Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 91, pp. 1–26.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

LOCAL EXISTENCE AND BLOW-UP CRITERION FOR THE

TWO AND THREE DIMENSIONAL IDEAL MAGNETIC

BÉNARD PROBLEM

UTPAL MANNA, AKASH ASHIRBAD PANDA

Abstract. In this article, we consider the ideal magnetic Bénard problem

in both two and three dimensions and prove the existence and uniqueness of
strong local-in-time solutions, in Hs for s > n

2
+ 1, n = 2, 3. In addition,

a necessary condition is derived for singularity development with respect to

the BMO-norm of the vorticity and electrical current, generalizing the Beale-
Kato-Majda condition for ideal hydrodynamics.

1. Introduction

The magnetic Bénard problem with full viscosity is

∂u

∂t
+ (u · ∇)u− ν∆u +∇p∗ = (b · ∇)b + θen, in Rn × (0,∞), (1.1)

∂θ

∂t
+ (u · ∇)θ − κ∆θ = u · en, in Rn × (0,∞), (1.2)

∂b

∂t
+ (u · ∇)b− µ∆b = (b · ∇)u, in Rn × (0,∞), (1.3)

∇ · u = 0 = ∇ · b, in Rn × (0,∞), (1.4)

with initial conditions

u(x, 0) = u0(x), θ(x, 0) = θ0(x), b(x, 0) = b0(x) in Rn,

where n = 2, 3. Here u : Rn× [0,∞)→ Rn is the velocity field, θ : Rn× [0,∞)→ R
is the temperature, b : Rn × [0,∞) → Rn is the magnetic field, p∗ is the total
pressure field where p∗ = p+ 1

2 |b|
2, p : Rn× [0,∞)→ R is the pressure. en denotes

the unit vector along the nth direction. The term θen represents buoyancy force
on fluid motion and u · en signifies the Rayleigh-Bénard convection in a heated
inviscid fluid. ν ≥ 0, µ ≥ 0 and κ ≥ 0 denote the coefficients of kinematic viscosity,
magnetic diffusion and thermal diffusion respectively.

The global-in-time regularity in two-dimensions of the above problem when ν, µ
and κ > 0 is known for a long time [14]. Because of the parabolic couplings, it is
indeed possible to rewrite the above system in the abstract framework of the Navier-
Stokes equations and then use the standard solvability techniques (see Temam [30]).

2010 Mathematics Subject Classification. 76D03, 35B44, 35A01.
Key words and phrases. Magnetic Bénard problem; commutator estimates;

blow-up criterion; logarithmic Sobolev inequality.
c©2020 Texas State University.

Submitted June 3, 2018. Published September 7, 2020.

1



2 U. MANNA, A. A. PANDA EJDE-2020/91

In three-dimensions, one can at-most expect local-in-time solvability result with
arbitrary initial data and global-in-time result for sufficiently small initial data,
much like the Navier-Stokes equations. In [9], the authors obtained the global well-
posedness of two-dimensional magnetic Bénard problem without thermal diffusivity
and with vertical or horizontal magnetic diffusion. Moreover, the authors prove
global regularity and some conditional regularity of strong solutions with mixed
partial viscosity. This work provides an extension of an earlier result [33] on the
global regularity with full dissipation and magnetic diffusion. It is worthwhile to
note that there are very few literatures available where the case ν = κ = µ = 0 has
been discussed in two and three dimensions for the magnetic Bénard problem.

However, for the ideal magneto hydrodynamic (MHD) equations, i.e. when θ ≡ 0
and ν = µ = 0, in (1.1)-(1.3), the local-in-time existence of strong solutions have
been proved by Schmidt [28] and Secchi [29], when the initial data is in Hm for
integer m > 1 + n/2. Schmidt [28] obtained the well-posedness and regularity of
maximal solutions and continuous dependence on forcing terms and initial data
(using a regularization procedure). Caflisch, Klapper and Steele [7] derived a cri-
teria for energy conservation and helicity conservation for weak solutions of ideal
MHD equations. The authors in [7] extended the Beale-Kato-Majda [3] criterion to
the three-dimensional ideal MHD equations by showing that for sufficiently regular
initial data the condition∫ T

0

(‖∇ × u(τ)‖L∞ + ‖∇ × b(τ)‖L∞) dτ <∞,

ensures that the solution can be continued beyond time T , where ∇×u is the fluid
vorticity, ∇× b is the electrical current.

On the other hand, for the ideal Boussinesq system, i.e. when b ≡ 0, ν = κ = 0,
and the Rayleigh-Bénard convection term u · en is absent in (1.1)-(1.3), only local-
in-time existence results are available even in two-dimensions. It was proved in
[8] that if the initial data (u0, θ0) ∈ H3

σ(R2)×H3(R2), then local-in-time classical
solutions exist and is unique. Moreover, Beale-Kato-Majda type criterion for blow-
up of smooth solutions is established in [8]. More precisely, they proved that the
smooth solution exists on [0, T ] if and only if ∇θ ∈ L1(0, T ;L∞(R2)). For the
three-dimensional Boussinesq system, a very few results on local-in-time existence
and blow-up criterion are available (e.g. see [15, 16, 26, 31]). However, in the very
particular case of the axisymmetric initial data, global-in-time well-posedness has
been proven in three-dimensions by Abidi et al [1]. In recent work [23], authors
proved local-in-time existence and uniqueness of strong solutions in Hs for real
s > n/2 + 1 for the ideal Boussinesq equations in Rn, n = 2, 3 and established
Beale-Kato-Majda type blow-up criterion with respect to the BMO-norm of the
vorticity.

In this work, we consider the ideal magnetic Bénard problem (i.e. when ν = κ =
µ = 0) in both two and three dimensions and prove local-in-time existence and
uniqueness of the strong solutions when the initial data (u0, θ0,b0) ∈ Hs

σ(Rn) ×
Hs(Rn)×Hs

σ(Rn), where s > n/2 + 1. We prove when s > n/2 + 1, BMO-norms
of the vorticity, electrical current and that of the gradient of the temperature (i.e.
∇×u,∇×b,∇θ ∈ L1(0, T ;BMO)) control the breakdown of smooth solutions of the
above systems. However, we later show that under suitable additional assumption
on θ0, one can completely relax the condition on gradient of the temperature and
the conditions ∇ × u,∇ × b ∈ L1(0, T ;BMO) are sufficient to ensure that the



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 3

smooth solution persists. To the best of authors’ knowledge, this work is new in
the literature and may be seen as an extension of the blow-up criterion for ideal
MHD equations due to Caflisch et al [7] and that of ideal Boussinesq equations due
to Manna et al [23].

We note that, in view of the recent work of Bourgain and Li [4] on the ill-
posedness of the two and three dimensional Euler equations in Hn/2+1, n = 2, 3,
it seems likely that the ideal magnetic Bénard problem is also ill-posed in Hn/2+1,
n = 2, 3, although it still remains an open problem.

To be precise, in this work, we consider the ideal magnetic Bénard problem

∂u

∂t
+ (u · ∇)u +∇p∗ = (b · ∇)b + θen, in Rn × (0,∞), (1.5)

∂θ

∂t
+ (u · ∇)θ = u · en, in Rn × (0,∞), (1.6)

∂b

∂t
+ (u · ∇)b = (b · ∇)u, in Rn × (0,∞), (1.7)

with

∇ · u = 0 = ∇ · b, in Rn × (0,∞), (1.8)

u(x, 0) = u0(x), θ(x, 0) = θ0(x), b(x, 0) = b0(x) in Rn, (1.9)

and prove the following main results.
First we state the result concerning the existence of strong local-in-time solutions.

Theorem 1.1. Let s ∈ R be such that s > n
2 + 1, n = 2, 3. Let

(u0, θ0,b0) ∈ Hs
σ(Rn)×Hs(Rn)×Hs

σ(Rn).

Then there exists a unique strong solution (u, θ,b) to the problem (1.5)-(1.9), with

u ∈ C([0, T ∗];Hs
σ(Rn)), θ ∈ C([0, T ∗];Hs(Rn)) b ∈ C([0, T ∗];Hs

σ(Rn))

for some finite time T ∗ = T ∗(s, ‖u0‖Hsσ , ‖θ0‖Hs , ‖b0‖Hsσ ) > 0.

To prove this result, we consider the Fourier truncated ideal magnetic Bénard
problem on the whole of Rn, n = 2, 3, and show that the solutions (uR, θR,bR)
of some smoothed version of the ideal magnetic Bénard system exist. We then
establish that the Hs-norm of (uR, θR,bR) are uniformly bounded up to a terminal

time T̃ , which is independent of R. We further show that up to the blowup time,
the solution (uR, θR,bR) is a Cauchy sequence in the L2-norm as R→∞, and by

using Sobolev interpolation, (uR, θR,bR) → (u, θ,b) in any Hs′ for 0 < s′ < s.
Finally we provide the proof of Theorem 1.1 in Theorem 3.10.

Next, we establish that the BMO norms of the vorticity and electrical current
control the breakdown of smooth solutions. Our main result concerning the blow-up
criterion is as follows.

Theorem 1.2. Let (u0, θ0,b0) have same regularity as above and s > n
2 + 1, n =

2, 3. If (u, θ,b) satisfy the condition∫ T∗

0

(‖∇ × u(τ)‖BMO + ‖∇θ(τ)‖BMO + ‖∇ × b(τ)‖BMO) dτ <∞,



4 U. MANNA, A. A. PANDA EJDE-2020/91

then the solution (u, θ,b) can be continuously extended to [0, T ] for some T > T ∗.
However, if θ0 ∈ Hs(Rn) ∩W 1,p(Rn), 2 ≤ p ≤ ∞, then the condition∫ T∗

0

(‖∇ × u(τ)‖BMO + ‖∇ × b(τ)‖BMO) dτ <∞

is sufficient to ensure that the solution (u, θ,b) can be extended continuously to
[0, T ] for some T > T ∗.

