoke	NOUN
ap-1688	206	29	equations	equation	NOUN
ap-1688	206	30	,	,	PUNCT
ap-1688	206	31	and	and	CCONJ
ap-1688	206	32	the	the	DET
ap-1688	206	33	related	related	ADJ
ap-1688	206	34	topic	topic	NOUN
ap-1688	206	35	of	of	ADP
ap-1688	206	36	the	the	DET
ap-1688	206	37	large	large	ADJ
ap-1688	206	38	time	time	NOUN
ap-1688	206	39	energy	energy	NOUN
ap-1688	206	40	concentration	concentration	NOUN
ap-1688	206	41	in	in	ADP
ap-1688	206	42	the	the	DET
ap-1688	206	43	frequency	frequency	NOUN
ap-1688	206	44	space	space	NOUN
ap-1688	206	45	.	.	PUNCT
ap-1688	207	1	in	in	ADP
ap-1688	207	2	the	the	DET
ap-1688	207	3	fourth	fourth	ADJ
ap-1688	207	4	section	section	NOUN
ap-1688	207	5	we	we	PRON
ap-1688	207	6	improved	improve	VERB
ap-1688	207	7	a	a	DET
ap-1688	207	8	result	result	NOUN
ap-1688	207	9	from	from	ADP
ap-1688	207	10	[	[	X
ap-1688	207	11	8	8	NUM
ap-1688	207	12	]	]	PUNCT
ap-1688	207	13	and	and	CCONJ
ap-1688	207	14	showed	show	VERB
ap-1688	207	15	that	that	SCONJ
ap-1688	207	16	some	some	DET
ap-1688	207	17	navier	navier	NOUN
ap-1688	207	18	-	-	PUNCT
ap-1688	207	19	stokes	stoke	NOUN
ap-1688	207	20	flows	flow	NOUN
ap-1688	207	21	do	do	AUX
ap-1688	207	22	not	not	PART
ap-1688	207	23	decay	decay	VERB
ap-1688	207	24	asymptotically	asymptotically	ADV
ap-1688	207	25	to	to	ADP
ap-1688	207	26	zero	zero	NUM
ap-1688	207	27	when	when	SCONJ
ap-1688	207	28	considered	consider	VERB
ap-1688	207	29	in	in	ADP
ap-1688	207	30	suitable	suitable	ADJ
ap-1688	207	31	besov	besov	NOUN
ap-1688	207	32	spaces	space	NOUN
ap-1688	207	33	.	.	PUNCT
ap-1688	208	1	acknowledgements	acknowledgement	NOUN
ap-1688	208	2	this	this	DET
ap-1688	208	3	work	work	NOUN
ap-1688	208	4	has	have	AUX
ap-1688	208	5	been	be	AUX
ap-1688	208	6	supported	support	VERB
ap-1688	208	7	by	by	ADP
ap-1688	208	8	the	the	DET
ap-1688	208	9	ministry	ministry	PROPN
ap-1688	208	10	of	of	ADP
ap-1688	208	11	education	education	NOUN
ap-1688	208	12	of	of	ADP
ap-1688	208	13	the	the	DET
ap-1688	208	14	czech	czech	PROPN
ap-1688	208	15	republic	republic	NOUN
ap-1688	208	16	,	,	PUNCT
ap-1688	208	17	under	under	ADP
ap-1688	208	18	project	project	NOUN
ap-1688	208	19	msm	msm	NOUN
ap-1688	208	20	6840770003	6840770003	NUM
ap-1688	208	21	.	.	PUNCT
ap-1688	209	1	103	103	NUM
ap-1688	209	2	acta	acta	PROPN
ap-1688	209	3	polytechnica	polytechnica	PROPN
ap-1688	209	4	vol	vol	NOUN
ap-1688	209	5	.	.	PROPN
ap-1688	210	1	52	52	NUM
ap-1688	210	2	no	no	NOUN
ap-1688	210	3	.	.	PUNCT
ap-1688	211	1	6/2012	6/2012	NUM
ap-1688	211	2	references	reference	NOUN
ap-1688	211	3	[	[	X
ap-1688	211	4	1	1	NUM
ap-1688	211	5	]	]	PUNCT
ap-1688	211	6	brandolese	brandolese	PROPN
ap-1688	211	7	,	,	PUNCT
ap-1688	211	8	l.	l.	PROPN
ap-1688	211	9	:	:	PUNCT
ap-1688	211	10	asymptotic	asymptotic	ADJ
ap-1688	211	11	bahavior	bahavior	NOUN
ap-1688	211	12	of	of	ADP
ap-1688	211	13	the	the	DET
