id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ap-6142	Kračmar, Stanislav; Neustupa, Jiří	MODELING OF FLOWS THROUGH A CHANNEL BY THE NAVIER–STOKES VARIATIONAL INEQUALITIES	2021	10	.pdf	application/pdf	7920	356	71	Substituting this to the first term in (16), we obtain∫ T 0 〈 ∂tw,w − v 〉 dt = ∫ t∗ 0 〈 ∂t(v∗ext + αu), (α− 1)u 〉 dt + ∫ t∗+δ t∗ 〈 ∂t(v∗ext + ηu), (η − 1)u 〉 dt = (α− 1) ∫ t∗ 0 〈 ∂tv ∗ ext, u 〉 dt + α(α− 1) ∫ t∗ 0 〈 ∂tu, u 〉 dt + ∫ t∗+δ t∗ 〈 ∂tv ∗ ext, (η − 1)u 〉 dt + ∫ t∗+δ t∗ 〈 ∂t(ηu), ηu 〉 dt− ∫ t∗+δ t∗ 〈 η̇u, u 〉 dt − ∫ t∗+δ t∗ 〈 η ∂tu, u 〉 dt = ∫ t∗+δ 0 (η − 1) 〈 ∂tv ∗ ext, u 〉 dt + α(α− 1) 2 ( ‖u(t∗)‖22 − ‖u(0)‖22 ) + 1 2 ( ‖η(t∗ + δ)u(t∗ + Considering δ → 0+, we get∫ T 0 〈 ∂tw,w − v 〉 dt = (α− 1) ∫ t∗ 0 〈∂tv∗ext, u〉 dt + α2 − 1 2 ‖u(t∗)‖22 − α(α− 1) 2 ‖u(0‖22.	cache/ap-6142.pdf	txt/ap-6142.txt