Remark 1.3. The above result still holds if we replace BMO with the Besov space
B0
∞,∞ used in Kozono et al [19] or if we replace the condition by the one introduced

in Planchon [25]. To be precise, the condition above can be weakened to∫ T∗

0

(
‖∇ × u(τ)‖B0

∞,∞
+ ‖∇ × b(τ)‖B0

∞,∞

)
dτ

=

∫ T∗

0

(
sup
j
‖4j(∇× u(τ))‖L∞ + sup

j
‖4j(∇× b(τ))‖L∞

)
dτ <∞,

or to

lim
δ→0

∫ T∗

T∗−δ

(
sup
j
‖4j(∇× u(τ))‖L∞ + sup

j
‖4j(∇× b(τ))‖L∞

)
dτ < ε,

for some sufficiently small ε > 0.

The rest of the article is organized as follows. We define various operators,
function spaces, and certain basic inequalities in Section 2. In Section 3, we start
investigating about the ideal magnetic Bénard problem and prove results concern-
ing energy estimates and convergence of the approximate solutions before proving
Theorem 1.1 and 3.10. In section 4, we prove Theorems 1.2, Theorem 4.1 and 4.3.

2. Preliminaries

2.1. Fractional derivative operator. Let us define Js (real s > 0), which de-
notes the Bessel potential of order s, in terms of Fourier transform as follows:

F [Jsf ](ξ) = (1 + |ξ|2)s/2f̂(ξ).

Js is also equivalent to the operator (I −∆)s/2.

Assume 0 < s < ∞ and f ∈ L2(Rn). Then f ∈ Hs(Rn) if (1 + |ξ|2)s/2f̂(ξ) ∈
L2(Rn). The norm on Hs(Rn) is

‖f‖Hs =
(∫

Rn
[(1 + |ξ|2)s/2|f̂(ξ)|]2

)1/2

= ‖(1 + |ξ|2)s/2f̂(ξ)‖L2 = ‖Jsf‖L2 (2.1)

and the inner product on Hs(Rn) is

(f, g)Hs =
(

(1 + |ξ|2)s/2f̂(ξ), (1 + |ξ|2)s/2ĝ(ξ)
)
L2

= (F [Jsf ](ξ),F [Jsg](ξ))L2

= (Jsf, Jsg)L2 .

Remark 2.1. It is trivial to show that

‖∇f‖Hs−1 ≤ ‖f‖Hs .



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 5

2.2. Fourier truncation operator. Let us define the Fourier truncation operator
SR as follows:

ŜRf(ξ) := 1BR(ξ)f̂(ξ),

where BR, a ball of radius R centered at the origin and 1BR is the indicator function.
Then we infer the following properties:

(1) ‖SRf‖Hs(Rn) ≤ C‖f‖Hs(Rn), where C is a constant independent of R;

(2) ‖SRf − f‖Hs(Rn) ≤ C
Rk
‖f‖Hs+k(Rn);

(3) ‖(SR − SR′)f‖Hs ≤ C max
{

( 1
R )k, ( 1

R′ )
k
}
‖f‖Hs+k .

For the proofs of these properties see [13]. We define the function spaces

Hs
σ(Rn) = {f ∈ Hs(Rn) : ∇ · f = 0}, Hs

σ(Rn) =
(
Hs
σ(Rn)

)n
.

Remark 2.2. If s > n/2, then each f ∈ Hs(Rn) is bounded and continuous and
hence

‖f‖L∞(Rn) ≤ C‖f‖Hs(Rn), for s > n/2.

Also, note that Hs is an algebra for s > n/2, i.e., if f, g ∈ Hs(Rn), then fg ∈
Hs(Rn), for s > n/2. Hence, we have

‖fg‖Hs ≤ C‖f‖Hs‖g‖Hs , for s > n/2.

Lemma 2.3. Fix s > n/2 and let f ∈ Hs
σ and g ∈ Hs. Then

‖(f · ∇)g‖Hs−1 ≤ C‖f‖Hs‖g‖Hs .

Proof. We being in Hs
σ, f is divergence free, and hence (f · ∇)g = ∇ · (f ⊗ g). Rest

of the proof is straightforward, since Hs is an algebra for s > n/2. �

Lemma 2.4 (Sobolev inequality). For f ∈ Hs(Rn), we have

‖f‖Lq(Rn) ≤ Cn,s,q‖f‖Hs(Rn)

provided that q lies in the following range

(i) if s < n/2, then 2 ≤ q ≤ 2n
n−2s .

(ii) if s = n/2, then 2 ≤ q <∞.
(iii) if s > n/2, then 2 ≤ q ≤ ∞.

For details see Kesavan [18].

Remark 2.5. We deduce the following result using Lemma 2.4. For n = 2, we use
Hölder’s inequality with exponents 2/ε and 2/(1 − ε), and Sobolev inequality for
0 < ε < s− 1 to obtain

‖fg‖L2 ≤ ‖f‖L2/ε‖g‖L2/1−ε ≤ C‖f‖Ḣ1−ε‖g‖Ḣε ≤ C‖f‖H1‖g‖Hs−1 .

For n = 3, we use Hölder’s inequality with exponents 6 and 3, and Sobolev inequal-
ity to obtain

‖fg‖L2 ≤ ‖f‖L6‖g‖L3 ≤ C‖f‖Ḣ1‖g‖Ḣ1/2 ≤ C‖f‖H1‖g‖H1/2 ≤ C‖f‖H1‖g‖Hs−1 .

We note that for both 2D and 3D we have the same estimate.

Lemma 2.6 (Interpolation in Sobolev spaces). Given s > 0, there exists a constant
C depending on s, so that for all f ∈ Hs(Rn) and 0 < s′ < s,

‖f‖Hs′ ≤ C‖f‖
1−s′/s
L2 ‖f‖s

′/s
Hs .

For details see [2] and for a proof see [22, Theorem 9.6, Remark 9.1].



6 U. MANNA, A. A. PANDA EJDE-2020/91

Lemma 2.7 (Gagliardo-Nirenberg interpolation inequality [24]). Let g ∈ Lq(Rn)
and its derivatives of order m, Dmg ∈ Lr(Rn), 1 ≤ q, r ≤ ∞. For the derivatives
Djg, 0 ≤ j < m, it holds

‖Djg‖Lp ≤ C‖Dmg‖aLr‖g‖1−aLq ,

where
1

p
=
j

n
+
(1

r
− m

n

)
a+

1− a
q

,

for all a in the interval j/m ≤ a < 1. The constant C depends only on n,m, j, q, r, a.

2.3. Commutator estimates. Let f and g are Schwartz class functions. Then
for s ≥ 0 we define

[Js, f ]g = Js(fg)− f(Jsg),

and

[Js, f ]∇g = Js((f · ∇)g)− (f · ∇)Jsg. (2.2)

where [Js, f ] = Jsf − fJs is the commutator, in which f is regarded as a multipli-
cation operator.

Lemma 2.8. For s ≥ 0, and 1 < p <∞, we have a basic estimate

‖[Js, f ]g‖Lp ≤ C
(
‖∇f‖L∞‖Js−1g‖Lp + ‖Jsf‖Lp‖g‖L∞

)
,

where C is a constant depending only on n, p, s.

For a proof of the above lemma see the appendix in [17].

2.4. BMO space and logarithmic Sobolev inequality.

Definition 2.9. The space BMO (Bounded Mean Oscillation) is the Banach space
of all functions f ∈ L1

loc(Rn) for which

‖f‖BMO = sup
Q

( 1

|Q|

∫
Q

|f(x)− fQ| dx
)
<∞,

where the sup ranges over all cubes Q ⊂ Rn, and fQ is the mean of f over Q. For
more details see [12].

The space BMO has two distinct advantageous properties compared to L∞. The
first being the Riesz transforms are bounded in BMO and secondly the singular
integral operators of the Calderon-Zygmund type are also bounded inBMO. Hence,
one can show that ‖∇u‖BMO ≤ C‖∇ × u‖BMO (see [20]).

It is well known that the Sobolev space W s,p is embedded continuously into
L∞ for sp > n. However this embedding is false in the space W k,r when kr = n.
Brezis-Gallouet [5] and Brezis-Wainger [6] provided the following inequality which
relates the function spaces L∞ and W s,p at the critical value and was used to prove
the existence of global solutions to the nonlinear Schrödinger equations.

Lemma 2.10. Let sp > n. Then

‖f‖L∞ ≤ C
(

1 + log
r−1
r (1 + ‖f‖W s,p)

)
,

provided ‖f‖Wk,r ≤ 1 for kr = n.

Similar embedding was investigated by Beale-Kato-Majda [3] for vector functions
to obtain the blow-up criterion of the solutions to the Euler equations.



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 7

Lemma 2.11. Let s > n
p + 1, then we have

‖∇f‖L∞ ≤ C
(

1 + ‖∇ · f‖L∞ + ‖∇ × f‖L∞ (1 + log(e+ ‖f‖W s,p))
)
,

for all f ∈W s,p(Rn).

Kozono and Taniuchi improved the above logarithmic Sobolev inequality in
BMO space, and applied the result to the three-dimensional Euler equations to
prove that BMO-norm of the vorticity controls breakdown of smooth solutions.

Lemma 2.12. Let 1 < p <∞ and let s > n
p , then we have

‖f‖L∞ ≤ C
(

1 + ‖f‖BMO(1 + log+ ‖f‖W s,p)
)
,

for all f ∈W s,p, where log+ a = log a if a ≥ 1 and zero otherwise.

For a proof of the above lemma, see [20, Theorem 1]. Throughout the following
sections, C denotes a generic constant.

3. Energy estimates, local existence and uniqueness for the
magnetic Bénard problem

We consider the following truncated ideal magnetic Bénard problem on the whole
of Rn, for n = 2, 3:

∂uR

∂t
+ SR[(uR · ∇)uR] +∇pR = θRen + SR[(bR · ∇)bR], (3.1)

∂θR

∂t
+ SR

[
(uR · ∇)θR

]
= uR · en, (3.2)

∂bR

∂t
+ SR[(uR · ∇)bR] = SR[(bR · ∇)uR], (3.3)

∇ · uR = 0 = ∇ · bR, (3.4)

uR(0) = SRu0, θ
R(0) = SRθ0,b

R(0) = SRb0. (3.5)

As the truncations are invariant under the flow of the equation, by taking the
truncated initial data we ensure that uR, bR lie in the space

V σR := {g ∈ L2(Rn) : supp(ĝ) ⊂ BR,∇ · g = 0}
and θR lies in the space

VR := {g ∈ L2(Rn) : supp(ĝ) ⊂ BR}.
The divergence free condition for uR can be obtained easily as

∇̂ · uR(ξ) = iξ · 1BR(ξ)û(ξ) = 1BR(ξ)iξ · û(ξ) = 1BR(ξ)∇̂ · u(ξ) = 0.

