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

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

UTPAL MANNA, AKASH ASHIRBAD PANDA

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, inH^s 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. e_n denotes the unit vector along the n^th 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 [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]).

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

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 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 [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 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 [28] and Secchi [29], when the initial data is in H^m 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

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 [8] that if the initial data (u0, θ0)∈H_σ³(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 H^s 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)∈ H_σ^s(Rⁿ)× H^s(Rⁿ)×H_σ^s(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

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 H^n/2+1, n= 2,3, it seems likely that the ideal magnetic B´enard problem is also ill-posed inH^n/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+θe_n, 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)∈H_σ^s(Rⁿ)×H^s(Rⁿ)×H_σ^s(Rⁿ).

Then there exists a unique strong solution(u, θ,b)to the problem (1.5)-(1.9), with u∈C([0, T^∗];H_σ^s(Rⁿ)), θ∈C([0, T^∗];H^s(Rⁿ)) b∈C([0, T^∗];H_σ^s(Rⁿ)) for some finite time T^∗=T^∗(s,ku₀k_Hs

σ,kθ₀k_Hs,kb₀k_Hs 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 (u^R, θ^R,b^R) of some smoothed version of the ideal magnetic B´enard system exist. We then establish that theH^s-norm of (u^R, θ^R,b^R) 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 (u^R, θ^R,b^R) is a Cauchy sequence in the L²-norm asR→ ∞, and by using Sobolev interpolation, (u^R, θ^R,b^R) → (u, θ,b) in any H^s0 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 <∞,

then the solution (u, θ,b)can be continuously extended to[0, T]for someT > T^∗. However, ifθ0∈H^s(Rⁿ)∩W^1,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 [19] or if we replace the condition by the one introduced in Planchon [25]. To be precise, the condition above can be weakened to

Z T^∗ 0

k∇ ×u(τ)k_B0

∞,∞+k∇ ×b(τ)k_B0

∞,∞

dτ

= 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 J^s (real s > 0), which de- notes the Bessel potential of orders, in terms of Fourier transform as follows:

F[J^sf](ξ) = (1 +|ξ|²)^s/2fb(ξ).

J^sis also equivalent to the operator (I−∆)^s/2.

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

kfkH^s =Z

Rⁿ

[(1 +|ξ|²)^s/2|fb(ξ)|]^21/2

=k(1 +|ξ|²)^s/2f(ξ)kb _L2 =kJ^sfk_L2 (2.1) and the inner product onH^s(Rⁿ) is

(f, g)_Hs =

(1 +|ξ|²)^s/2fb(ξ),(1 +|ξ|²)^s/2bg(ξ)

L²

= (F[J^sf](ξ),F[J^sg](ξ))_L2

= (J^sf, J^sg)_L2. Remark 2.1. It is trivial to show that

k∇fkH^s−1 ≤ kfkH^s.

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

Sd_Rf(ξ) :=1_BR(ξ)fb(ξ),

whereBR, a ball of radiusRcentered at the origin and1B_Ris the indicator function.

Then we infer the following properties:

(1) kSRfkH^s(Rⁿ)≤CkfkH^s(Rⁿ), whereC is a constant independent ofR;

(2) kS_Rf−fk_Hs(Rⁿ)≤ _R^C_kkfk_Hs+k(Rⁿ); (3) k(SR− SR⁰)fkH^s ≤Cmax

(_R¹)^k,(_R¹0)^k kfk_Hs+k.

For the proofs of these properties see [13]. We define the function spaces H_σ^s(Rⁿ) ={f ∈H^s(Rⁿ) :∇ ·f = 0}, H_σ^s(Rⁿ) = H_σ^s(Rⁿ)ⁿ

.

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

kfk_L∞(Rⁿ)≤Ckfk_Hs(Rⁿ), fors > n/2.

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

kf gkH^s ≤CkfkH^skgkH^s, fors > n/2.