ap-1688	211	14	energy	energy	NOUN
ap-1688	211	15	and	and	CCONJ
ap-1688	211	16	pointwise	pointwise	NOUN
ap-1688	211	17	estimates	estimate	NOUN
ap-1688	211	18	for	for	ADP
ap-1688	211	19	solutions	solution	NOUN
ap-1688	211	20	to	to	ADP
ap-1688	211	21	the	the	DET
ap-1688	211	22	navier	navier	NOUN
ap-1688	211	23	-	-	PUNCT
ap-1688	211	24	stokes	stoke	NOUN
ap-1688	211	25	equations	equation	NOUN
ap-1688	211	26	,	,	PUNCT
ap-1688	211	27	rev	rev	PROPN
ap-1688	211	28	.	.	PROPN
ap-1688	211	29	mat	mat	PROPN
ap-1688	211	30	.	.	PROPN
ap-1688	211	31	iberoamericana	iberoamericana	PROPN
ap-1688	211	32	,	,	PUNCT
ap-1688	211	33	20	20	NUM
ap-1688	211	34	,	,	PUNCT
ap-1688	211	35	2004	2004	NUM
ap-1688	211	36	,	,	PUNCT
ap-1688	211	37	223–256	223–256	NUM
ap-1688	211	38	.	.	PUNCT
ap-1688	212	1	[	[	X
ap-1688	212	2	2	2	NUM
ap-1688	212	3	]	]	PUNCT
ap-1688	212	4	bahouri	bahouri	NOUN
ap-1688	212	5	,	,	PUNCT
ap-1688	212	6	h.	h.	PROPN
ap-1688	212	7	,	,	PUNCT
ap-1688	212	8	chemin	chemin	PROPN
ap-1688	212	9	,	,	PUNCT
ap-1688	212	10	j.y	j.y	PROPN
ap-1688	212	11	.	.	PROPN
ap-1688	212	12	,	,	PUNCT
ap-1688	212	13	danchin	danchin	PROPN
ap-1688	212	14	,	,	PUNCT
ap-1688	212	15	r.	r.	PROPN
ap-1688	212	16	:	:	PUNCT
ap-1688	212	17	fourier	fourier	NOUN
ap-1688	212	18	analysis	analysis	NOUN
ap-1688	212	19	and	and	CCONJ
ap-1688	212	20	nonlinear	nonlinear	ADJ
ap-1688	212	21	partial	partial	ADJ
ap-1688	212	22	differential	differential	NOUN
ap-1688	212	23	equations	equation	NOUN
ap-1688	212	24	.	.	PUNCT
ap-1688	213	1	[	[	X
ap-1688	213	2	fundamental	fundamental	ADJ
ap-1688	213	3	principles	principle	NOUN
ap-1688	213	4	of	of	ADP
ap-1688	213	5	mathematical	mathematical	ADJ
ap-1688	213	6	sciences	science	NOUN
ap-1688	213	7	343	343	NUM
ap-1688	213	8	]	]	PUNCT
ap-1688	213	9	,	,	PUNCT
ap-1688	213	10	heidelberg	heidelberg	NOUN
ap-1688	213	11	:	:	PUNCT
ap-1688	213	12	springer	springer	NOUN
ap-1688	213	13	,	,	PUNCT
ap-1688	213	14	2011	2011	NUM
ap-1688	213	15	.	.	PUNCT
ap-1688	214	1	[	[	X
ap-1688	214	2	3	3	NUM
ap-1688	214	3	]	]	X
ap-1688	214	4	chemin	chemin	X
ap-1688	214	5	,	,	PUNCT
ap-1688	214	6	j.y	j.y	PROPN
ap-1688	214	7	.	.	PROPN
ap-1688	214	8	:	:	PUNCT
ap-1688	214	9	localization	localization	NOUN
ap-1688	214	10	in	in	ADP
ap-1688	214	11	fourier	fourier	ADJ
ap-1688	214	12	space	space	NOUN
ap-1688	214	13	and	and	CCONJ
ap-1688	214	14	navier	navier	NOUN
ap-1688	214	15	-	-	PUNCT
ap-1688	214	16	stokes	stoke	NOUN
ap-1688	214	17	system	system	NOUN
ap-1688	214	18	,	,	PUNCT
ap-1688	214	19	pubbl	pubbl	PROPN
ap-1688	214	20	.	.	PUNCT
ap-1688	215	1	cent	cent	NOUN
ap-1688	215	2	.	.	PUNCT
ap-1688	216	1	ric	ric	INTJ
ap-1688	216	2	.	.	PUNCT
ap-1688	216	3	mat	mat	PROPN
ap-1688	216	4	.	.	PUNCT
ap-1688	217	1	ennio	ennio	PROPN
ap-1688	217	2	giorgi	giorgi	PROPN
ap-1688	217	3	,	,	PUNCT
ap-1688	217	4	scuola	scuola	NOUN
ap-1688	217	5	norm	norm	NOUN