Similarly we obtain divergence free condition for bR.

Proposition 3.1. Let (uR, bR) ∈ Hs
σ(Rn) ×Hs

σ(Rn), for s > n/2 + 1. Then the
nonlinear operator F (uR,bR) := SR[(uR · ∇)bR] is locally Lipschitz in uR and bR

on the space V σR .

Proof. Let bR ∈ Hs
σ(Rn), for s > n/2 + 1. Then for proving F (·, ·) to be locally

Lipschitz in uR, we use integration by parts, Hölder’s inequality and Lemma 2.4 to
obtain

|
(
F (uR1 ,b

R)− F (uR2 ,b
R),uR1 − uR2

)
L2 |



8 U. MANNA, A. A. PANDA EJDE-2020/91

= |
(
SR[(uR1 − uR2 ) · ∇bR],uR1 − uR2

)
L2 |

= |
((

(uR1 − uR2 ) · ∇
)
bR,SR(uR1 − uR2 )

)
L2 |

= | −
((

(uR1 − uR2 ) · ∇
)
(uR1 − uR2 ),SRbR

)
L2

≤ ‖uR1 − uR2 ‖L2
σ
‖∇(uR1 − uR2 )‖L2

σ
‖SRbR‖L∞

≤ ‖uR1 − uR2 ‖L2
σ
‖uR1 − uR2 ‖H1

σ
‖bR‖L∞

≤ C‖uR1 − uR2 ‖Hsσ‖b
R‖Hsσ‖u

R
1 − uR2 ‖L2

σ
.

For bR ∈ Hs
σ(Rn), this gives

‖F (uR1 ,b
R)− F (uR2 ,b

R)‖L2 ≤ C‖bR‖Hsσ‖u
R
1 − uR2 ‖Hsσ

And hence F (·, ·) is locally Lipschitz in uR. To prove F to be locally Lipschitz in
bR, we use Remark 2.3. For s > n/2 + 1 and uR ∈ Hs

σ(Rn), we have

|
(
F (uR,bR1 )− F (uR,bR2 ),bR1 − bR2

)
L2 |

= |
(
SR(uR · ∇)(bR1 − bR2 ),bR1 − bR2

)
L2 |

= |
(
(uR · ∇)(bR1 − bR2 ),SR(bR1 − bR2 )

)
L2 |

≤ ‖(uR · ∇)(bR1 − bR2 )‖L2
σ
‖SR(bR1 − bR2 )‖L2

σ

≤ C‖uR‖H1
σ
‖∇(bR1 − bR2 )‖Hs−1

σ
‖bR1 − bR2 ‖L2

σ

≤ C‖uR‖Hsσ‖b
R
1 − bR2 ‖Hsσ‖b

R
1 − bR2 ‖L2

σ

Hence for uR ∈ Hs
σ(Rn), we have

‖
(
F (uR,bR1 )− F (uR,bR2

)
‖L2 ≤ C‖uR‖Hsσ‖b

R
1 − bR2 ‖Hsσ

And hence F (·, ·) is locally Lipschitz in bR. �

Similarly one can show that F (bR,uR) is locally Lipschitz in bR and uR on
the space V σR × V σR and F (uR, θR) is locally Lipschitz in uR and θR on the space
V σR × VR.

Hence by Picard’s theorem for infinite dimensional ordinary differential equa-
tions, there exist a solution (uR, θR,bR) in V σR × V R × V σR for some interval [0, T ],
where T depends on R. Moreover, the solution will exist as long as ‖uR‖Hsσ , ‖θR‖Hs
and ‖bR‖Hsσ remain finite.

3.1. Energy estimates. In this section we obtain L2 and Hs, s > n/2 + 1, energy
estimates for uR, θR and bR. In the course of proving the ‖uR‖Hsσ , ‖θR‖Hs and

‖bR‖Hsσ are uniformly bounded, we will pick up a blow-up time T ∗.

Proposition 3.2 (L2-Energy Estimate). Given (u0, θ0,b0) ∈ L2
σ(Rn)× L2(Rn)×

L2
σ(Rn) with s > n/2 + 1, then for any t ∈ [0, T ], where 0 < T <∞, we have

sup
t∈[0,T ]

(
‖uR(t)‖2L2

σ
+ ‖θR(t)‖2L2 + ‖bR(t)‖2L2

σ

)
< C

where C depends only on ‖u0‖L2
σ
, ‖θ0‖L2 , ‖b0‖L2

σ
and T .

Proof. Consider the equations (3.1)-(3.3). Taking L2-inner product of (3.1), (3.2)
and (3.3) with uR, θR and bR respectively, and adding we obtain

1

2

d

dt

(
‖uR‖2L2

σ
+ ‖θR‖2L2 + ‖bR‖2L2

σ

)
=
(
θRen,u

R
)
L2 +

(
(uR · en), θR

)
L2 . (3.6)



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 9

In the above calculation, we have used that
(
(uR ·∇)uR,uR

)
L2 ,
(
(uR ·∇)θR, θR

)
L2

and
(
(uR ·∇)bR,bR

)
L2 vanish and

(
(bR ·∇)bR,uR

)
L2 = −

(
(bR ·∇)uR,bR

)
L2 . It

is easy to see that

|(θRen,uR)L2 | ≤ ‖θRen‖L2‖uR‖L2 ≤ ‖θR‖L2‖uR‖L2

≤ 1

2

(
‖uR‖2L2 + ‖θR‖2L2 + ‖bR‖2L2

)
,

and

|
(
(uR · en), θR

)
L2 | ≤ ‖uR‖L2‖θR‖L2 ≤ 1

2

(
‖uR‖2L2 + ‖θR‖2L2 + ‖bR‖2L2

)
.

Using the above estimates in (3.6) and letting Y (t) = ‖uR(t)‖2L2
σ

+ ‖θR(t)‖2L2 +

‖bR(t)‖2L2
σ
, we obtain

dY (t)

dt
≤ 2Y (t).

Straightforward integration and the fact that ‖uR(0)‖L2
σ
≤ ‖u0‖L2

σ
, ‖θR(0)‖L2 ≤

‖θ0‖L2 and ‖bR(0)‖L2
σ
≤ ‖b0‖L2

σ
yield

sup
t∈[0,T ]

Y (t) ≤ C(‖u0‖L2
σ
, ‖θ0‖L2 , ‖b0‖L2

σ
, T )

So we have the desired result. �

Proposition 3.3. Let (u0, θ0,b0) ∈ Hs
σ(Rn)×Hs(Rn)×Hs

σ(Rn) with s > n/2+1.
Then there exists a time T ∗ = T ∗(s, ‖u0‖Hsσ , ‖θ0‖Hs , ‖b0‖Hsσ ) > 0 such that

sup
t∈[0,T∗]

‖uR(t)‖Hsσ , sup
t∈[0,T∗]

‖θR(t)‖Hs , sup
t∈[0,T∗]

‖bR(t)‖Hsσ

are bounded uniformly in R.

Proof. Let Js denote the fractional derivative operator as defined earlier. Now for
s > n/2 + 1, apply Js to all the equations (3.1)-(3.3):

∂(JsuR)

∂t
+ SRJs[(uR · ∇)uR] +∇JspR = Js(θRen) + SRJs[(bR · ∇)bR], (3.7)

∂(JsθR)

∂t
+ SRJs[(uR · ∇)θR] = Js(uR · en), (3.8)

∂(JsbR)

∂t
+ SRJs[(uR · ∇)bR] = SRJs[(bR · ∇)uR] (3.9)

Taking the L2-inner product of (3.7), (3.8) and (3.9) with JsuR, JsθR and JsbR

respectively, we obtain(∂(JsuR)

∂t
, JsuR

)
L2

+
(
SRJs[(uR · ∇)uR], JsuR

)
L2 +

(
∇JspR, JsuR

)
L2

=
(
Js(θRen), JsuR

)
L2 +

(
SRJs[(bR · ∇)bR], JsuR

)
L2 ,

(3.10)

(∂(JsθR)

∂t
, JsθR

)
L2 +

(
SRJs[(uR · ∇)θR], JsθR

)
L2

=
(
Js(uR · en), JsθR

)
L2 ,

(3.11)

(∂(JsbR)

∂t
, JsbR

)
L2 +

(
SRJs[(uR · ∇)bR], JsbR

)
L2

=
(
SRJs[(bR · ∇)uR], JsbR

)
L2 .

(3.12)



10 U. MANNA, A. A. PANDA EJDE-2020/91

We estimate each term of (3.10), (3.11) and (3.12) separately. (1)(∂(JsuR)

∂t
, JsuR

)
L2 =

∫
BR

∂JsuR

∂t
JsuR dx =

1

2

∫
BR

∂|JsuR|2

∂t

=
1

2

d

dt
‖JsuR‖2L2

σ
=

1

2

d

dt
‖uR‖2Hsσ .

(2) Applying weak Parseval’s identity and using the fact that SRuR = uR, since
uR ∈ V σR we obtain(

SRJs[(uR · ∇)uR], JsuR
)
L2 =

(
Js[(uR · ∇)uR], JsuR

)
L2 .

(3) Using the definition of commutator and incompressibility of uR, we obtain(
[Js,uR]∇uR, JsuR

)
L2 =

(
Js[(uR · ∇)uR]−

(
uR · ∇

)
JsuR, JsuR

)
L2

=
(
Js[(uR · ∇)uR], JsuR

)
L2 .

Now using Lemma 2.8 and Hölder’s inequality we obtain∣∣([Js,uR]∇uR, JsuR
)
L2

∣∣
≤ ‖[Js,uR]∇uR‖L2‖JsuR‖L2

≤ C
(
‖∇uR‖L∞‖Js−1∇uR‖L2

σ
+ ‖JsuR‖L2

σ
‖∇uR‖L∞

)
‖uR‖Hsσ

≤ C
(
‖∇uR‖Hs−1

σ
‖∇uR‖Hs−1

σ
+ ‖uR‖Hsσ‖∇uR‖Hs−1

σ

)
‖uR‖Hsσ

≤ C
(
‖uR‖2Hsσ + ‖uR‖2Hsσ

)
‖uR‖Hsσ

≤ C
(
‖uR‖2Hsσ + ‖θR‖2Hs + ‖bR‖2Hsσ

)
‖uR‖Hsσ .