Lemma 2.3. Fix s > n/2 and letf ∈H_σ^s andg∈H^s. Then k(f· ∇)gk_Hs−1≤Ckfk_Hskgk_Hs.

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

Lemma 2.4 (Sobolev inequality). Forf ∈H^s(Rⁿ), we have kfkL^q(Rⁿ)≤C_n,s,qkfkH^s(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 [18].

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≤ kfk_L2/kgk_L2/1−≤CkfkH˙¹⁻kgkH˙≤CkfkH¹kgkH^s−1.

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

kf gk_L2≤ kfk_L6kgk_L3≤CkfkH˙¹kgkH˙^1/2≤Ckfk_H1kgk_H1/2≤Ckfk_H1kgk_Hs−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 ∈H^s(Rⁿ)and0< s⁰ < s,

kfk_Hs0 ≤Ckfk^1−s_L2 ^0/skfk^s_H^0/ss .

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

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

kD^jgkL^p≤CkD^mgk^a_Lrkgk^1−a_Lq , 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

[J^s, f]g=J^s(f g)−f(J^sg), and

[J^s, f]∇g=J^s((f· ∇)g)−(f· ∇)J^sg. (2.2) where [J^s, f] =J^sf−f J^sis the commutator, in whichf is regarded as a multipli- cation operator.

Lemma 2.8. Fors≥0, and1< p <∞, we have a basic estimate k[J^s, f]gkL^p≤C k∇fkL^∞kJ^s−1gkL^p+kJ^sfkL^pkgkL^∞

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

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

2.4. BM Ospace and logarithmic Sobolev inequality.

Definition 2.9. The spaceBM O(Bounded Mean Oscillation) is the Banach space of all functionsf ∈L¹_loc(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 f_Q is the mean off overQ. For more details see [12].

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 [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 spaceW^k,r when kr=n.

Brezis-Gallouet [5] and Brezis-Wainger [6] provided the following inequality which relates the function spacesL^∞andW^s,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 + log^r−1r (1 +kfkW^s,p) , providedkfk_Wk,r ≤1forkr=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.

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

1 +k∇ ·fkL^∞+k∇ ×fkL^∞(1 +log(e+kfkW^s,p)) , for allf ∈W^s,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 >ⁿ_p, then we have kfkL^∞ ≤C

1 +kfkBM O(1 + log⁺kfkW^s,p) , for allf ∈W^s,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:

∂u^R

∂t +SR[(u^R· ∇)u^R] +∇p^R=θ^Ren+SR[(b^R· ∇)b^R], (3.1)

∂θ^R

∂t +SR

(u^R· ∇)θ^R

=u^R·en, (3.2)

∂b^R

∂t +SR[(u^R· ∇)b^R] =SR[(b^R· ∇)u^R], (3.3)

∇ ·u^R= 0 =∇ ·b^R, (3.4)

u^R(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 thatu^R,b^R lie in the space

V_R^σ:={g∈L²(Rⁿ) : supp(bg)⊂B_R,∇ ·g= 0}

andθ^R lies in the space

VR:={g∈L²(Rⁿ) : supp(bg)⊂BR}.

The divergence free condition foru^Rcan be obtained easily as

∇ ·\u^R(ξ) =iξ·1_BR(ξ)u(ξ) =b 1_BR(ξ)iξ·u(ξ) =b 1_BR(ξ)∇ ·[u(ξ) = 0.

Similarly we obtain divergence free condition forb^R.

Proposition 3.1. Let (u^R,b^R)∈H_σ^s(Rⁿ)×H_σ^s(Rⁿ), for s > n/2 + 1. Then the nonlinear operatorF(u^R,b^R) :=S_R[(u^R· ∇)b^R]is locally Lipschitz inu^R andb^R on the spaceV_R^σ.