ap-1688	217	6	.	.	PUNCT
ap-1688	218	1	sup	sup	NOUN
ap-1688	218	2	.	.	PROPN
ap-1688	218	3	,	,	PUNCT
ap-1688	218	4	vol	vol	NOUN
ap-1688	218	5	.	.	PUNCT
ap-1688	219	1	i	i	PRON
ap-1688	219	2	,	,	PUNCT
ap-1688	219	3	53	53	NUM
ap-1688	219	4	,	,	PUNCT
ap-1688	219	5	2004	2004	NUM
ap-1688	219	6	.	.	PUNCT
ap-1688	220	1	[	[	X
ap-1688	220	2	4	4	NUM
ap-1688	220	3	]	]	SYM
ap-1688	220	4	farwig	farwig	PROPN
ap-1688	220	5	,	,	PUNCT
ap-1688	220	6	r.	r.	PROPN
ap-1688	220	7	,	,	PUNCT
ap-1688	220	8	kozono	kozono	PROPN
ap-1688	220	9	,	,	PUNCT
ap-1688	220	10	h.	h.	PROPN
ap-1688	220	11	,	,	PUNCT
ap-1688	220	12	sohr	sohr	PROPN
ap-1688	220	13	,	,	PUNCT
ap-1688	220	14	h.	h.	PROPN
ap-1688	220	15	:	:	PUNCT
ap-1688	220	16	an	an	DET
ap-1688	220	17	lq−	lq−	PROPN
ap-1688	220	18	approach	approach	NOUN
ap-1688	220	19	to	to	ADP
ap-1688	220	20	stokes	stoke	NOUN
ap-1688	220	21	and	and	CCONJ
ap-1688	220	22	navier	navier	NOUN
ap-1688	220	23	-	-	PUNCT
ap-1688	220	24	stokes	stoke	NOUN
ap-1688	220	25	equations	equation	NOUN
ap-1688	220	26	in	in	ADP
ap-1688	220	27	general	general	ADJ
ap-1688	220	28	domains	domain	NOUN
ap-1688	220	29	,	,	PUNCT
ap-1688	220	30	acta	acta	PROPN
ap-1688	220	31	math	math	PROPN
ap-1688	220	32	.	.	PUNCT
ap-1688	220	33	,	,	PUNCT
ap-1688	220	34	195	195	NUM
ap-1688	220	35	,	,	PUNCT
ap-1688	220	36	2005	2005	NUM
ap-1688	220	37	,	,	PUNCT
ap-1688	220	38	21–53	21–53	NUM
ap-1688	220	39	.	.	PUNCT
ap-1688	221	1	[	[	X
ap-1688	221	2	5	5	NUM
ap-1688	221	3	]	]	SYM
ap-1688	221	4	kajikiya	kajikiya	PROPN
ap-1688	221	5	,	,	PUNCT
ap-1688	221	6	r.	r.	PROPN
ap-1688	221	7	,	,	PUNCT
ap-1688	221	8	miyakawa	miyakawa	NOUN
ap-1688	221	9	,	,	PUNCT
ap-1688	221	10	t.	t.	PROPN
ap-1688	221	11	:	:	PUNCT
ap-1688	221	12	on	on	ADP
ap-1688	221	13	l2	l2	NOUN
ap-1688	221	14	decay	decay	NOUN
ap-1688	221	15	of	of	ADP
ap-1688	221	16	weak	weak	ADJ
ap-1688	221	17	solutions	solution	NOUN
ap-1688	221	18	of	of	ADP
ap-1688	221	19	the	the	DET
ap-1688	221	20	navier	navier	NOUN
ap-1688	221	21	-	-	PUNCT
ap-1688	221	22	stokes	stoke	NOUN
ap-1688	221	23	equations	equation	NOUN
ap-1688	221	24	in	in	ADP
ap-1688	221	25	rn	rn	PROPN
ap-1688	221	26	,	,	PUNCT
ap-1688	221	27	math	math	NOUN
ap-1688	221	28	.	.	PUNCT
ap-1688	222	1	z.	z.	PROPN
ap-1688	222	2	,	,	PUNCT
ap-1688	222	3	192	192	NUM
ap-1688	222	4	,	,	PUNCT
ap-1688	222	5	1986	1986	NUM
ap-1688	222	6	,	,	PUNCT
ap-1688	222	7	135–148	135–148	NUM
ap-1688	222	8	.	.	PUNCT
ap-1688	223	1	[	[	X
ap-1688	223	2	6	6	NUM
ap-1688	223	3	]	]	X
ap-1688	223	4	kato	kato	PROPN
ap-1688	223	5	,	,	PUNCT
ap-1688	223	6	t.	t.	PROPN
ap-1688	223	7	:	:	PUNCT
ap-1688	223	8	strong	strong	ADJ
ap-1688	223	9	lp−	lp−	PROPN
ap-1688	223	10	solutions	solution	NOUN
ap-1688	223	11	of	of	ADP
ap-1688	223	12	the	the	DET
ap-1688	223	13	navierstokes	navierstoke	NOUN
ap-1688	223	14	equations	equation	NOUN
ap-1688	223	15	in	in	ADP