(3) Using integration by parts we infer(
∇JspR, JsuR

)
L2 =

(
JspR, Js∇ · uR

)
L2 = 0.

(4) Using the Hölder’s inequality and then Young’s inequality we infer∣∣(Js(θRen), JsuR
)
L2

∣∣ ≤ ‖Js(θRen)‖L2‖JsuR‖L2
σ

≤ ‖θRen‖Hs‖uR‖Hsσ
≤ C

(
‖uR‖2Hsσ + ‖θR‖2Hs + ‖bR‖2Hsσ

)
.

(5) Using the property of the bilinear operator, we have(
SRJs[(bR · ∇)bR], JsuR

)
L2 =

(
Js[(bR · ∇)bR], JsSRuR

)
L2

=
(
Js[(bR · ∇)bR], JsuR

)
L2

= −
(
Js[(bR · ∇)uR], JsbR

)
L2 .

(6) Similarly referring to (1) we infer(
∂(JsθR)

∂t
, JsθR

)
L2

=
1

2

d

dt
‖θR‖2Hs .

(7) Similar calculations as in (2) and using Lemma 2.8 we obtain(
SRJs[(uR · ∇)θR], JsθR

)
L2

and ∣∣([Js,uR]∇θR, JsθR
)
L2

∣∣ ≤ ‖[Js,uR]∇θR‖L2‖JsθR‖L2



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 11

≤ C
(
‖∇uR‖L∞‖Js−1∇θR‖L2 + ‖JsuR‖L2

σ
‖∇θR‖L∞

)
‖θR‖Hs

≤ C
(
‖∇uR‖Hs−1

σ
‖∇θR‖Hs−1 + ‖uR‖Hsσ‖∇θ

R‖Hs−1

)
‖θR‖Hs

≤ C
(
‖uR‖2Hsσ + ‖θR‖2Hs + ‖bR‖2Hsσ

)
‖θR‖Hs .

(8) Using the Hölder’s inequality and the Young’s inequality we obtain∣∣(Js(uR · en), JsθR
)
L2

∣∣ ≤ ‖Js(uR · en)‖L2
σ
‖JsθR‖L2

≤ ‖uR‖Hsσ‖θ
R‖Hs

≤ C
(
‖uR‖2Hsσ + ‖θR‖2Hs + ‖bR‖2Hsσ

)
.

(9) Similarly, (∂(JsbR)

∂t
, JsbR

)
L2

=
1

2

d

dt
‖bR‖2Hsσ .

(10) Following similar steps as in (7), replacing θR by bR we obtain∣∣(SRJs[(uR · ∇)bR], JsbR
)
L2

∣∣ ≤ C (‖uR‖2Hsσ + ‖θR‖2Hs + ‖bR‖2Hsσ
)
‖bR‖Hsσ .

(11) Weak Parseval’s identity gives(
SRJs[(bR · ∇)uR], JsbR

)
L2 =

(
Js[(bR · ∇)uR], JsbR

)
L2 .

Now adding (3.10), (3.11) and (3.12) (using the estimates obtained through (1)
to (11)) we have

1

2

d

dt

(
‖uR‖2Hsσ + ‖θR‖2Hs + ‖bR‖2Hsσ

)
≤ C

(
‖uR‖2Hsσ + ‖θR‖2Hs + ‖bR‖2Hsσ

) (
‖uR‖Hsσ + ‖θR‖Hs + ‖bR‖Hsσ

)
≤ C

2

(
‖uR‖2Hsσ + ‖θR‖2Hs + ‖bR‖2Hsσ

)2

+
3C

2

(
‖uR‖2Hsσ + ‖θR‖2Hs + ‖bR‖2Hsσ

)
.

Now letting X(t) = ‖uR(t)‖2Hsσ + ‖θR(t)‖2Hs + ‖bR(t)‖2Hsσ we obtain

d

dt
X(t) ≤ 3CX(t) +X(t)2 ≤ 3

2
C2 +

(3

2
+ C

)
X(t)2.

So for all 0 ≤ t ≤ T ,

X(t) ≤ X0 +
3

2
C2 +

(3

2
+ C

)∫ t

0

X(s)2 ds.

Now applying Bihari’s inequality [10], we have

X(t) ≤
3
2C

2 +X0

1− ( 3
2C

2 +X0)( 3
2 + C)T

.

Note that ‖uR(0)‖Hsσ ≤ ‖u0‖Hsσ , ‖θR(0)‖Hs ≤ ‖θ0‖Hs and ‖bR(0)‖Hsσ ≤ ‖b0‖Hsσ .
So provided we choose

T ∗ <
1

( 3
2C

2 +X0)( 3
2 + C)

,

then the norms ‖uR‖Hsσ , ‖θ
R‖Hs and ‖bR‖Hsσ remain bounded on [0, T ∗] indepen-

dent of R. �



12 U. MANNA, A. A. PANDA EJDE-2020/91

3.2. Local existence and uniqueness. In this subsection, we prove existence
and uniqueness of the local-in time strong solution of the magnetic Bénard problem
(1.5)-(1.8).

At first, we show that the family (uR, θR,bR) is Cauchy in a suitable space.

Proposition 3.4. The family (uR, θR,bR) of solutions of the magnetic Bénard
problem (3.1)-(3.5) are Cauchy in the space

L∞
(
[0, T ∗];L2

σ(Rn)
)
× L∞

(
[0, T ∗];L2(Rn)

)
× L∞

(
[0, T ∗];L2

σ(Rn)
)
,

as R→∞.

Proof. We consider equations (3.1), (3.2) and (3.3). Then taking the difference
between the equations for R and R′ with R′ > R we obtain

∂

∂t

(
uR − uR

′)
+∇

(
pR − pR

′)
= θRen − θR

′
en − SR[(uR · ∇)uR] + SR′ [(uR

′
· ∇)uR

′
]

+ SR[(bR · ∇)bR]− SR′ [(bR
′
· ∇)bR

′
],

(3.13)

∂

∂t

(
θR − θR

′)
+ SR[(uR · ∇)θR]− SR′ [(uR

′
· ∇)θR

′
] = uR · en − uR

′
· en, (3.14)

∂

∂t

(
bR − bR

′)
+ SR[(uR · ∇)bR]− SR′ [(uR

′
· ∇)bR

′
]

= SR[(bR · ∇)uR]− SR′ [(bR
′
· ∇)uR

′
].

(3.15)

Taking the inner product of (3.13), (3.14) and (3.15) with uR − uR
′
, θR − θR′ and

bR − bR
′

respectively, and then adding we obtain

1

2

d

dt

(
‖uR − uR

′
‖2L2

σ
+ ‖θR − θR

′
‖2L2 + ‖bR − bR

′
‖2L2

σ

)
=
(
θRen − θR

′
en,u

R − uR
′)
−
(
uR · en − uR

′
· en, θR − θR

′)
−
(
SR[(uR · ∇)uR]− SR′ [(uR

′
· ∇)uR

′
],uR − uR

′)︸ ︷︷ ︸
I1

+
(
SR[(bR · ∇)bR]− SR′ [(bR

′
· ∇)bR

′
],uR − uR

′)︸ ︷︷ ︸
I2

−
(
SR[(uR · ∇)θR]− SR′ [(uR

′
· ∇)θR

′
], θR − θR

′)︸ ︷︷ ︸
I3

+
(
SR[(bR · ∇)uR]− SR′ [(bR

′
· ∇)uR

′
],bR − bR

′)︸ ︷︷ ︸
I4

−
(
SR[(uR · ∇)bR]− SR′ [(uR

′
· ∇)bR

′
],bR − bR

′)︸ ︷︷ ︸
I5

(3.16)

We will calculate each term on the right hand side of (3.16) separately. First observe
that ∣∣∣(θRen − θR′en,uR − uR

′)∣∣∣ ≤ ‖θRen − θR′en‖L2‖uR − uR
′
‖L2

σ

≤ ‖θR − θR
′
‖L2‖uR − uR

′
‖L2

σ
,

(3.17)



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 13

and∣∣∣(uR · en − uR
′
· en, θR − θR

′)∣∣∣ ≤ ‖uR · en − uR
′
· en‖L2

σ
‖θR − θR

′
‖L2

≤ ‖uR − uR
′
‖L2

σ
‖θR − θR

′
‖L2 .

(3.18)

We split I1 =
(
SR[(uR · ∇)uR]− SR′ [(uR

′ · ∇)uR
′
],uR − uR

′)
in to three parts:

I1∗ =
(

(SR − SR′)[(uR · ∇)uR],uR − uR
′
)

+
(
SR′ [((uR − uR

′
) · ∇)uR],uR − uR

′
)

+
(
SR′ [(uR

′
· ∇)(uR − uR

′
)],uR − uR

′
)
.

(3.19)

For R′ > R, using the property of Fourier truncation operator provided 0 < ε <
s− 1, the first term of (3.19) becomes∣∣∣((SR − SR′)[(uR · ∇)uR],uR − uR

′)∣∣∣
≤ ‖(SR − SR′)[(uR · ∇)uR]‖L2

σ
‖uR − uR

′
‖L2

σ

≤ C

Rε
‖(uR · ∇)uR‖Hεσ‖u

R − uR
′
‖L2

σ

=
C

Rε
‖∇ · (uR ⊗ uR)‖Hεσ‖u

R − uR
′
‖L2

σ

≤ C

Rε
‖uR ⊗ uR‖Hsσ‖u

R − uR
′
‖L2

σ

≤ C

Rε
‖uR‖2Hsσ‖u

R − uR
′
‖L2

σ
.

