Global Well-Posedness for the d-Dimensional Magnetic Bénard Problem without Thermal Diffusion

This paper focuses on the global existence of strong solutions to the magnetic Bénard problem with fractional dissipation and without thermal diffusion in 
 
 
 
 ℝ
 
 
 d
 
 
 
 with 
 
 d
 ≥
 3
 
 . By using the energy method and the regularization of generalized heat operators, we obtain the global regularity for this model under minimal amount dissipation.


Introduction
Consider the global well-posedness problem to the d-dimensional (dD) magnetic Bénard problem z t u + u · ∇u + μΛ 2α u � − ∇p + b · ∇b + θe d , x ∈ R d , t > 0, where u(x, t), b(x, t), θ(x, t), and p(x, t) denote the velocity field, the magnetic field, the temperature, and the pressure, respectively. μ ≥ 0 is kinematic viscosity and ] ≥ 0 is magnetic diffusion. θe d and u · e d model the acting of the buoyancy force on fluid motion and the Rayleigh-Bénard convection in a heated inviscid fluid, respectively. e parameters α and β are nonnegative, and Λ r f with r ≥ 0 is defined via Λ r f(ξ) � |ξ| r f(ξ). e magnetic Bénard problem can be used to model the behavior of the thermal instability under the influence of the magnetic field. One can refer to [1][2][3][4][5][6][7] for more physical background. e global regularity problem of the magnetic Bénard problem (1) has caught much attention. For the 2D case, Zhou et al. in [8] obtained the global regularity for the case α � β � 1. When α � 2 and ] � 0, the global existence and uniqueness of strong solutions were established by Yamazaki in [9]. Recently, Shang in [10] showed the global wellposedness of solutions for the case α � 1/2 and β � 1. Compared to the magnitude results on the 2D case, it appears that there are only few global regularity results on dD (d ≥ 3) magnetic Bénard problem (1).
is paper focuses its attention on the global regularity problem of (1) with minimal amount dissipation. More precisely, we are able to establish the following global regularity result. (2) Then, (1) has a unique global strong solution (u, b, θ) satisfying, for any T > 0, In the subsequent section, we prove eorem 1. Moreover, the definitions of the Besov spaces are provided in the appendix. We shall separately write L p � L p (R d ), H s � H s (R d ), and � R d for notational convenience.

Proof of Theorem 1
is section proves eorem 1. e key step is to establish global a priori H s -bound for (u, b, θ). More specifically, we shall establish the following result.
Then, the corresponding solution of (1) is globally bounded in H s (R d ).

Preparations.
To prove the main theorem, as preparations we give three lemmas in this section. e first contains two calculus inequalities.
e second is the property of the generalized heat operator.
en, there exists C > 0 such that

Global H s Bound.
is section gives the proof of Proposition 1. We first prove the following global L 2 -bound.

Complexity
Proof. Applying Λ σ to (1.1) 2 and dotting the result with Λ σ b, we have Applying Lemma 1, we arrive at Note that en, by again applying Lemma 1, we find that I 2 obeys the same bound as I 1 . Combing the above estimates up, together with (10), Gronwall's inequality yields (12).
e last preparation is stated as follows.

□
Proof. of Proposition 1. Applying Λ s to (1) and taking the L 2 -inner product with (Λ s u, Λ s b, Λ s θ) yields where Complexity 5 Taking advantage of Lemma 1, By Young's inequality, Using ∇ · b � 0, we have Similarly, by Lemma 1, K 4 and K 5 are bounded by By Lemma 1, Combining the above bounds with (38), we infer that We bound ‖∇u‖ L ∞ by Lemma 3. By Lemma 6, (10), and (17), we obtain en, the desired global H s -bound follows from these bounds and Osgood's inequality. is completes the proof of Proposition 1.
Let S be the usual Schwartz class and S ′ be its dual. Write for each j ∈ Z, e Littlewood-Paley decomposition means that there exist functions Φ j j∈Z ∈ S such that en, for all ψ ∈ S, and hence in S ′ for any f ∈ S ′ . In addition, set where Δ j f is as defined in (A.2).
An important tool in dealing with Fourier localized functions is the following Bernstein's inequality.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.