ap-1688	223	16	rm	rm	PROPN
ap-1688	223	17	,	,	PUNCT
ap-1688	223	18	with	with	ADP
ap-1688	223	19	applications	application	NOUN
ap-1688	223	20	to	to	ADP
ap-1688	223	21	weak	weak	ADJ
ap-1688	223	22	solutions	solution	NOUN
ap-1688	223	23	,	,	PUNCT
ap-1688	223	24	math	math	NOUN
ap-1688	223	25	.	.	PUNCT
ap-1688	224	1	z.	z.	PROPN
ap-1688	224	2	187	187	NUM
ap-1688	224	3	1984	1984	NUM
ap-1688	224	4	,	,	PUNCT
ap-1688	224	5	1471–480	1471–480	NOUN
ap-1688	224	6	.	.	PUNCT
ap-1688	225	1	[	[	X
ap-1688	225	2	7	7	X
ap-1688	225	3	]	]	X
ap-1688	225	4	leray	leray	ADV
ap-1688	225	5	,	,	PUNCT
ap-1688	225	6	j.	j.	PROPN
ap-1688	225	7	:	:	PUNCT
ap-1688	225	8	sur	sur	PROPN
ap-1688	225	9	le	le	X
ap-1688	225	10	mouvement	mouvement	PROPN
ap-1688	225	11	d’un	d’un	PROPN
ap-1688	225	12	liquide	liquide	PROPN
ap-1688	225	13	visqueux	visqueux	PROPN
ap-1688	225	14	emplissant	emplissant	PROPN
ap-1688	225	15	l’éspace	l’éspace	NOUN
ap-1688	225	16	,	,	PUNCT
ap-1688	225	17	acta	acta	PROPN
ap-1688	225	18	math	math	PROPN
ap-1688	225	19	.	.	PUNCT
ap-1688	226	1	63	63	NUM
ap-1688	226	2	,	,	PUNCT
ap-1688	226	3	1934	1934	NUM
ap-1688	226	4	,	,	PUNCT
ap-1688	226	5	193–248	193–248	NUM
ap-1688	226	6	.	.	PUNCT
ap-1688	227	1	[	[	X
ap-1688	227	2	8	8	NUM
ap-1688	227	3	]	]	PUNCT
ap-1688	227	4	miyakawa	miyakawa	NOUN
ap-1688	227	5	,	,	PUNCT
ap-1688	227	6	t.	t.	PROPN
ap-1688	227	7	:	:	PUNCT
ap-1688	227	8	on	on	ADP
ap-1688	227	9	upper	upper	ADJ
ap-1688	227	10	and	and	CCONJ
ap-1688	227	11	lower	low	ADJ
ap-1688	227	12	bounds	bound	NOUN
ap-1688	227	13	of	of	ADP
ap-1688	227	14	rates	rate	NOUN
ap-1688	227	15	of	of	ADP
ap-1688	227	16	decay	decay	NOUN
ap-1688	227	17	for	for	ADP
ap-1688	227	18	nonstationary	nonstationary	ADJ
ap-1688	227	19	navier	navier	NOUN
ap-1688	227	20	-	-	PUNCT
ap-1688	227	21	stokes	stoke	NOUN
ap-1688	227	22	flows	flow	NOUN
ap-1688	227	23	in	in	ADP
ap-1688	227	24	the	the	DET
ap-1688	227	25	whole	whole	ADJ
ap-1688	227	26	space	space	NOUN
ap-1688	227	27	,	,	PUNCT
ap-1688	227	28	hiroshima	hiroshima	PROPN
ap-1688	227	29	math	math	PROPN
ap-1688	227	30	.	.	PUNCT
ap-1688	228	1	j.	j.	PROPN
ap-1688	228	2	,	,	PUNCT
ap-1688	228	3	126	126	NUM
ap-1688	228	4	,	,	PUNCT
ap-1688	228	5	2002	2002	NUM
ap-1688	228	6	,	,	PUNCT
ap-1688	228	7	431–462	431–462	NUM
ap-1688	228	8	.	.	PUNCT
ap-1688	229	1	[	[	X
ap-1688	229	2	9	9	NUM
ap-1688	229	3	]	]	PUNCT
ap-1688	229	4	miyakawa	miyakawa	NOUN
ap-1688	229	5	,	,	PUNCT
ap-1688	229	6	t.	t.	PROPN
ap-1688	229	7	,	,	PUNCT
ap-1688	229	8	schonbek	schonbek	PROPN
ap-1688	229	9	,	,	PUNCT
ap-1688	229	10	m.	m.	NOUN
ap-1688	229	11	:	:	PUNCT
ap-1688	229	12	on	on	ADP
ap-1688	229	13	optimal	optimal	ADJ
ap-1688	229	14	decay	decay	NOUN
ap-1688	229	15	rates	rate	NOUN
ap-1688	229	16	for	for	ADP
ap-1688	229	17	weak	weak	ADJ
ap-1688	229	18	solutions	solution	NOUN
ap-1688	229	19	to	to	ADP