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

| F(u^R₁,b^R)−F(u^R₂,b^R),u^R₁ −u^R₂

L²|

=| SR[(u^R₁ −u^R₂)· ∇b^R],u^R₁ −u^R₂

L²|

=| (u^R₁ −u^R₂)· ∇

b^R,SR(u^R₁ −u^R₂)

L²|

=| − (u^R₁ −u^R₂)· ∇

(u^R₁ −u^R₂),SRb^R

L²

≤ ku^R₁ −u^R₂k_L2

σk∇(u^R₁ −u^R₂)k_L2

σkSRb^RkL^∞

≤ ku^R₁ −u^R₂kL²_σku^R₁ −u^R₂kH_σ¹kb^RkL^∞

≤Cku^R₁ −u^R₂kH_σ^skb^RkH_σ^sku^R₁ −u^R₂k_L2 σ. Forb^R ∈H_σ^s(Rⁿ), this gives

kF(u^R₁,b^R)−F(u^R₂,b^R)k_L2 ≤Ckb^RkH_σ^sku^R₁ −u^R₂kH_σ^s

And henceF(·,·) is locally Lipschitz inu^R. To prove F to be locally Lipschitz in b^R, we use Remark 2.3. Fors > n/2 + 1 andu^R∈H_σ^s(Rⁿ), we have

| F(u^R,b^R₁)−F(u^R,b^R₂),b^R₁ −b^R₂

L²|

=| S_R(u^R· ∇)(b^R₁ −b^R₂),b^R₁ −b^R₂

L²|

=| (u^R· ∇)(b^R₁ −b^R₂),SR(b^R₁ −b^R₂)

L²|

≤ k(u^R· ∇)(b^R₁ −b^R₂)k_L2

σkSR(b^R₁ −b^R₂)k_L2 σ

≤Cku^RkH_σ¹k∇(b^R₁ −b^R₂)k_H^s−1

σ kb^R₁ −b^R₂kL²_σ

≤Cku^RkH_σ^skb^R₁ −b^R₂kH_σ^skb^R₁ −b^R₂k_L2 σ

Hence foru^R ∈H_σ^s(Rⁿ), we have k F(u^R,b^R₁)−F(u^R,b^R₂

k_L2 ≤Cku^RkH_σ^skb^R₁ −b^R₂kH_σ^s

And henceF(·,·) is locally Lipschitz inb^R.

Similarly one can show that F(b^R,u^R) is locally Lipschitz in b^R and u^R on the space V_R^σ×V_R^σ and F(u^R, θ^R) is locally Lipschitz inu^R and θ^R on the space V_R^σ×VR.

Hence by Picard’s theorem for infinite dimensional ordinary differential equa- tions, there exist a solution (u^R, θ^R,b^R) inV_R^σ×V^R×V_R^σ for some interval [0, T], whereTdepends onR. Moreover, the solution will exist as long asku^RkH_σ^s,kθ^RkH^s

andkb^RkH_σ^s remain finite.

3.1. Energy estimates. In this section we obtainL²andH^s, s > n/2 + 1, energy estimates for u^R, θ^R and b^R. In the course of proving the ku^Rk_Hs

σ, kθ^Rk_Hs and kb^RkH_σ^s are uniformly bounded, we will pick up a blow-up timeT^∗.

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], where0< T <∞, we have

sup

t∈[0,T]

ku^R(t)k²_L2

σ+kθ^R(t)k²_L2+kb^R(t)k²_L2 σ

< C

whereC depends only on ku0kL²_σ,kθ0kL²,kb0kL²_σ and T.