(3.20)

Now for s > n/2 + 1, the second term of (3.19),∣∣∣(SR′ [((uR − uR
′
) · ∇)uR],uR − uR

′)∣∣∣
≤ ‖((uR − uR

′
) · ∇)uR‖L2

σ
‖uR − uR

′
‖L2

σ

≤ ‖uR − uR
′
‖L2

σ
‖∇uR‖L∞‖uR − uR

′
‖L2

σ

≤ ‖∇uR‖Hs−1
σ
‖uR − uR

′
‖2L2

σ

≤ ‖uR‖Hsσ‖u
R − uR

′
‖2L2

σ
.

(3.21)

Using weak Parseval’s identity, integration by parts and divergence free condition
on uR and uR

′
to the third term of (3.19) we obtain(

SR′ [(uR
′
· ∇)(uR − uR

′
)],uR − uR

′)
= 0.

Therefore, using (3.20), (3.21) in (3.19), we obtain

|I1| ≤
C

Rε
‖uR‖2Hsσ‖u

R − uR
′
‖L2

σ
+ ‖uR‖Hsσ‖u

R − uR
′
‖2L2

σ
. (3.22)

Similarly we split I3 and I5 to obtain

|I3| ≤
C

Rε ‖uR‖Hsσ‖θ
R‖Hs‖θR−θR

′
‖L2 +‖uR−uR

′
‖L2

σ
‖θR‖Hs‖θR−θR

′
‖L2 . (3.23)



14 U. MANNA, A. A. PANDA EJDE-2020/91

and

|I5| ≤
C

Rε ‖uR‖Hsσ‖b
R‖Hsσ‖b

R − bR
′
‖L2

σ

+ ‖uR − uR
′
‖L2

σ
‖bR‖Hsσ‖b

R − bR
′
‖L2

σ
.

(3.24)

We spilt I2 and I4 in the similar manner. However, note that one term of I2 will
cancel with one term of I4 because(

SR′ [(bR
′
· ∇)(bR − bR

′
)],uR − uR

′)
= −

(
(bR

′
· ∇)(uR − uR

′
),bR − bR

′)
.

Therefore,

I2 ≤
C

Rε ‖bR‖2Hsσ‖u
R − uR

′
‖L2

σ
+ ‖bR‖Hsσ‖b

R − bR
′
‖L2

σ
‖uR − uR

′
‖L2

σ
, (3.25)

I4 ≤
C

Rε ‖bR‖Hsσ‖u
R‖Hsσ‖b

R − bR
′
‖L2

σ
+ ‖uR‖Hsσ‖b

R − bR
′
‖2L2

σ
. (3.26)

Using the estimates obtained in (3.17), (3.18), (3.22)-(3.26) in (3.16), we have

1

2

d

dt

(
‖uR − uR

′
‖2L2

σ
+ ‖θR − θR

′
‖2L2 + ‖bR − bR

′
‖2L2

σ

)
≤ C

Rε ‖uR‖2Hsσ‖u
R − uR

′
‖L2

σ
+ ‖uR‖Hsσ‖u

R − uR
′
‖2L2

σ

+
C

Rε ‖uR‖Hsσ‖θ
R‖Hs‖θR − θR

′
‖L2 + ‖bR‖Hsσ‖u

R − uR
′
‖L2

σ
‖bR − bR

′
‖L2

σ

+ ‖θR‖Hs‖uR − uR
′
‖L2

σ
‖θR − θR

′
‖L2 +

C

Rε ‖bR‖Hsσ‖u
R‖Hsσ‖b

R − bR
′
‖L2

σ

+
C

Rε ‖bR‖2Hsσ‖u
R − uR

′
‖L2

σ
+ ‖bR‖Hsσ‖u

R − uR
′
‖L2

σ
‖bR − bR

′
‖L2

σ

+
C

Rε ‖bR‖Hsσ‖u
R‖Hsσ‖b

R − bR
′
‖L2

σ
+ ‖uR‖Hsσ‖b

R − bR
′
‖2L2

σ

+ 2‖θR − θR
′
‖L2‖uR − uR

′
‖L2

σ
.

Applying Proposition 3.3 and Young’s inequality and rearranging the terms we
obtain

d

dt

(
‖uR − uR

′
‖2L2

σ
+ ‖θR − θR

′
‖2L2 + ‖bR − bR

′
‖2L2

σ

)
≤ C1

Rε ‖uR − uR
′
‖L2

σ
+ C2‖uR − uR

′
‖2L2

σ
+ 2
(
‖θR − θR

′
‖2L2 + ‖uR − uR

′
‖2L2

σ

)
+
C3

Rε ‖θR − θR
′
‖L2 + C4

(
‖θR − θR

′
‖2L2 + ‖uR − uR

′
‖2L2

σ

)
+
C5

Rε ‖bR − bR
′
‖L2

σ
+ C6

(
‖uR − uR

′
‖2L2

σ
+ ‖bR − bR

′
‖2L2

σ

)
+
C7

Rε ‖uR − uR
′
‖L2

σ
+ C8

(
‖uR − uR

′
‖2L2

σ
+ ‖bR − bR

′
‖2L2

σ

)
+ C10‖bR − bR

′
‖2L2

σ

≤ M

Rε

(
‖uR − uR

′
‖L2

σ
+ ‖θR − θR

′
‖L2 + ‖bR − bR

′
‖L2

σ

)
+M

(
‖uR − uR

′
‖2L2

σ
+ ‖θR − θR

′
‖2L2 + ‖bR − bR

′
‖2L2

σ

)
.

(3.27)



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 15

Let Y (t) = ‖uR − uR
′‖L2

σ
+ ‖θR − θR′‖L2 + ‖bR − bR

′‖L2
σ
, then we infer

‖uR − uR
′
‖2L2

σ
+ ‖θR − θR

′
‖2L2 + ‖bR − bR

′
‖2L2

σ

≤ Y (t)2 ≤ 3
(
‖uR − uR

′
‖2L2

σ
+ ‖θR − θR

′
‖2L2 + ‖bR − bR

′
‖2L2

σ

)
.

Thus, we obtain

d

dt

(
Y (t)2

)
≤ 3

d

dt

(
‖uR − uR

′
‖2L2

σ
+ ‖θR − θR

′
‖2L2 + ‖bR − bR

′
‖2L2

σ

)
,

which implies

2Y
dY

dt
≤ 3

d

dt

(
‖uR − uR

′
‖2L2

σ
+ ‖θR − θR

′
‖2L2 + ‖bR − bR

′
‖2L2

σ

)
From (3.27), we obtain

dY

dt
≤MY +

M

Rε
.

Finally, applying the Gronwall’s lemma we infer

sup
t∈[0,T∗]

Y (t) ≤ C(M,T ∗)

Rε
→ 0, (3.28)

as R→∞ (as R′ > R, R′ →∞ as well), concluding that (uR, θR,bR) are Cauchy in
L∞

(
[0, T ∗];L2

σ(Rn)
)
×L∞

(
[0, T ∗];L2(Rn)

)
×L∞

(
[0, T ∗];L2

σ(Rn)
)

as R→∞. �

Now we show the following convergence result.

Proposition 3.5. For any s′ > n/2 + 1 with s′ < s, (uR, θR,bR) → (u, θ,b) in

L∞
(
[0, T ∗];Hs′

σ (Rn)
)
× L∞

(
[0, T ∗];Hs′(Rn)

)
× L∞

(
[0, T ∗];Hs′

σ (Rn)
)
.

Proof. From Proposition 3.4 we conclude that (uR, θR,bR)→ (u, θ,b) strongly in
L∞

(
[0, T ∗];L2

σ(Rn)
)
× L∞

(
[0, T ∗];L2(Rn)

)
× L∞

(
[0, T ∗];L2

σ(Rn)
)
.

Using Lemma 2.6 for s′ < s and s′ > n/2 + 1,

sup
t∈[0,T∗]

‖bR − b‖Hs′σ ≤ C sup
t∈[0,T∗]

(
‖bR − b‖1−s

′/s
L2
σ
‖bR − b‖s

′/s
Hsσ

)
≤ C

(
sup

t∈[0,T∗]

‖bR − b‖L2
σ

) 1−s′
s
(

sup
t∈[0,T∗]

‖bR − b‖Hsσ
)s′/s

.

From Propositions 3.3 and 3.4 we obtain

sup
t∈[0,T∗]

‖bR − b‖Hs′σ ≤M
(

sup
t∈[0,T∗]

‖bR − b‖L2
σ

)1−s′/s
→ 0 as R→∞.

So we obtain

bR → b in L∞
(

[0, T ∗];Hs′

σ (Rn)
)
, (3.29)

Similarly we can show that

θR → θ in L∞
(

[0, T ∗];Hs′(Rn)
)
, uR → u in L∞

(
[0, T ∗];Hs′

σ (Rn)
)
,

which gives the desired result. �

Now we use the above result to show the term-wise convergence of the non-linear
terms.



16 U. MANNA, A. A. PANDA EJDE-2020/91

Proposition 3.6. For any s′ > n/2 + 1, we have

SR[(uR · ∇)uR]→ (u · ∇)u,

SR[(bR · ∇)bR]→ (b · ∇)b,

SR[(uR · ∇)bR]→ (u · ∇)b,

SR[(bR · ∇)uR]→ (b · ∇)u, strongly in L∞
(

[0, T ∗];Hs′−1
σ (Rn)

)
,

SR[(uR · ∇)θR]→ (u · ∇)θ, strongly in L∞
(

[0, T ∗];Hs′−1(Rn)
)
,

as R→∞.

Proof. We prove the result for the non-linear terms SR[(bR · ∇)bR], when s′ >
n/2 + 1. One can follow the similar steps to show the convergence of the other
non-linear terms in the respective spaces as claimed.

Using the properties of Fourier truncation operator and Remark 2.2 we have

sup
t∈[0,T∗]

‖SR[(bR · ∇)bR]− (b · ∇)b‖
Hs
′−1
σ

≤ sup
t∈[0,T∗]

(
‖SR[(bR − b) · ∇)bR]‖

Hs
′−1
σ

+ ‖SR[(b · ∇)(bR − b)]‖
Hs
′−1
σ

)
≤ sup
t∈[0,T∗]

(
C‖[(bR − b) · ∇)bR]‖

Hs
′−1
σ

+ C‖[(b · ∇)(bR − b)]‖
Hs
′−1
σ

)
≤ sup
t∈[0,T∗]

(
C‖bR − b‖Hs′σ ‖b

R‖Hs′σ + C‖b‖Hs′σ ‖b
R − b‖Hs′σ

)
Clearly from (3.29), Propositions 3.3 and 3.4, the right-hand side tends to 0 as
R→∞. �

Next we show the convergence of time derivatives.