ap-1688	229	20	the	the	DET
ap-1688	229	21	navier	navier	NOUN
ap-1688	229	22	-	-	PUNCT
ap-1688	229	23	stokes	stoke	NOUN
ap-1688	229	24	equations	equation	NOUN
ap-1688	229	25	in	in	ADP
ap-1688	229	26	rn	rn	PROPN
ap-1688	229	27	,	,	PUNCT
ap-1688	229	28	math	math	NOUN
ap-1688	229	29	.	.	PUNCT
ap-1688	229	30	bohem	bohem	PROPN
ap-1688	229	31	.	.	PUNCT
ap-1688	229	32	,	,	PUNCT
ap-1688	229	33	126	126	NUM
ap-1688	229	34	,	,	PUNCT
ap-1688	229	35	2001	2001	NUM
ap-1688	229	36	,	,	PUNCT
ap-1688	229	37	443	443	NUM
ap-1688	229	38	–	–	PUNCT
ap-1688	229	39	455	455	NUM
ap-1688	229	40	.	.	PUNCT
ap-1688	230	1	[	[	X
ap-1688	230	2	10	10	NUM
ap-1688	230	3	]	]	X
ap-1688	230	4	scarpellini	scarpellini	PROPN
ap-1688	230	5	,	,	PUNCT
ap-1688	230	6	b.	b.	NOUN
ap-1688	230	7	:	:	PUNCT
ap-1688	230	8	solutions	solution	NOUN
ap-1688	230	9	of	of	ADP
ap-1688	230	10	evolution	evolution	NOUN
ap-1688	230	11	equations	equation	NOUN
ap-1688	230	12	of	of	ADP
ap-1688	230	13	slow	slow	ADJ
ap-1688	230	14	exponential	exponential	ADJ
ap-1688	230	15	decay	decay	NOUN
ap-1688	230	16	,	,	PUNCT
ap-1688	230	17	analysis	analysis	NOUN
ap-1688	230	18	20	20	NUM
ap-1688	230	19	2000	2000	NUM
ap-1688	230	20	,	,	PUNCT
ap-1688	230	21	255	255	NUM
ap-1688	230	22	–	–	PUNCT
ap-1688	230	23	283	283	NUM
ap-1688	230	24	.	.	PUNCT
ap-1688	231	1	[	[	X
ap-1688	231	2	11	11	NUM
ap-1688	231	3	]	]	X
ap-1688	231	4	schonbek	schonbek	NOUN
ap-1688	231	5	,	,	PUNCT
ap-1688	231	6	m.	m.	NOUN
ap-1688	231	7	:	:	PUNCT
ap-1688	231	8	asymptotic	asymptotic	ADJ
ap-1688	231	9	behavior	behavior	NOUN
ap-1688	231	10	of	of	ADP
ap-1688	231	11	solutions	solution	NOUN
ap-1688	231	12	to	to	ADP
ap-1688	231	13	the	the	DET
ap-1688	231	14	three	three	NUM
ap-1688	231	15	-	-	PUNCT
ap-1688	231	16	dimensional	dimensional	ADJ
ap-1688	231	17	navier	navier	NOUN
ap-1688	231	18	-	-	PUNCT
ap-1688	231	19	stokes	stoke	NOUN
ap-1688	231	20	equations	equation	NOUN
ap-1688	231	21	,	,	PUNCT
ap-1688	231	22	indiana	indiana	PROPN
ap-1688	231	23	university	university	PROPN
ap-1688	231	24	mathematics	mathematics	PROPN
ap-1688	231	25	journal	journal	NOUN
ap-1688	231	26	,	,	PUNCT
ap-1688	231	27	41	41	NUM
ap-1688	231	28	,	,	PUNCT
ap-1688	231	29	1992	1992	NUM
ap-1688	231	30	,	,	PUNCT
ap-1688	231	31	809–823	809–823	NUM
ap-1688	231	32	.	.	PUNCT
ap-1688	232	1	[	[	X
ap-1688	232	2	12	12	NUM
ap-1688	232	3	]	]	X
ap-1688	232	4	schonbek	schonbek	NOUN
ap-1688	232	5	,	,	PUNCT
ap-1688	232	6	m.	m.	NOUN
ap-1688	232	7	:	:	PUNCT
ap-1688	232	8	large	large	ADJ
ap-1688	232	9	time	time	NOUN
ap-1688	232	10	behavior	behavior	NOUN
ap-1688	232	11	of	of	ADP
ap-1688	232	12	solutions	solution	NOUN
ap-1688	232	13	to	to	ADP
ap-1688	232	14	the	the	DET
ap-1688	232	15	navier	navier	NOUN
ap-1688	232	16	-	-	PUNCT
ap-1688	232	17	stokes	stoke	NOUN
ap-1688	232	18	equations	equation	NOUN
ap-1688	232	19	,	,	PUNCT
ap-1688	232	20	comm	comm	NOUN