Proof. Consider the equations (3.1)-(3.3). TakingL²-inner product of (3.1), (3.2) and (3.3) withu^R, θ^R andb^Rrespectively, and adding we obtain

1 2

d dt

ku^Rk²_L2

σ+kθ^Rk²_L2+kb^Rk²_L2 σ

= θ^Re_n,u^R

L²+ (u^R·e_n), θ^R

L². (3.6)

In the above calculation, we have used that (u^R· ∇)u^R,u^R

L², (u^R· ∇)θ^R, θ^R

L²

and (u^R· ∇)b^R,b^R

L² vanish and (b^R· ∇)b^R,u^R

L² =− (b^R· ∇)u^R,b^R

L². It is easy to see that

|(θ^Ren,u^R)_L2| ≤ kθ^Renk_L2ku^Rk_L2 ≤ kθ^Rk_L2ku^Rk_L2

≤ 1

2 ku^Rk²_L2+kθ^Rk²_L2+kb^Rk²_L2

, and

| (u^R·en), θ^R

L²| ≤ ku^RkL²kθ^RkL² ≤1

2 ku^Rk²_L2+kθ^Rk²_L2+kb^Rk²_L2

. Using the above estimates in (3.6) and letting Y(t) = ku^R(t)k²_L2

σ +kθ^R(t)k²_L2+ kb^R(t)k²_L2

σ, we obtain

dY(t)

dt ≤2Y(t).

Straightforward integration and the fact that ku^R(0)k_L2

σ ≤ ku₀k_L2

σ, kθ^R(0)k_L2 ≤ kθ0k_L2 andkb^R(0)k_L2

σ ≤ kb0k_L2 σ yield sup

t∈[0,T]

Y(t)≤C(ku0kL²_σ,kθ0kL²,kb0kL²_σ, T)

So we have the desired result.

Proposition 3.3. Let(u0, θ0,b0)∈H_σ^s(Rⁿ)×H^s(Rⁿ)×H_σ^s(Rⁿ)withs > n/2 + 1.

Then there exists a timeT^∗=T^∗(s,ku0kH_σ^s,kθ0kH^s,kb0kH^s_σ)>0 such that sup

t∈[0,T^∗]

ku^R(t)k_Hs

σ, sup

t∈[0,T^∗]

kθ^R(t)k_Hs, sup

t∈[0,T^∗]

kb^R(t)k_Hs σ

are bounded uniformly in R.

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

∂(J^su^R)

∂t +S_RJ^s[(u^R· ∇)u^R] +∇J^sp^R=J^s(θ^Re_n) +S_RJ^s[(b^R· ∇)b^R], (3.7)

∂(J^sθ^R)

∂t +SRJ^s[(u^R· ∇)θ^R] =J^s(u^R·en), (3.8)

∂(J^sb^R)

∂t +SRJ^s[(u^R· ∇)b^R] =SRJ^s[(b^R· ∇)u^R] (3.9) Taking theL²-inner product of (3.7), (3.8) and (3.9) withJ^su^R,J^sθ^RandJ^sb^R respectively, we obtain

∂(J^su^R)

∂t , J^su^R

L²+ SRJ^s[(u^R· ∇)u^R], J^su^R

L²+ ∇J^sp^R, J^su^R

L²

= J^s(θ^Ren), J^su^R

L²+ SRJ^s[(b^R· ∇)b^R], J^su^R

L²,

(3.10)

∂(J^sθ^R)

∂t , J^sθ^R

L²+ SRJ^s[(u^R· ∇)θ^R], J^sθ^R

L²

= J^s(u^R·en), J^sθ^R

L²,

(3.11)

∂(J^sb^R)

∂t , J^sb^R

L²+ SRJ^s[(u^R· ∇)b^R], J^sb^R

L²

= S_RJ^s[(b^R· ∇)u^R], J^sb^R

L².

(3.12)

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

∂(J^su^R)

∂t , J^su^R

L² = Z

B_R

∂J^su^R

∂t J^su^Rdx= 1 2 Z

B_R

∂|J^su^R|²

∂t

= 1 2

d

dtkJ^su^Rk²_L2 σ = 1

2 d

dtku^Rk²_Hs σ.