Proposition 3.7. For any s′ > n/2 + 1,

∂uR

∂t
→ ∂u

∂t
and

∂bR

∂t
→ ∂b

∂t
strongly in the space L∞

(
[0, T ∗];Hs′−1

σ (Rn)
)

and
∂θR

∂t
→ ∂θ

∂t
strongly in L∞

(
[0, T ∗];Hs′−1(Rn)

)
as R→∞.

Proof. Takingthe Hs′−1-norm on both sides of (3.1)-(3.3) and using the properties
of the Fourier truncation operator, and Remarks 2.2 and 2.3, we obtain for s′ >
n/2 + 1,

‖∂uR

∂t
‖
Hs
′−1
σ
≤ ‖θRen‖Hs′−1 + ‖SR[(bR · ∇)bR]‖

Hs
′−1
σ

+ ‖SR[(uR · ∇)uR]‖
Hs
′−1
σ

≤ C
(
‖θR‖Hs′ + ‖bR‖2

Hs′σ
+ ‖uR‖2

Hs′σ

)
,

‖∂θ
R

∂t
‖
Hs
′−1
σ
≤ ‖uR · en‖Hs′−1

σ
+ ‖SR[(uR · ∇)θR]‖

Hs
′−1
σ

≤ C
(
‖uR‖Hs′σ + ‖uR‖Hs′σ ‖θ

R‖Hs′
)



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 17

and

‖∂bR

∂t
‖
Hs
′−1
σ
≤ ‖SR[(bR · ∇)uR]‖

Hs
′−1
σ

+ ‖SR[(uR · ∇)bR]‖
Hs
′−1
σ

≤ C‖bR‖Hs′σ ‖u
R‖Hs′σ

After adding these inequalities

‖∂uR

∂t
‖
Hs
′−1
σ

+ ‖∂θ
R

∂t
‖
Hs
′−1
σ

+ ‖∂bR

∂t
‖
Hs
′−1
σ

≤ C(‖uR‖Hs′σ + ‖θR‖Hs′ + ‖uR‖2
Hs′σ

+ ‖bR‖2
Hs′σ

+ ‖uR‖Hs′σ ‖θ
R‖Hs′ + ‖bR‖Hs′σ ‖u

R‖Hs′σ )

(3.30)

Now taking supremum in both side over t ∈ [0, T ∗], then using Proposition 3.3 and
dropping the first two terms of left hand side we obtain

sup
t∈[0,T∗]

‖∂bR

∂t
‖
Hs
′−1
σ
≤ C(T ∗) <∞.

Using the Banach-Alaoglu Theorem (see Robinson [27], Yosida [32]) we can extract
a subsequence Rm → +∞ such that

∂bRm

∂t

∗
⇀

∂b

∂t
in L∞

(
[0, T ∗];Hs′−1

σ (Rn)
)
. (3.31)

Similar argument works for ∂uR

∂t and ∂θR

∂t as well. Note that ‖uR‖Hsσ‖θ
R‖Hs →

‖u‖Hsσ‖θ‖Hs and ‖bR‖Hs′σ ‖u
R‖Hs′σ → ‖b‖Hs′σ ‖u‖Hs′σ because of the strong conver-

gences of (uR, θR,bR) to (u, θ,b) in L∞
(
[0, T ∗];Hs′

σ (Rn)
)
×L∞

(
[0, T ∗];Hs′(Rn)

)
×

L∞
(
[0, T ∗];Hs′

σ (Rn)
)
. Hence all the terms on the right-hand side of (3.30) converge

strongly (from Proposition 3.4), we observe that the convergence of the time deriva-
tives are strong. �

Finally, we are ready to prove the solution lies in the desired space.

Proposition 3.8. For s > n/2 + 1, (u, θ,b) lie in the space

L∞ ([0, T ∗];Hs
σ(Rn))× L∞ ([0, T ∗];Hs(Rn))× L∞ ([0, T ∗];Hs

σ(Rn)) .

Proof. By the Banach-Alaoglu Theorem, the uniform bounds in Proposition 3.3
guarantee the existence of a subsequence such that

uRm
∗
⇀ u in L∞ ([0, T ∗];Hs

σ(Rn)) ; (3.32)

θRm
∗
⇀ θ in L∞ ([0, T ∗];Hs(Rn)) ; (3.33)

bRm
∗
⇀ b in L∞ ([0, T ∗];Hs

σ(Rn)) ; (3.34)

which guarantees that the limit satisfies

u ∈ L∞ ([0, T ∗];Hs
σ(Rn)) , θ ∈ L∞ ([0, T ∗];Hs(Rn)) , (3.35)

b ∈ L∞ ([0, T ∗];Hs
σ(Rn)) , (3.36)

which is the desired result. �

Now we prove the uniqueness of solutions in the suitable space.



18 U. MANNA, A. A. PANDA EJDE-2020/91

Proposition 3.9. Let (u0, θ0,b0) ∈ Hs
σ(Rn)×Hs(Rn)×Hs

σ(Rn) for s > n/2 + 1.
Let the solutions (u, θ,b) of the ideal magnetic Bénard problem (1.5)-(1.8) have the
regularity

u ∈ L∞ ([0, T ∗];Hs
σ(Rn)) , θ ∈ L∞ ([0, T ∗];Hs(Rn)) ,b ∈ L∞ ([0, T ∗];Hs

σ(Rn)) .

Then the solution (u, θ,b) is unique in [0, T ∗].

Proof. The proof of the uniqueness is very similar to the proof of Proposition 3.4.
Let (uR, θR,bR) and (uR

′
, θR

′
,bR

′
) be two solutions of the truncated ideal magnetic

Bénard problem (3.1)-(3.3) for R′ > R. Then from (3.28), we have

sup
t∈[0,T∗]

(
‖uR − uR

′
‖L2

σ
+ ‖θR − θR

′
‖L2 + ‖bR − bR

′
‖L2

σ

)
≤ C

Rε
.

Now letting R→ R′ then letting R→∞ we observe that

uR → uR
′
, θR → θR

′
and bR → bR

′
.

Thus we have the uniqueness of the limits (u, θ,b). �

We finally prove that the solutions (u, θ,b) are continuous in time.

Theorem 3.10. Let s > n
2 + 1, u0 ∈ Hs

σ(Rn), θ0 ∈ Hs(Rn) and b0 ∈ Hs
σ(Rn).

Then there exists a unique strong solution

(u, θ,b) ∈ C([0, T ∗];Hs
σ(Rn))× C([0, T ∗];Hs(Rn))× C([0, T ∗];Hs

σ(Rn))

to the system (1.5)-(1.8).

Proof. We shall prove u ∈ C ([0, T ∗];Hs
σ(Rn)). Proofs for θ and b will follow in the

similar manner. Let us first recall that for s ∈ R, 1 ≤ p, q <∞, the inhomogeneous
Besov space Bsp,q is defined as the space of all tempered distributions f ∈ S′(Rn)
such that

Bsp,q = {f ∈ S′(Rn) : ‖f‖Bsp,q <∞},
where

‖f‖Bsp,q =
( ∑
j≥−1

2jqs‖∆jf‖qLp
) 1
q

,

where ∆j is the inhomogeneous Littlewood-Paley operator. We note that ‖f‖Bs2,2 ≈
‖f‖Hs . For details see Chapter 3 of [21].

We consider t1, t2 ∈ [0, T ∗] such that 0 ≤ t1 < t2 ≤ T ∗. Then

‖u(t2)−u(t1)‖Hsσ ≈ ‖u(t2)−u(t1)‖Bs2,2 =
{∑
j∈Z

(
2js‖∆ju(t2)−∆ju(t1)‖L2

σ

)2 }1/2

.

Let ε > 0 be arbitrarily small. As u ∈ L∞ ([0, T ∗];Hs(Rn)), there exists an integer
N > 0 such that {∑

j≥N

(
2js‖∆ju(t2)−∆ju(t1)‖L2

σ

)2}1/2

<
ε

2
. (3.37)

But we have {∑
j∈Z

(
2js‖∆ju(t2)−∆ju(t1)‖L2

σ

)2 }1/2

=
{(∑

j<N

+
∑
j≥N

) (
2js‖∆ju(t2)−∆ju(t1)‖L2

σ

)2 }1/2

.



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 19

Now for 0 ≤ t1 < t2 ≤ T ∗ we have

∆ju(t2)−∆ju(t1) =

∫ t2

t1

∂

∂τ
∆ju(τ) dτ

=

∫ t2

t1

∆jP[(b · ∇)b + θen − (u · ∇)u](τ) dτ.

So we obtain∑
j<N

22js
∥∥∆ju(t2)−∆ju(t1)

∥∥2

L2
σ

=
∑
j<N

22js
∥∥∫ t2

t1

∆jP[(b · ∇)b + θen − (u · ∇)u](τ) dτ
∥∥2

L2

≤
∑
j<N

22js
(∫ t2

t1

[‖∆j(b · ∇b)‖L2
σ

+ ‖∆jθ‖L2 + ‖∆j(u · ∇u)‖L2
σ
] dτ
)2

(3.38)

=
∑
j<N

22j
(∫ t2

t1

2j(s−1)
[
‖∆j(b · ∇b)‖L2

σ
+ ‖∆jθ‖L2 + ‖∆j(u · ∇u)‖L2

σ

]
dτ
)2

≤
∑
j<N

22j

∫ t2

t1

(
‖(b · ∇)b‖2

Hs−1
σ

+ ‖θ‖2Hs−1 + ‖(u · ∇)u‖2
Hs−1
σ

)
dτ .

Clearly the individual terms of right-hand side of (3.38) are far less than their
L∞([0, T ∗];Hs−1)-norm.

As (u, θ,b) ∈ L∞ ([0, T ∗];Hs
σ) × L∞ ([0, T ∗];Hs) × L∞ ([0, T ∗];Hs

σ) and from
Remark 2.2 and Remark 2.3 we obtain

‖(u · ∇)u‖2
L∞([0,T∗];Hs−1

σ ) =
(

sup
t∈[0,T∗]

‖(u · ∇)u‖Hs−1
σ

)2

≤
(

sup
t∈[0,T∗]

‖u‖Hsσ · sup
t∈[0,T∗]

‖u‖Hsσ
)2

< C1 <∞.

