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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

B ´ENARD PROBLEM

Abstract. In this article, we consider the ideal magnetic B´enard problem in both two and three dimensions and prove the existence and uniqueness of strong local-in-time solutions, inHs for s > n2 + 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 B´enard problem with full viscosity is

∂u

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

∂θ

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

∂b

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

∇ ·u= 0 =∇ ·b, inRn×(0,∞), (1.4) with initial conditions

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

wheren= 2,3. Hereu:Rn×[0,∞)→Rnis the velocity field,θ:Rn×[0,∞)→R is the temperature, b : Rn ×[0,∞) → Rn is the magnetic field, p is the total pressure field wherep=p+12|b|2,p:Rn×[0,∞)→Ris the pressure. en denotes the unit vector along the nth direction. The termθen represents buoyancy force on fluid motion and u·en signifies the Rayleigh-B´enard convection in a heated inviscid fluid. ν ≥0,µ≥0 andκ≥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 whenν, µ and κ >0 is known for a long time . 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 ).

2010Mathematics Subject Classification. 76D03, 35B44, 35A01.

Key words and phrases. Magnetic B´enard problem; commutator estimates;

blow-up criterion; logarithmic Sobolev inequality.

c

2020 Texas State University.

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

1

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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 , the authors obtained the global well- posedness of two-dimensional magnetic B´enard 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  on the global regularity with full dissipation and magnetic diffusion. It is worthwhile to note that there are very few literatures available where the caseν =κ=µ= 0 has been discussed in two and three dimensions for the magnetic B´enard problem.

However, for the ideal magneto hydrodynamic (MHD) equations, i.e. whenθ≡0 and ν =µ= 0, in (1.1)-(1.3), the local-in-time existence of strong solutions have been proved by Schmidt  and Secchi , when the initial data is in Hm for integer m > 1 +n/2. Schmidt  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  derived a cri- teria for energy conservation and helicity conservation for weak solutions of ideal MHD equations. The authors in  extended the Beale-Kato-Majda  criterion to the three-dimensional ideal MHD equations by showing that for sufficiently regular initial data the condition

Z T 0

(k∇ ×u(τ)kL+k∇ ×b(τ)kL)dτ <∞,

ensures that the solution can be continued beyond timeT, where∇ ×uis the fluid vorticity,∇ ×bis the electrical current.

On the other hand, for the ideal Boussinesq system, i.e. whenb≡0,ν=κ= 0, and the Rayleigh-B´enard convection termu·en is absent in (1.1)-(1.3), only local- in-time existence results are available even in two-dimensions. It was proved in  that if the initial data (u0, θ0)∈Hσ3(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 . 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 . In recent work , 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 BM O-norm of the vorticity.

In this work, we consider the ideal magnetic B´enard problem (i.e. whenν =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)∈ Hσs(Rn)× Hs(Rn)×Hσs(Rn), where s > n/2 + 1. We prove when s > n/2 + 1,BM O-norms of the vorticity, electrical current and that of the gradient of the temperature (i.e.

∇×u,∇×b,∇θ∈L1(0, T;BM O)) control the breakdown of smooth solutions of the above systems. However, we later show that under suitable additional assumption onθ0, one can completely relax the condition on gradient of the temperature and the conditions ∇ ×u,∇ ×b ∈ L1(0, T;BM O) are sufficient to ensure that the

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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  and that of ideal Boussinesq equations due to Manna et al .

We note that, in view of the recent work of Bourgain and Li  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 B´enard problem is also ill-posed inHn/2+1, n= 2,3, although it still remains an open problem.

To be precise, in this work, we consider the ideal magnetic B´enard problem

∂u

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

∂θ

∂t + (u· ∇)θ=u·en, inRn×(0,∞), (1.6)

∂b

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

∇ ·u= 0 =∇ ·b, inRn×(0,∞), (1.8) u(x,0) =u0(x), θ(x,0) =θ0(x), b(x,0) =b0(x) inRn, (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∈Rbe such thats > n2+ 1, n= 2,3. Let (u0, θ0,b0)∈Hσs(Rn)×Hs(Rn)×Hσs(Rn).