(2) Applying weak Parseval’s identity and using the fact that SRu^R = u^R, since u^R∈V_R^σ we obtain

SRJ^s[(u^R· ∇)u^R], J^su^R

L² = J^s[(u^R· ∇)u^R], J^su^R

L² .

(3) Using the definition of commutator and incompressibility ofu^R, we obtain [J^s,u^R]∇u^R, J^su^R

L² = J^s[(u^R· ∇)u^R]− u^R· ∇

J^su^R, J^su^R

L²

= J^s[(u^R· ∇)u^R], J^su^R

L² . Now using Lemma 2.8 and H¨older’s inequality we obtain

[J^s,u^R]∇u^R, J^su^R

L²

≤ k[J^s,u^R]∇u^RkL²kJ^su^RkL²

≤C k∇u^RkL^∞kJ^s−1∇u^Rk_L2

σ+kJ^su^Rk_L2

σk∇u^RkL^∞

ku^RkH_σ^s

≤C

k∇u^Rk_H^s−1

σ k∇u^Rk_H^s−1

σ +ku^RkH_σ^sk∇u^Rk_H^s−1

σ

ku^RkH^s_σ

≤C ku^Rk²_Hs

σ+ku^Rk²_Hs σ

ku^Rk_Hs σ

≤C ku^Rk²_Hs

σ+kθ^Rk²_Hs+kb^Rk²_Hs σ

ku^Rk_Hs σ. (3) Using integration by parts we infer

∇J^sp^R, J^su^R

L² = J^sp^R, J^s∇ ·u^R

L² = 0.

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

J^s(θ^Re_n), J^su^R

L²

≤ kJ^s(θ^Re_n)k_L2kJ^su^Rk_L2 σ

≤ kθ^RenkH^sku^RkH^s_σ

≤C ku^Rk²_Hs

σ+kθ^Rk²_Hs+kb^Rk²_Hs σ

. (5) Using the property of the bilinear operator, we have

SRJ^s[(b^R· ∇)b^R], J^su^R

L² = J^s[(b^R· ∇)b^R], J^sSRu^R

L²

= J^s[(b^R· ∇)b^R], J^su^R

L²

=− J^s[(b^R· ∇)u^R], J^sb^R

L² . (6) Similarly referring to (1) we infer

∂(J^sθ^R)

∂t , J^sθ^R

L²

= 1 2

d

dtkθ^Rk²_Hs. (7) Similar calculations as in (2) and using Lemma 2.8 we obtain

SRJ^s[(u^R· ∇)θ^R], J^sθ^R

L²

and

[J^s,u^R]∇θ^R, J^sθ^R

L²

≤ k[J^s,u^R]∇θ^Rk_L2kJ^sθ^Rk_L2

≤C k∇u^RkL^∞kJ^s−1∇θ^RkL²+kJ^su^RkL²_σk∇θ^RkL^∞

kθ^RkH^s

≤C

k∇u^Rk_Hs−1

σ k∇θ^Rk_Hs−1+ku^RkH_σ^sk∇θ^Rk_Hs−1

kθ^RkH^s

≤C ku^Rk²_Hs

σ+kθ^Rk²_Hs+kb^Rk²_Hs σ

kθ^RkH^s.

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

J^s(u^R·en), J^sθ^R

L²

≤ kJ^s(u^R·en)kL²_σkJ^sθ^RkL²

≤ ku^RkH^s_σkθ^RkH^s

≤C ku^Rk²_Hs

σ+kθ^Rk²_Hs+kb^Rk²_Hs σ

.

(9) Similarly,

∂(J^sb^R)

∂t , J^sb^R

L²

= 1 2

d

dtkb^Rk²_Hs σ.

(10) Following similar steps as in (7), replacingθ^Rbyb^R we obtain

SRJ^s[(u^R· ∇)b^R], J^sb^R

L²

≤C

ku^Rk²_Hs

σ +kθ^Rk²_Hs+kb^Rk²_Hs σ

kb^RkH^s_σ.