Similarly, ‖(u · ∇)u‖2
L∞([0,T∗];Hs−1

σ )
< C2 and from Proposition 3.3, we have

‖θ‖2
L∞([0,T∗];Hs−1

σ )
< C3. Choosing M = C ·max{C1, C2, C3}, for |t2−t1| < ε

M22N+1 ,

from (3.38) we obtain∑
j<N

22js‖∆ju(t2)−∆ju(t1)‖2L2
σ
≤M

∑
j<N

22j |t2− t1| ≤M22N |t2− t1| <
ε

2
. (3.39)

Finally combining (3.37) and (3.39), we conclude that u ∈ C ([0, T ∗];Hs
σ(Rn)). �

4. Blow-up criterion

In this section, we will establish the Blow-up criterion of the local-in-time solution
obtained in the previous section. We show that the BMO norms of the vorticity and
electrical current inhibit the breakdown of smooth solutions, relaxing the condition
on the gradient of temperature, under suitable assumption on the regularity of the
initial data.

Theorem 4.1. Let (u0, θ0,b0) ∈ Hs
σ(Rn)×Hs(Rn)×Hs

σ(Rn), s > n
2 +1, n = 2, 3.

Let (u, θ,b) ∈ C ([0, T ∗];Hs
σ(Rn)) × C ([0, T ∗];Hs(Rn)) × C ([0, T ∗];Hs

σ(Rn)) be a



20 U. MANNA, A. A. PANDA EJDE-2020/91

strong solution of the magnetic Bénard problem (1.5)-(1.8). If (u, θ,b) satisfies∫ T∗

0

(‖∇ × u(τ)‖BMO + ‖∇θ(τ)‖BMO + ‖∇ × b(τ)‖BMO) dτ <∞, (4.1)

then the solution (u, θ,b) can be continuously extended to [0, T ] for some T > T ∗.

Proof. Applying Js to (1.5)-(1.7) and then taking L2-inner product with Jsu, Jsθ
and Jsb respectively, we obtain, for s > n

2 + 1,(∂(Jsu)

∂t
, Jsu

)
L2

= (Js[(b · ∇)b], Jsu)L2 − (Js[(u · ∇)u], Jsu)L2

− (∇Jsp∗, Jsu)L2 + (Js(θen), Jsu)L2 ,
(4.2)

(∂(Jsθ)

∂t
, Jsθ

)
L2

= − (Js[(u · ∇)θ], Jsθ)L2 + (Jsun, J
sθ)L2 , (4.3)(∂(Jsb)

∂t
, Jsb

)
L2

= (Js[(b · ∇)u], Jsb)L2 − (Js[(u · ∇)b], Jsb)L2 . (4.4)

Using the definition of commutator, (4.2)-(4.4) become

1

2

d

dt
‖Jsu‖2L2

σ
= ([Js,b]∇b, Jsu)L2 + ((b · ∇)Jsb, Jsu)L2

− ([Js,u]∇u, Jsu)L2 − ((u · ∇)Jsu, Jsu)L2

− (Jsp∗, J
s∇ · u)L2 + (Js(θen), Jsu)L2 ,

(4.5)

1

2

d

dt
‖Jsθ‖2L2 = − ([Js,u]∇θ, Jsθ)L2 − ((u · ∇)Jsθ, Jsθ)L2 + (Jsun, J

sθ)L2 , (4.6)

1

2

d

dt
‖Jsb‖2L2

σ

= ([Js,b]∇u, Jsb)L2 + ((b · ∇)Jsu, Jsb)L2

− ([Js,u]∇b, Jsb)L2 − ((u · ∇)Jsb, Jsb)L2 .

(4.7)

Now adding (4.5), (4.6) and (4.7), then applying integration by parts, divergence
free condition on u and b and the fact ((b · ∇)Jsb, Jsu)L2 = − ((b · ∇)Jsu, Jsb)L2 ,
we obtain

1

2

d

dt

(
‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ

)
= ([Js,b]∇b, Jsu)L2 − ([Js,u]∇u, Jsu)L2 + (Js(θen), Jsu)L2

− ([Js,u]∇θ, Jsθ)L2 + (Jsun, J
sθ)L2 + ([Js,b]∇u, Jsb)L2

− ([Js,u]∇b, Jsb)L2 .

(4.8)

We estimate each term on the right-hand side of (4.8) separately. Using Lemma
2.8, Remark 2.1, Young’s inequality and finally rearranging, we obtain

| ([Js,b]∇b, Jsu)L2 |
≤ ‖[Js,b]∇b‖L2

σ
‖Jsu‖L2

σ

≤ C
(
‖∇b‖L∞‖Js−1∇b‖L2

σ
+ ‖Jsb‖L2

σ
‖∇b‖L∞

)
‖Jsu‖L2

σ

≤ C
(
‖∇b‖L∞‖∇b‖Hs−1

σ
+ ‖b‖Hsσ‖∇b‖L∞

)
‖u‖Hsσ

≤ C
(
‖∇b‖L∞‖b‖Hsσ‖u‖Hsσ + ‖b‖Hsσ‖u‖Hsσ‖∇b‖L∞

)
≤ C‖∇b‖L∞(‖u‖Hsσ‖b‖Hsσ )



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 21

≤ C(‖∇u‖L∞ + ‖∇θ‖L∞ + ‖∇b‖L∞)(‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ ).

Similarly the other commutator terms on the right hand side of (4.8) can be
estimated.

|([Js,u]∇u, Jsu)L2 |
≤ C(‖∇u‖L∞ + ‖∇θ‖L∞ + ‖∇b‖L∞)(‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ ),

|([Js,u]∇θ, Jsθ)L2 |
≤ C(‖∇u‖L∞ + ‖∇θ‖L∞ + ‖∇b‖L∞)(‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ ),

|([Js,b]∇u, Jsb)L2 |
≤ C(‖∇u‖L∞ + ‖∇θ‖L∞ + ‖∇b‖L∞)(‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ ),

|([Js,u]∇b, Jsb)L2 |
≤ C(‖∇u‖L∞ + ‖∇θ‖L∞ + ‖∇b‖L∞)(‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ ).

Now

| (Js(θen), Jsu)L2 | ≤ ‖Js(θen)‖L2
σ
‖Jsu‖L2

σ
≤ ‖θ‖Hs‖u‖Hsσ

≤ C(‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ ),

and

| (Jsun, Jsθ)L2 | ≤ C(‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ ).

Combining all the estimates above, from (4.8) after taking X(t) = ‖u(t)‖2Hsσ +

‖θ(t)‖2Hs + ‖b(t)‖2Hsσ , for t ∈ [0, T ∗], we obtain

d

dt
X(t) ≤ C(‖∇u‖L∞ + ‖∇θ‖L∞ + ‖∇b‖L∞ + 2)X(t).

Standard Gronwall’s inequality yields

X(t) ≤ X(0) exp
(
C

∫ t

0

(‖∇u(τ)‖L∞ + ‖∇θ(τ)‖L∞ + ‖∇b(τ)‖L∞ + 2) dτ
)
.

Hence

X(t) ≤ X(0) exp
(
C

∫ t

0

(‖∇u(τ)‖L∞ + ‖∇θ(τ)‖L∞ + ‖∇b(τ)‖L∞ + 2) dτ
)
. (4.9)

By the logarithmic Sobolev inequality in Lemma 2.12, and the fact that sin-
gular integral operators of Calderon-Zygmund type are bounded in BMO (i.e.
‖∇u‖BMO ≤ ‖∇× u‖BMO), for s > n

2 + 1 we obtain

‖∇u‖L∞ ≤ C
[
1 + ‖∇u‖BMO

(
1 + log+ ‖∇u‖Hs−1

σ

)]
≤ C

[
1 + ‖∇ × u‖BMO

(
1 + log+ ‖u‖Hsσ

)]
≤ C

[
1 + ‖∇ × u‖BMO

(
1 +

1

2
log+ ‖u‖2Hsσ

)]
≤ C

[
1 + ‖∇ × u‖BMO

(
1 +

1

2
log+

(
‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ

))]
≤ C

[
1 + ‖∇ × u‖BMO

(
1 + log+

(
‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ

))]
.

(4.10)

Similarly we obtain

‖∇θ‖L∞ ≤ C
[
1 + ‖∇θ‖BMO

(
1 + log+

(
‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ

))]
, (4.11)



22 U. MANNA, A. A. PANDA EJDE-2020/91

and

‖∇b‖L∞ ≤ C
[
1 + ‖∇ × b‖BMO

(
1 + log+

(
‖u‖2Hsσ + ‖θ‖2Hs + ‖b‖2Hsσ

))]
. (4.12)

Now using (4.10), (4.11) and (4.12) in (4.9), for all t ∈ [0, T ∗] we obtain

X(t) ≤ X(0) exp
[
C

∫ t

0

{
5 + (‖∇ × u(τ)‖BMO + ‖∇θ(τ)‖BMO

+ ‖∇ × b(τ)‖BMO)
(
1 + log+X(τ)

) }
dτ
]
.

Taking “log” on both sides we obtain for all t ∈ [0, T ∗],

logX(t) ≤ logX(0) + C

∫ t

0

{
5 + (‖∇ × u(τ)‖BMO + ‖∇θ(τ)‖BMO

+ ‖∇ × b(τ)‖BMO)(1 + log+X(τ))
}
dτ.

Rearranging the terms we have

log(eX(t)) ≤ log(eX(0)) + CT ∗ +

∫ t

0

{
(‖∇ × u(τ)‖BMO + ‖∇θ(τ)‖BMO

+ ‖∇ × b(τ)‖BMO)(log(eX(τ)))
}
dτ.

Now Gronwall’s inequality yields

log(eX(t)) ≤
{

(log(eX(0)) + CT ∗) exp
(
C

∫ t

0

(‖∇ × u(τ)‖BMO

+ ‖∇θ(τ)‖BMO + ‖∇ × b(τ)‖BMO) dτ
)}
.