Then there exists a unique strong solution(u, θ,b)to the problem (1.5)-(1.9), with u∈C([0, T];Hσs(Rn)), θ∈C([0, T];Hs(Rn)) b∈C([0, T];Hσs(Rn)) for some finite time T=T(s,ku0kHs

σ,kθ0kHs,kb0kHs σ)>0.

To prove this result, we consider the Fourier truncated ideal magnetic B´enard 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 B´enard system exist. We then establish that theHs-norm of (uR, θR,bR) are uniformly bounded up to a terminal time ˜T, 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 asR→ ∞, and by using Sobolev interpolation, (uR, θR,bR) → (u, θ,b) in any Hs0 for 0 < s0 < s.

Finally we provide the proof of Theorem 1.1 in Theorem 3.10.

Next, we establish that the BM Onorms 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 ands > n2 + 1, n= 2,3. If(u, θ,b) satisfy the condition

Z T 0

(k∇ ×u(τ)kBM O+k∇θ(τ)kBM O+k∇ ×b(τ)kBM O)dτ <∞,

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then the solution (u, θ,b)can be continuously extended to[0, T]for someT > T. However, ifθ0∈Hs(Rn)∩W1,p(Rn),2≤p≤ ∞, then the condition

Z T 0

(k∇ ×u(τ)kBM O+k∇ ×b(τ)kBM O)dτ <∞

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

Remark 1.3. The above result still holds if we replaceBM Owith the Besov space B∞,∞0 used in Kozono et al  or if we replace the condition by the one introduced in Planchon . To be precise, the condition above can be weakened to

Z T 0

k∇ ×u(τ)kB0

∞,∞+k∇ ×b(τ)kB0

∞,∞

= Z T

0

sup

j

k4j(∇ ×u(τ))kL+ sup

j

k4j(∇ ×b(τ))kL

dτ <∞,

or to

δ→0lim Z T

T−δ

sup

j

k4j(∇ ×u(τ))kL+ sup

j

k4j(∇ ×b(τ))kL

dτ < ,

for some sufficiently small >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 B´enard 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 orders, in terms of Fourier transform as follows:

F[Jsf](ξ) = (1 +|ξ|2)s/2fb(ξ).

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

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

kfkHs =Z

Rn

[(1 +|ξ|2)s/2|fb(ξ)|]21/2

=k(1 +|ξ|2)s/2f(ξ)kb L2 =kJsfkL2 (2.1) and the inner product onHs(Rn) is

(f, g)Hs =

(1 +|ξ|2)s/2fb(ξ),(1 +|ξ|2)s/2bg(ξ)

L2

= (F[Jsf](ξ),F[Jsg](ξ))L2

= (Jsf, Jsg)L2. Remark 2.1. It is trivial to show that

k∇fkHs−1 ≤ kfkHs.

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

SdRf(ξ) :=1BR(ξ)fb(ξ),

whereBR, a ball of radiusRcentered at the origin and1BRis the indicator function.

Then we infer the following properties:

(1) kSRfkHs(Rn)≤CkfkHs(Rn), whereC is a constant independent ofR;

(2) kSRf−fkHs(Rn)RCkkfkHs+k(Rn); (3) k(SR− SR0)fkHs ≤Cmax

(R1)k,(R10)k kfkHs+k.

For the proofs of these properties see . We define the function spaces Hσs(Rn) ={f ∈Hs(Rn) :∇ ·f = 0}, Hσs(Rn) = Hσs(Rn)n

.

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

kfkL(Rn)≤CkfkHs(Rn), fors > n/2.

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

kf gkHs ≤CkfkHskgkHs, fors > n/2.

Lemma 2.3. Fix s > n/2 and letf ∈Hσs andg∈Hs. Then k(f· ∇)gkHs−1≤CkfkHskgkHs.

Proof. We being inHσs,f is divergence free, and hence (f· ∇)g=∇ ·(f⊗g). Rest of the proof is straightforward, sinceHs is an algebra fors > n/2.