ap-1688	232	21	.	.	PUNCT
ap-1688	233	1	partial	partial	ADJ
ap-1688	233	2	differential	differential	NOUN
ap-1688	233	3	equations	equation	NOUN
ap-1688	233	4	,	,	PUNCT
ap-1688	233	5	11	11	NUM
ap-1688	233	6	1986	1986	NUM
ap-1688	233	7	,	,	PUNCT
ap-1688	233	8	753–763	753–763	NUM
ap-1688	233	9	.	.	PUNCT
ap-1688	234	1	[	[	X
ap-1688	234	2	13	13	NUM
ap-1688	234	3	]	]	X
ap-1688	234	4	schonbek	schonbek	NOUN
ap-1688	234	5	,	,	PUNCT
ap-1688	234	6	m.	m.	NOUN
ap-1688	234	7	:	:	PUNCT
ap-1688	234	8	lower	low	ADJ
ap-1688	234	9	bounds	bound	NOUN
ap-1688	234	10	of	of	ADP
ap-1688	234	11	rates	rate	NOUN
ap-1688	234	12	of	of	ADP
ap-1688	234	13	decay	decay	NOUN
ap-1688	234	14	for	for	ADP
ap-1688	234	15	solutions	solution	NOUN
ap-1688	234	16	to	to	ADP
ap-1688	234	17	the	the	DET
ap-1688	234	18	navier	navier	NOUN
ap-1688	234	19	-	-	PUNCT
ap-1688	234	20	stokes	stoke	NOUN
ap-1688	234	21	equations	equation	NOUN
ap-1688	234	22	,	,	PUNCT
ap-1688	234	23	journal	journal	NOUN
ap-1688	234	24	of	of	ADP
ap-1688	234	25	the	the	DET
ap-1688	234	26	american	american	PROPN
ap-1688	234	27	mathematical	mathematical	PROPN
ap-1688	234	28	society	society	NOUN
ap-1688	234	29	,	,	PUNCT
ap-1688	234	30	4	4	NUM
ap-1688	234	31	,	,	PUNCT
ap-1688	234	32	1991	1991	NUM
ap-1688	234	33	,	,	PUNCT
ap-1688	234	34	423–449	423–449	NUM
ap-1688	234	35	.	.	PUNCT
ap-1688	235	1	[	[	X
ap-1688	235	2	14	14	NUM
ap-1688	235	3	]	]	X
ap-1688	235	4	schonbek	schonbek	ADJ
ap-1688	235	5	,	,	PUNCT
ap-1688	235	6	m.	m.	NOUN
ap-1688	235	7	,	,	PUNCT
ap-1688	235	8	wiegner	wiegner	NOUN
ap-1688	235	9	,	,	PUNCT
ap-1688	235	10	m.	m.	NOUN
ap-1688	235	11	:	:	PUNCT
ap-1688	235	12	on	on	ADP
ap-1688	235	13	the	the	DET
ap-1688	235	14	decay	decay	NOUN
ap-1688	235	15	of	of	ADP
ap-1688	235	16	higher	high	ADJ
ap-1688	235	17	-	-	PUNCT
ap-1688	235	18	order	order	NOUN
ap-1688	235	19	norms	norm	NOUN
ap-1688	235	20	of	of	ADP
ap-1688	235	21	the	the	DET
ap-1688	235	22	solutions	solution	NOUN
ap-1688	235	23	of	of	ADP
ap-1688	235	24	navierstokes	navierstoke	NOUN
ap-1688	235	25	equations	equation	NOUN
ap-1688	235	26	,	,	PUNCT
ap-1688	235	27	proceedings	proceeding	NOUN
ap-1688	235	28	of	of	ADP
ap-1688	235	29	the	the	DET
ap-1688	235	30	royal	royal	ADJ
ap-1688	235	31	society	society	NOUN
ap-1688	235	32	of	of	ADP
ap-1688	235	33	edinburgh	edinburgh	PROPN
ap-1688	235	34	,	,	PUNCT
ap-1688	235	35	126a	126a	NOUN
ap-1688	235	36	,	,	PUNCT
ap-1688	235	37	1996	1996	NUM
ap-1688	235	38	,	,	PUNCT
ap-1688	235	39	677–685	677–685	NUM
ap-1688	235	40	.	.	PUNCT
ap-1688	236	1	[	[	X
ap-1688	236	2	15	15	NUM
ap-1688	236	3	]	]	X
ap-1688	236	4	skalák	skalák	PROPN
ap-1688	236	5	,	,	PUNCT
ap-1688	236	6	z.	z.	PROPN
ap-1688	236	7	:	:	PUNCT
ap-1688	236	8	some	some	DET
ap-1688	236	9	aspects	aspect	NOUN
ap-1688	236	10	of	of	ADP
ap-1688	236	11	the	the	DET
ap-1688	236	12	asymptotic	asymptotic	ADJ