Taking supremum over all t ∈ [0, T ∗] we obtain

sup
t∈[0,T∗]

logX(t) ≤ sup
t∈[0,T∗]

log(eX(t))

≤ (log(eX(0)) + CT ∗) exp
(
C

∫ T∗

0

(‖∇ × u(τ)‖BMO

+ ‖∇θ(τ)‖BMO + ‖∇ × b(τ)‖BMO) dτ
)
.

So finally we acquire

sup
t∈[0,T∗]

X(t) ≤ e(1+CT∗)X(0) exp
{

exp
(
C

∫ T∗

0

(‖∇ × u(τ)‖BMO

+ ‖∇θ(τ)‖BMO + ‖∇ × b(τ)‖BMO) dτ
)}
.

This implies that if∫ T∗

0

(‖∇ × u(τ)‖BMO + ‖∇θ(τ)‖BMO + ‖∇ × b(τ)‖BMO) dτ <∞,

then by continuation of local solutions, we can extend the solution to [0, T ] for some
T > T ∗. �

We now show that the assumption we made in Theorem 4.1 for∇θ can be relaxed
completely. In other words, provided θ0 ∈ Hs(Rn) ∩W 1,p(Rn), p ≥ 2, the bound
on curl of u and curl of b are enough to extend the solution continuously to some
time T > T ∗. Before proving this result let us note the following vector identity.



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 23

Remark 4.2. Using the vector product we have the following identity:

∇(u · ∇θ) = (u · ∇)∇θ + (∇θ · ∇)u + u× (∇×∇θ) +∇θ × (∇× u)

= (u · ∇)∇θ + (∇θ · ∇)u +∇θ × (∇× u)

= (u · ∇)∇θ + (∇u)t · ∇θ

where we have that the curl of the gradient of a scalar function is zero (i.e., u ×
(∇×∇θ) = 0) and (∇u)t · ∇θ = (∇θ · ∇)u +∇θ × (∇× u).

Theorem 4.3. Let s > n
2 + 1, u0 ∈ Hs

σ(Rn), b0 ∈ Hs
σ(Rn) and θ0 ∈ Hs(Rn) ∩

W 1,p(Rn), for 2 ≤ p ≤ ∞, n = 2, 3. Let (u, θ,b) ∈ C ([0, T ∗];Hs
σ(Rn)) ×

C ([0, T ∗];Hs(Rn))× C ([0, T ∗];Hs(Rn)) be a strong solution of the ideal magnetic
Bénard problem (1.5)-(1.8). Then∫ T∗

0

‖∇ × u(τ)‖BMO + ‖∇ × b(τ)‖BMO dτ <∞

guarantees that the solution can be extended continuously to [0, T ] for some T > T ∗.

Proof. We rewrite equation (1.6) as

∂θ

∂t
+ (u · ∇)θ = un,

and apply the gradient operator ∇ = (∂x1
, . . . , ∂xn) on both sides and take L2-inner

product with ∇θ|∇θ|p−2 to obtain( ∂
∂t

(∇θ),∇θ|∇θ|p−2
)

+
(
∇(u · ∇θ),∇θ|∇θ|p−2

)
=
(
∇un,∇θ|∇θ|p−2

)
.

Using the vector identity in Remark 4.2 we obtain( ∂
∂t

(∇θ),∇θ|∇θ|p−2
)

+
(
(∇u)t · ∇θ,∇θ|∇θ|p−2

)
+
(
(u · ∇)∇θ,∇θ|∇θ|p−2

)
=
(
∇un,∇θ|∇θ|p−2

)
.

(4.13)
We calculate each term separately. The first term of left-hand side of (4.13) gives( ∂

∂t
(∇θ),∇θ|∇θ|p−2

)
=

1

p

∫
Rn

∂

∂t
|∇θ|p dx =

1

p

d

dt
‖∇θ‖pLp ,

and (
(∇u)t · ∇θ,∇θ|∇θ|p−2

)
=

∫
Rn

(∇u)t · ∇θ · ∇θ|∇θ|p−2 dx

≤
∫
Rn

(∇u)t · |∇θ|p ≤ ‖∇u‖L∞‖∇θ‖pLp .

Integrating by parts and the divergence free condition of u, from the third term of
(4.13), we have(

(u · ∇)∇θ,∇θ|∇θ|p−2
)

=

∫
Rn

(u · ∇)∇θ · ∇θ|∇θ|p−2 dx

=
1

p

∫
Rn

u · ∇|∇θ|p dx

= −1

p

∫
Rn

(∇ · u) · |∇θ|p dx = 0.



24 U. MANNA, A. A. PANDA EJDE-2020/91

Now

|(∇un,∇θ|∇θ|p−2)| =
∣∣∣ ∫

Rn
∇un · ∇θ|∇θ|p−2 dx

∣∣∣ ≤ ∫
Rn
|∇un||∇θ|p−1 dx

≤
(∫

Rn
|∇un|p

)1/p(∫
Rn

(|∇θ|p−1)
p
p−1

) p−1
p

≤ ‖∇un‖Lp‖∇θ‖p−1
Lp

≤ 1

p

(
‖∇un‖pLp + (p− 1)‖∇θ‖pLp

)
.

From the Gagliardo-Nirenberg interpolation inequality (see Lemma 2.7), while j =
1, m = 3, r = 2, q = 2, we have: for n = 2,

‖∇u2‖Lp ≤ C‖u2‖
2+p
3p

L2 ‖u2‖
2p−2
3p

H3 , p ≥ 2, (4.14)

and for n = 3,

‖∇u3‖Lp ≤ C‖u3‖
6+p
6p

L2 ‖u3‖
5p−6
6p

H3 , p ≥ 2. (4.15)

From the term-wise estimates of (4.13), when n = 3, we obtain

d

dt
‖∇θ‖pLp ≤ p‖∇u‖L∞‖∇θ‖pLp + Cp‖u3‖

6+p
6

L2 ‖u3‖
5p−6

6

H3 + (p− 1)‖∇θ‖pLp ,

which further gives by Gronwall’s inequality,

‖∇θ‖pLp ≤
(
‖∇θ0‖pLp + Cp

∫ t

0

‖u3(τ)‖
6+p
6

L2 ‖u3(τ)‖
5p−6

6

H3 dτ
)

× exp
(∫ t

0

(p‖∇u(τ)‖L∞ + p− 1) dτ
)

≤
[
‖∇θ0‖pLp + Cp T ∗

(
sup

t∈[0,T∗]

‖u3‖L2

) 6+p
6
(

sup
t∈[0,T∗]

‖u3‖H3

) 5p−6
6
]

× exp(pT ∗) exp
(
p

∫ t

0

‖∇u(τ)‖L∞ dτ
)
.

Therefore, when n = 3, we obtain

‖∇θ‖Lp ≤
[
‖∇θ0‖pLp + Cp T ∗

(
sup

t∈[0,T∗]

‖u3‖L2

) 6+p
6
(

sup
t∈[0,T∗]

‖u3‖H3

) 5p−6
6
]1/p

× exp(T ∗) exp
(∫ t

0

‖∇u(τ)‖L∞ dτ
)

≤
[
‖∇θ0‖Lp + CT ∗

1/p
(

sup
t∈[0,T∗]

‖u3‖L2

) 6+p
6p
(

sup
t∈[0,T∗]

‖u3‖H3

) 5p−6
6p
]

× exp(T ∗) exp
(∫ t

0

‖∇u(τ)‖L∞ dτ
)
.

Since the L2-energy estimate and H3-energy estimate of u are finite by Propositions
3.2 and 3.3, letting p→∞, we finally obtain

‖∇θ‖L∞ ≤ C(T ∗)‖∇θ0‖L∞ exp
(∫ t

0

‖∇u(τ)‖L∞ dτ
)
.



EJDE-2020/91 IDEAL MAGNETIC BÉNARD PROBLEM 25

Similarly, the case n = 2 (from (4.14)) will also yield the same above estimate.
Note that, by Lemma 2.12, and properties of the BMO space, we further have

‖∇θ‖L∞ ≤ ‖∇θ0‖L∞ exp
(
C

∫ t

0

(
1 + ‖∇ × u(τ)‖BMO

(
1 + log+ ‖u(τ)‖Hsσ

))
dτ
)
.

As θ0 ∈ Hs(Rn) ∩ W 1,p(Rn), 2 ≤ p ≤ ∞ and supt∈[0,T∗] ‖u‖Hsσ is bounded for

s > n/2 + 1, we have

‖∇θ‖L∞ ≤ C exp
(∫ T∗

0

‖∇ × u(τ)‖BMO dτ
)
, (4.16)

where C = C(‖∇θ0‖L∞ , ‖u‖Hsσ , T
∗). From the assumption∫ T∗

0

‖∇ × u(τ)‖BMO dτ <∞,

the estimate in (4.16) is bounded. Hence,

‖∇θ‖BMO ≤ 2‖∇θ‖L∞ ≤ C <∞.
So the bound on BMO norms of vorticity and electrical current are enough to
guarantee that the solution can be extended to [0, T ] for some T > T ∗ provided

θ0 ∈ Hs(Rn) ∩W 1,p(Rn).

This proves the desired result. �

Acknowledgements. U Manna was supported by the National Board of Higher
Mathematics (NBHM), Govt. of India. Both authors would like to thank Indian
Institute of Science Education and Research Thiruvananthapuram for providing
stimulating scientific environment and resources.

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Utpal Manna

School of Mathematics, Indian Institute of Science Education and Research Thiru-
vananthapuram, 695551, Kerala, India

Email address: manna.utpal@iisertvm.ac.in

Akash Ashirbad Panda
School of Mathematics, Indian Institute of Science Education and Research Thiru-

vananthapuram, 695551, Kerala, India

Email address: akash.panda13@iisertvm.ac.in, akashp595@gmail.com


	1. Introduction
	2. Preliminaries
	2.1. Fractional derivative operator
	2.2. Fourier truncation operator
	2.3. Commutator estimates
	2.4. BMO space and logarithmic Sobolev inequality

	3. Energy estimates, local existence and uniqueness for the magnetic Bénard problem
	3.1. Energy estimates
	3.2. Local existence and uniqueness

	4. Blow-up criterion
	Acknowledgements

	References