Lemma 2.4 (Sobolev inequality). Forf ∈Hs(Rn), we have kfkLq(Rn)≤Cn,s,qkfkHs(Rn)

provided that qlies in the following range (i) ifs < n/2, then2≤q≤ n−2s2n . (ii) ifs=n/2, then2≤q <∞.

(iii) ifs > n/2, then2≤q≤ ∞.

For details see Kesavan .

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

kf gkL2≤ kfkL2/kgkL2/1−≤CkfkH˙1−kgkH˙≤CkfkH1kgkHs−1.

Forn= 3, we use H¨older’s inequality with exponents 6 and 3, and Sobolev inequal- ity to obtain

kf gkL2≤ kfkL6kgkL3≤CkfkH˙1kgkH˙1/2≤CkfkH1kgkH1/2≤CkfkH1kgkHs−1. We note that for both 2D and 3D we have the same estimate.

Lemma 2.6(Interpolation in Sobolev spaces). Givens >0, there exists a constant C depending ons, so that for all f ∈Hs(Rn)and0< s0 < s,

kfkHs0 ≤Ckfk1−sL2 0/skfksH0/ss .

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

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Lemma 2.7 (Gagliardo-Nirenberg interpolation inequality ). 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

kDjgkLp≤CkDmgkaLrkgk1−aLq , where

1 p = j

n+1 r −m

n

a+1−a q ,

for allain the intervalj/m≤a <1. The constantCdepends only onn, m, j, q, r, a.

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

[Js, f]g=Js(f g)−f(Jsg), and

[Js, f]∇g=Js((f· ∇)g)−(f· ∇)Jsg. (2.2) where [Js, f] =Jsf−f Jsis the commutator, in whichf is regarded as a multipli- cation operator.

Lemma 2.8. Fors≥0, and1< p <∞, we have a basic estimate k[Js, f]gkLp≤C k∇fkLkJs−1gkLp+kJsfkLpkgkL

, whereC is a constant depending only onn, p, s.

For a proof of the above lemma see the appendix in .

2.4. BM Ospace and logarithmic Sobolev inequality.

Definition 2.9. The spaceBM O(Bounded Mean Oscillation) is the Banach space of all functionsf ∈L1loc(Rn) for which

kfkBM O = sup

Q

1

|Q|

Z

Q

|f(x)−fQ|dx

<∞,

where the sup ranges over all cubes Q⊂Rn, and fQ is the mean off overQ. For more details see .

The spaceBM Ohas two distinct advantageous properties compared toL. The first being the Riesz transforms are bounded in BM O and secondly the singular integral operators of the Calderon-Zygmund type are also bounded inBM O. Hence, one can show thatk∇ukBM O ≤Ck∇ ×ukBM O (see ).

It is well known that the Sobolev space Ws,p is embedded continuously into L for sp > n. However this embedding is false in the spaceWk,r when kr=n.

Brezis-Gallouet  and Brezis-Wainger  provided the following inequality which relates the function spacesLandWs,pat the critical value and was used to prove the existence of global solutions to the nonlinear Schr¨odinger equations.

Lemma 2.10. Let sp > n. Then kfkL ≤C

1 + logr−1r (1 +kfkWs,p) , providedkfkWk,r ≤1forkr=n.

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

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Lemma 2.11. Let s >np + 1, then we have k∇fkL ≤C

1 +k∇ ·fkL+k∇ ×fkL(1 +log(e+kfkWs,p)) , for allf ∈Ws,p(Rn).

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

Lemma 2.12. Let 1< p <∞ and lets >np, then we have kfkL ≤C

1 +kfkBM O(1 + log+kfkWs,p) , for allf ∈Ws,p, wherelog+a= logaif 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 B´enard problem

We consider the following truncated ideal magnetic B´enard problem on the whole ofRn, forn= 2,3:

∂uR

∂t +SR[(uR· ∇)uR] +∇pRRen+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, θR(0) =SRθ0,bR(0) =SRb0. (3.5) As the truncations are invariant under the flow of the equation, by taking the truncated initial data we ensure thatuR,bR lie in the space

VRσ:={g∈L2(Rn) : supp(bg)⊂BR,∇ ·g= 0}

andθR lies in the space

VR:={g∈L2(Rn) : supp(bg)⊂BR}.