ap-1688	236	13	dynamics	dynamic	NOUN
ap-1688	236	14	of	of	ADP
ap-1688	236	15	solutions	solution	NOUN
ap-1688	236	16	of	of	ADP
ap-1688	236	17	the	the	DET
ap-1688	236	18	homogeneous	homogeneous	ADJ
ap-1688	236	19	navierstokes	navierstoke	NOUN
ap-1688	236	20	equations	equation	NOUN
ap-1688	236	21	in	in	ADP
ap-1688	236	22	general	general	ADJ
ap-1688	236	23	domains	domain	NOUN
ap-1688	236	24	,	,	PUNCT
ap-1688	236	25	j.	j.	PROPN
ap-1688	236	26	math	math	PROPN
ap-1688	236	27	.	.	PUNCT
ap-1688	237	1	fluid	fluid	ADJ
ap-1688	237	2	mech	mech	NOUN
ap-1688	237	3	.	.	PUNCT
ap-1688	237	4	,	,	PUNCT
ap-1688	237	5	12	12	NUM
ap-1688	237	6	,	,	PUNCT
ap-1688	237	7	2010	2010	NUM
ap-1688	237	8	,	,	PUNCT
ap-1688	237	9	503–535	503–535	NUM
ap-1688	237	10	.	.	PUNCT
ap-1688	238	1	[	[	X
ap-1688	238	2	16	16	NUM
ap-1688	238	3	]	]	X
ap-1688	238	4	skalák	skalák	PROPN
ap-1688	238	5	,	,	PUNCT
ap-1688	238	6	z.	z.	PROPN
ap-1688	238	7	:	:	PUNCT
ap-1688	238	8	on	on	ADP
ap-1688	238	9	the	the	DET
ap-1688	238	10	asymptotic	asymptotic	ADJ
ap-1688	238	11	decay	decay	NOUN
ap-1688	238	12	of	of	ADP
ap-1688	238	13	higherorder	higherorder	NOUN
ap-1688	238	14	norms	norm	NOUN
ap-1688	238	15	of	of	ADP
ap-1688	238	16	the	the	DET
ap-1688	238	17	solutions	solution	NOUN
ap-1688	238	18	to	to	ADP
ap-1688	238	19	the	the	DET
ap-1688	238	20	navier	navier	NOUN
ap-1688	238	21	-	-	PUNCT
ap-1688	238	22	stokes	stoke	NOUN
ap-1688	238	23	equations	equation	NOUN
ap-1688	238	24	in	in	ADP
ap-1688	238	25	r3	r3	PROPN
ap-1688	238	26	,	,	PUNCT
ap-1688	238	27	discrete	discrete	ADJ
ap-1688	238	28	contin	contin	NOUN
ap-1688	238	29	.	.	PUNCT
ap-1688	239	1	dyn	dyn	NOUN
ap-1688	239	2	.	.	PUNCT
ap-1688	240	1	syst	syst	PROPN
ap-1688	240	2	.	.	PUNCT
ap-1688	241	1	ser	ser	PROPN
ap-1688	241	2	.	.	PUNCT
ap-1688	242	1	s	s	PROPN
ap-1688	242	2	,	,	PUNCT
ap-1688	242	3	3	3	NUM
ap-1688	242	4	,	,	PUNCT
ap-1688	242	5	2010	2010	NUM
ap-1688	242	6	,	,	PUNCT
ap-1688	242	7	361–370	361–370	NUM
ap-1688	242	8	.	.	PUNCT
ap-1688	243	1	[	[	X
ap-1688	243	2	17	17	NUM
ap-1688	243	3	]	]	X
ap-1688	243	4	skalák	skalák	PROPN
ap-1688	243	5	,	,	PUNCT
ap-1688	243	6	z.	z.	PROPN
ap-1688	243	7	:	:	PUNCT
ap-1688	244	1	large	large	ADJ
ap-1688	244	2	time	time	NOUN
ap-1688	244	3	behavior	behavior	NOUN
ap-1688	244	4	of	of	ADP
ap-1688	244	5	energy	energy	NOUN
ap-1688	244	6	in	in	ADP
ap-1688	244	7	exponentially	exponentially	ADV
ap-1688	244	8	decreasing	decrease	VERB
ap-1688	244	9	solutions	solution	NOUN
ap-1688	244	10	of	of	ADP
ap-1688	244	11	the	the	DET
ap-1688	244	12	navierstokes	navierstoke	NOUN
ap-1688	244	13	equations	equation	NOUN
ap-1688	244	14	,	,	PUNCT
ap-1688	244	15	nonlinear	nonlinear	ADJ
ap-1688	244	16	analysis	analysis	NOUN
ap-1688	244	17	,	,	PUNCT
ap-1688	244	18	71	71	NUM
ap-1688	244	19	,	,	PUNCT
ap-1688	244	20	2009	2009	NUM
ap-1688	244	21	,	,	PUNCT