(11) Weak Parseval’s identity gives SRJ^s[(b^R· ∇)u^R], J^sb^R

L² = J^s[(b^R· ∇)u^R], J^sb^R

L² .

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

1 2

d dt

ku^Rk²_Hs

σ +kθ^Rk²_Hs+kb^Rk²_Hs σ

≤C ku^Rk²_Hs

σ+kθ^Rk²_Hs+kb^Rk²_Hs σ

ku^RkH_σ^s+kθ^RkH^s+kb^RkH_σ^s

≤ C 2

ku^Rk²_Hs

σ+kθ^Rk²_Hs+kb^Rk²_Hs σ

² +3C

2

ku^Rk²_Hs

σ+kθ^Rk²_Hs+kb^Rk²_Hs σ

. Now lettingX(t) =ku^R(t)k²_Hs

σ+kθ^R(t)k²_Hs+kb^R(t)k²_Hs

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 [10], we have X(t)≤

3

2C²+X0

1−(³₂C²+X0)(³₂+C)T .

Note that ku^R(0)kH_σ^s ≤ ku0kH_σ^s, kθ^R(0)kH^s ≤ kθ0kH^s and kb^R(0)kH_σ^s ≤ kb0kH_σ^s. So provided we choose

T^∗< 1

(³₂C²+X₀)(³₂+C),

then the normsku^RkH_σ^s,kθ^RkH^s andkb^RkH^s_σ remain bounded on [0, T^∗] indepen-

dent ofR.

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 (u^R, θ^R,b^R) is Cauchy in a suitable space.

Proposition 3.4. The family (u^R, θ^R,b^R) of solutions of the magnetic B´enard 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) , 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 u^R−u^R0

+∇ p^R−p^R0

=θ^Ren−θ^R0en− SR[(u^R· ∇)u^R] +SR⁰[(u^R0· ∇)u^R0] +SR[(b^R· ∇)b^R]− SR⁰[(b^R0· ∇)b^R0],

(3.13)

∂

∂t θ^R−θ^R0

+SR[(u^R· ∇)θ^R]− SR⁰[(u^R0· ∇)θ^R0] =u^R·en−u^R0·en, (3.14)

∂

∂t b^R−b^R0

+SR[(u^R· ∇)b^R]− SR⁰[(u^R0· ∇)b^R0]

=SR[(b^R· ∇)u^R]− SR⁰[(b^R0· ∇)u^R0].

(3.15)

Taking the inner product of (3.13), (3.14) and (3.15) withu^R−u^R0,θ^R−θ^R0 and b^R−b^R0 respectively, and then adding we obtain

1 2

d dt

ku^R−u^R0k²_L2

σ+kθ^R−θ^R0k²_L2+kb^R−b^R0k²_L2 σ

= θ^Ren−θ^R0en,u^R−u^R0

− u^R·en−u^R0·en, θ^R−θ^R0

− SR[(u^R· ∇)u^R]− SR⁰[(u^R0· ∇)u^R0],u^R−u^R0

| {z }

I₁

+ SR[(b^R· ∇)b^R]− SR⁰[(b^R0· ∇)b^R0],u^R−u^R0

| {z }

I2

− S_R[(u^R· ∇)θ^R]− S_R⁰[(u^R0· ∇)θ^R0], θ^R−θ^R0

| {z }

I₃

+ SR[(b^R· ∇)u^R]− SR⁰[(b^R0· ∇)u^R0],b^R−b^R0

| {z }

I₄

− S_R[(u^R· ∇)b^R]− S_R⁰[(u^R0· ∇)b^R0],b^R−b^R0

| {z }

I5

(3.16)

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

θ^Ren−θ^R0en,u^R−u^R0

≤ kθ^Ren−θ^R0enk_L2ku^R−u^R0k_L2 σ

≤ kθ^R−θ^R0k_L2ku^R−u^R0k_L2 σ,

(3.17)