The divergence free condition foruRcan be obtained easily as

∇ ·\uR(ξ) =iξ·1BR(ξ)u(ξ) =b 1BR(ξ)iξ·u(ξ) =b 1BR(ξ)∇ ·[u(ξ) = 0.

Similarly we obtain divergence free condition forbR.

Proposition 3.1. Let (uR,bR)∈Hσs(Rn)×Hσs(Rn), for s > n/2 + 1. Then the nonlinear operatorF(uR,bR) :=SR[(uR· ∇)bR]is locally Lipschitz inuR andbR on the spaceVRσ.

Proof. LetbR ∈ Hσs(Rn), for s > n/2 + 1. Then for provingF(·,·) to be locally Lipschitz inuR, we use integration by parts, H¨older’s inequality and Lemma 2.4 to obtain

| F(uR1,bR)−F(uR2,bR),uR1 −uR2

L2|

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=| SR[(uR1 −uR2)· ∇bR],uR1 −uR2

L2|

=| (uR1 −uR2)· ∇

bR,SR(uR1 −uR2)

L2|

=| − (uR1 −uR2)· ∇

(uR1 −uR2),SRbR

L2

≤ kuR1 −uR2kL2

σk∇(uR1 −uR2)kL2

σkSRbRkL

≤ kuR1 −uR2kL2σkuR1 −uR2kHσ1kbRkL

≤CkuR1 −uR2kHσskbRkHσskuR1 −uR2kL2 σ. ForbR ∈Hσs(Rn), this gives

kF(uR1,bR)−F(uR2,bR)kL2 ≤CkbRkHσskuR1 −uR2kHσs

And henceF(·,·) is locally Lipschitz inuR. To prove F to be locally Lipschitz in bR, we use Remark 2.3. Fors > n/2 + 1 anduR∈Hσs(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|

≤ k(uR· ∇)(bR1 −bR2)kL2

σkSR(bR1 −bR2)kL2 σ

≤CkuRkHσ1k∇(bR1 −bR2)kHs−1

σ kbR1 −bR2kL2σ

≤CkuRkHσskbR1 −bR2kHσskbR1 −bR2kL2 σ

Hence foruR ∈Hσs(Rn), we have k F(uR,bR1)−F(uR,bR2

kL2 ≤CkuRkHσskbR1 −bR2kHσs

And henceF(·,·) is locally Lipschitz inbR.

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

Hence by Picard’s theorem for infinite dimensional ordinary differential equa- tions, there exist a solution (uR, θR,bR) inVRσ×VR×VRσ for some interval [0, T], whereTdepends onR. Moreover, the solution will exist as long askuRkHσs,kθRkHs

andkbRkHσs remain finite.

3.1. Energy estimates. In this section we obtainL2andHs, s > n/2 + 1, energy estimates for uR, θR and bR. In the course of proving the kuRkHs

σ, kθRkHs and kbRkHσs are uniformly bounded, we will pick up a blow-up timeT.

Proposition 3.2 (L2-Energy Estimate). Given(u0, θ0,b0)∈L2σ(Rn)×L2(Rn)× L2σ(Rn)with s > n/2 + 1, then for any t∈[0, T], where0< T <∞, we have

sup

t∈[0,T]

kuR(t)k2L2

σ+kθR(t)k2L2+kbR(t)k2L2 σ

< C

whereC depends only on ku0kL2σ,kθ0kL2,kb0kL2σ and T.

Proof. Consider the equations (3.1)-(3.3). TakingL2-inner product of (3.1), (3.2) and (3.3) withuR, θR andbRrespectively, and adding we obtain

1 2

d dt

kuRk2L2

σ+kθRk2L2+kbRk2L2 σ

= θRen,uR

L2+ (uR·en), θR

L2. (3.6)

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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| ≤ kθRenkL2kuRkL2 ≤ kθRkL2kuRkL2

≤ 1

2 kuRk2L2+kθRk2L2+kbRk2L2

, and

| (uR·en), θR

L2| ≤ kuRkL2RkL2 ≤1

2 kuRk2L2+kθRk2L2+kbRk2L2

. Using the above estimates in (3.6) and letting Y(t) = kuR(t)k2L2

σ +kθR(t)k2L2+ kbR(t)k2L2

σ, we obtain

dY(t)

dt ≤2Y(t).