ap-1688	244	22	593–603	593–603	NUM
ap-1688	244	23	.	.	PUNCT
ap-1688	245	1	[	[	X
ap-1688	245	2	18	18	NUM
ap-1688	245	3	]	]	X
ap-1688	245	4	skalák	skalák	PROPN
ap-1688	245	5	,	,	PUNCT
ap-1688	245	6	z.	z.	PROPN
ap-1688	245	7	:	:	PUNCT
ap-1688	245	8	solutions	solution	NOUN
ap-1688	245	9	to	to	ADP
ap-1688	245	10	the	the	DET
ap-1688	245	11	navier	navier	NOUN
ap-1688	245	12	-	-	PUNCT
ap-1688	245	13	stokes	stoke	NOUN
ap-1688	245	14	equations	equation	NOUN
ap-1688	245	15	with	with	ADP
ap-1688	245	16	large	large	ADJ
ap-1688	245	17	time	time	NOUN
ap-1688	245	18	energy	energy	NOUN
ap-1688	245	19	concentration	concentration	NOUN
ap-1688	245	20	in	in	ADP
ap-1688	245	21	low	low	ADJ
ap-1688	245	22	frequencies	frequency	NOUN
ap-1688	245	23	,	,	PUNCT
ap-1688	245	24	zamm	zamm	PROPN
ap-1688	245	25	,	,	PUNCT
ap-1688	245	26	91	91	NUM
ap-1688	245	27	,	,	PUNCT
ap-1688	245	28	2011	2011	NUM
ap-1688	245	29	,	,	PUNCT
ap-1688	245	30	733–742	733–742	NUM
ap-1688	245	31	.	.	PUNCT
ap-1688	246	1	[	[	X
ap-1688	246	2	19	19	NUM
ap-1688	246	3	]	]	X
ap-1688	246	4	sohr	sohr	NOUN
ap-1688	246	5	,	,	PUNCT
ap-1688	246	6	h.	h.	PROPN
ap-1688	246	7	:	:	PUNCT
ap-1688	246	8	the	the	DET
ap-1688	246	9	navier	navier	NOUN
ap-1688	246	10	-	-	PUNCT
ap-1688	246	11	stokes	stoke	NOUN
ap-1688	246	12	equations	equation	NOUN
ap-1688	246	13	,	,	PUNCT
ap-1688	246	14	an	an	DET
ap-1688	246	15	elementary	elementary	ADJ
ap-1688	246	16	functional	functional	ADJ
ap-1688	246	17	analytic	analytic	ADJ
ap-1688	246	18	approach	approach	NOUN
ap-1688	246	19	.	.	PUNCT
ap-1688	247	1	basel	basel	PROPN
ap-1688	247	2	,	,	PUNCT
ap-1688	247	3	boston	boston	PROPN
ap-1688	247	4	,	,	PUNCT
ap-1688	247	5	berlin	berlin	PROPN
ap-1688	247	6	:	:	PUNCT
ap-1688	247	7	birkhäuser	birkhäuser	PROPN
ap-1688	247	8	verlag	verlag	PROPN
ap-1688	247	9	,	,	PUNCT
ap-1688	247	10	2001	2001	NUM
ap-1688	247	11	.	.	PUNCT
ap-1688	248	1	[	[	X
ap-1688	248	2	20	20	NUM
ap-1688	248	3	]	]	SYM
ap-1688	248	4	wiegner	wiegner	NOUN
ap-1688	248	5	,	,	PUNCT
ap-1688	248	6	m.	m.	NOUN
ap-1688	248	7	:	:	PUNCT
ap-1688	248	8	decay	decay	VERB
ap-1688	248	9	results	result	NOUN
ap-1688	248	10	for	for	ADP
ap-1688	248	11	weak	weak	ADJ
ap-1688	248	12	solutions	solution	NOUN
ap-1688	248	13	of	of	ADP
ap-1688	248	14	the	the	DET
ap-1688	248	15	navier	navier	NOUN
ap-1688	248	16	-	-	PUNCT
ap-1688	248	17	stokes	stoke	NOUN
ap-1688	248	18	equations	equation	NOUN
ap-1688	248	19	on	on	ADP
ap-1688	248	20	rn	rn	PROPN
ap-1688	248	21	,	,	PUNCT
ap-1688	248	22	j.	j.	PROPN
ap-1688	248	23	london	london	PROPN
ap-1688	248	24	math	math	PROPN
ap-1688	248	25	.	.	PUNCT
ap-1688	249	1	soc	soc	PROPN
ap-1688	249	2	.	.	PROPN
ap-1688	249	3	,	,	PUNCT
ap-1688	249	4	35	35	NUM
ap-1688	249	5	,	,	PUNCT
ap-1688	249	6	1987	1987	NUM
ap-1688	249	7	,	,	PUNCT
ap-1688	249	8	303–313	303–313	NUM
ap-1688	249	9	.	.	PUNCT
ap-1688	250	1	104	104	NUM