Straightforward integration and the fact that kuR(0)kL2

σ ≤ ku0kL2

σ, kθR(0)kL2 ≤ kθ0kL2 andkbR(0)kL2

σ ≤ kb0kL2 σ yield sup

t∈[0,T]

Y(t)≤C(ku0kL2σ,kθ0kL2,kb0kL2σ, T)

So we have the desired result.

Proposition 3.3. Let(u0, θ0,b0)∈Hσs(Rn)×Hs(Rn)×Hσs(Rn)withs > n/2 + 1.

Then there exists a timeT=T(s,ku0kHσs,kθ0kHs,kb0kHsσ)>0 such that sup

t∈[0,T]

kuR(t)kHs

σ, sup

t∈[0,T]

R(t)kHs, sup

t∈[0,T]

kbR(t)kHs σ

are bounded uniformly in R.

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

∂(JsuR)

∂t +SRJs[(uR· ∇)uR] +∇JspR=JsRen) +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 theL2-inner product of (3.7), (3.8) and (3.9) withJsuR,JsθRandJsbR respectively, we obtain

∂(JsuR)

∂t , JsuR

L2+ SRJs[(uR· ∇)uR], JsuR

L2+ ∇JspR, JsuR

L2

= JsRen), 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)

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

∂(JsuR)

∂t , JsuR

L2 = Z

BR

∂JsuR

∂t JsuRdx= 1 2 Z

BR

∂|JsuR|2

∂t

= 1 2

d

dtkJsuRk2L2 σ = 1

2 d

dtkuRk2Hs σ.

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

SRJs[(uR· ∇)uR], JsuR

L2 = Js[(uR· ∇)uR], JsuR

L2 .

(3) Using the definition of commutator and incompressibility ofuR, 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 H¨older’s inequality we obtain

[Js,uR]∇uR, JsuR

L2

≤ k[Js,uR]∇uRkL2kJsuRkL2

≤C k∇uRkLkJs−1∇uRkL2

σ+kJsuRkL2

σk∇uRkL

kuRkHσs

≤C

k∇uRkHs−1

σ k∇uRkHs−1

σ +kuRkHσsk∇uRkHs−1

σ

kuRkHsσ

≤C kuRk2Hs

σ+kuRk2Hs σ

kuRkHs σ

≤C kuRk2Hs

σ+kθRk2Hs+kbRk2Hs σ

kuRkHs σ. (3) Using integration by parts we infer

∇JspR, JsuR

L2 = JspR, Js∇ ·uR

L2 = 0.

(4) Using the H¨older’s inequality and then Young’s inequality we infer

JsRen), JsuR

L2

≤ kJsRen)kL2kJsuRkL2 σ

≤ kθRenkHskuRkHsσ

≤C kuRk2Hs

σ+kθRk2Hs+kbRk2Hs σ

. (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

dtkθRk2Hs. (7) Similar calculations as in (2) and using Lemma 2.8 we obtain

SRJs[(uR· ∇)θR], JsθR

L2

and

[Js,uR]∇θR, JsθR

L2

≤ k[Js,uR]∇θRkL2kJsθRkL2

(11)

≤C k∇uRkLkJs−1∇θRkL2+kJsuRkL2σk∇θRkL

RkHs

≤C

k∇uRkHs−1

σ k∇θRkHs−1+kuRkHσsk∇θRkHs−1

RkHs

≤C kuRk2Hs

σ+kθRk2Hs+kbRk2Hs σ

RkHs.

(8) Using the H¨older’s inequality and the Young’s inequality we obtain

Js(uR·en), JsθR

L2

≤ kJs(uR·en)kL2σkJsθRkL2

≤ kuRkHsσRkHs

≤C kuRk2Hs

σ+kθRk2Hs+kbRk2Hs σ

.

(9) Similarly,

∂(JsbR)

∂t , JsbR

L2

= 1 2

d

dtkbRk2Hs σ.

(10) Following similar steps as in (7), replacingθRbybR we obtain

SRJs[(uR· ∇)bR], JsbR

L2

≤C

kuRk2Hs

σ +kθRk2Hs+kbRk2Hs σ

kbRkHsσ.

(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

kuRk2Hs

σ +kθRk2Hs+kbRk2Hs σ

≤C kuRk2Hs

σ+kθRk2Hs+kbRk2Hs σ

kuRkHσs+kθRkHs+kbRkHσs

≤ C 2

kuRk2Hs

σ+kθRk2Hs+kbRk2Hs σ

2 +3C

2

kuRk2Hs

σ+kθRk2Hs+kbRk2Hs σ

. Now lettingX(t) =kuR(t)k2Hs

σ+kθR(t)k2Hs+kbR(t)k2Hs

σ we obtain d

dtX(t)≤3CX(t) +X(t)2≤ 3

2C2+3 2+C

X(t)2. So for all 0≤t≤T,

X(t)≤X0+3

2C2+3

2 +CZ t 0

X(s)2ds.

Now applying Bihari’s inequality , we have X(t)≤

3

2C2+X0

1−(32C2+X0)(32+C)T .

Note that kuR(0)kHσs ≤ ku0kHσs, kθR(0)kHs ≤ kθ0kHs and kbR(0)kHσs ≤ kb0kHσs. So provided we choose

T< 1

(32C2+X0)(32+C),

then the normskuRkHσs,kθRkHs andkbRkHsσ remain bounded on [0, T] indepen-

dent ofR.

(12)

3.2. Local existence and uniqueness. In this subsection, we prove existence and uniqueness of the local-in time strong solution of the magnetic B´enard 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 B´enard problem (3.1)-(3.5)are Cauchy in the space

L [0, T];L2σ(Rn)

×L [0, T];L2(Rn)

×L [0, T];L2σ(Rn) , asR→ ∞.

Proof. We consider equations (3.1), (3.2) and (3.3). Then taking the difference between the equations forR andR0 withR0> Rwe obtain

∂t uR−uR0

+∇ pR−pR0

Ren−θR0en− SR[(uR· ∇)uR] +SR0[(uR0· ∇)uR0] +SR[(bR· ∇)bR]− SR0[(bR0· ∇)bR0],

(3.13)

∂t θR−θR0

+SR[(uR· ∇)θR]− SR0[(uR0· ∇)θR0] =uR·en−uR0·en, (3.14)

∂t bR−bR0

+SR[(uR· ∇)bR]− SR0[(uR0· ∇)bR0]

=SR[(bR· ∇)uR]− SR0[(bR0· ∇)uR0].

(3.15)

Taking the inner product of (3.13), (3.14) and (3.15) withuR−uR0R−θR0 and bR−bR0 respectively, and then adding we obtain

1 2

d dt

kuR−uR0k2L2

σ+kθR−θR0k2L2+kbR−bR0k2L2 σ

= θRen−θR0en,uR−uR0

− uR·en−uR0·en, θR−θR0

− SR[(uR· ∇)uR]− SR0[(uR0· ∇)uR0],uR−uR0

| {z }

I1

+ SR[(bR· ∇)bR]− SR0[(bR0· ∇)bR0],uR−uR0

| {z }

I2

− SR[(uR· ∇)θR]− SR0[(uR0· ∇)θR0], θR−θR0

| {z }

I3

+ SR[(bR· ∇)uR]− SR0[(bR0· ∇)uR0],bR−bR0

| {z }

I4

− SR[(uR· ∇)bR]− SR0[(uR0· ∇)bR0],bR−bR0

| {z }

I5

(3.16)

We will calculate each term on the right hand side of (3.16) separately. First observe that

θRen−θR0en,uR−uR0

≤ kθRen−θR0enkL2kuR−uR0kL2 σ

≤ kθR−θR0kL2kuR−uR0kL2 σ,

(3.17)

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