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

Abstract

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

1. Introduction

Consider the global well-posedness problem to the -dimensional (D) magnetic Bénard problemwhere , , , and denote the velocity field, the magnetic field, the temperature, and the pressure, respectively. is kinematic viscosity and is magnetic diffusion. and model the acting of the buoyancy force on fluid motion and the Rayleigh–Bénard convection in a heated inviscid fluid, respectively. The parameters and are nonnegative, and with is defined via . The 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 [1–7] for more physical background.

The 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 . When and , the global existence and uniqueness of strong solutions were established by Yamazaki in [9]. Recently, Shang in [10] showed the global well-posedness of solutions for the case and . Compared to the magnitude results on the 2D case, it appears that there are only few global regularity results on D () magnetic Bénard problem (1).

This 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.

Theorem 1. Consider (1) with and . Suppose that with , , and . Assume also thatThen, (1) has a unique global strong solution satisfying, for any ,

In the subsequent section, we prove Theorem 1. Moreover, the definitions of the Besov spaces are provided in the appendix. We shall separately write , , and for notational convenience.

2. Proof of Theorem 1

This section proves Theorem 1. The key step is to establish global a priori-bound for . More specifically, we shall establish the following result.

Proposition 1. Consider (1) with and . Suppose that with and , andThen, the corresponding solution of (1) is globally bounded in .

We only prove Proposition 1 for the case . In fact, the case is even simpler.

2.1. Preparations

To prove the main theorem, as preparations we give three lemmas in this section. The first contains two calculus inequalities.

Lemma 1. (see [11, 12]). Let . Let and with and . Then,with andwhere are constants.

The second is the property of the generalized heat operator.

Lemma 2. (see, e.g., [13, 14]). Let with being constants. Then, there exist two positive constants and such that, forand we have, for , , and ,

The last is the following logarithmic type interpolation inequality.

Lemma 3. (see [14, 15]). Let and . Then, there exists such that

2.2. Global Bound

This section gives the proof of Proposition 1. We first prove the following global -bound.

Lemma 4. For any , the solution of (1) obeys

Proof. Taking the -inner product of (1) with , together with the Young inequality, we obtainThen, (10) follows from this and Gronwall’s inequality.
The second one is a better regularity for .

Lemma 5. Let be the solution of (1). Then, for all and ,

Proof. Applying to and dotting the result with , we haveApplying Lemma 1, we arrive atNote thatThen, by again applying Lemma 1, we find that obeys the same bound as . Combing the above estimates up, together with (10), Gronwall’s inequality yields (12).
The last preparation is stated as follows.

Lemma 6. Let be the solution of (1). Then, for all and ,where .
In addition, for all ,

Proof. Taking curl on the both sides of , we havewith . Write (18) into the integral formWe further localize it by applying with :For , taking the -norm to this equation, we derive thatChoose such thatBy Sobolev’s inequality,Similarly, we haveWe multiply the second equation in (1) by and integrate the result in space domain to obtainwhich yieldsNote that ; then, inserting (23)–(26) into (21), we derive thatIntegrating this in time, together with (12), we obtainNext, we apply to the first equation of (1) to obtainUsing ,where we have chosen satisfyingSimilarly, we haveInserting (30)–(33) into (29),Then, Gronwall’s inequality and (10)–(12) yieldsThis together with (26) yields (17). Substituting (35) into (28), we obtainThus, for ,This is (16).
With Lemmas 4–6 at our disposal, we are ready to prove Proposition 1.

Proof. of Proposition 1. Applying to (1) and taking the -inner product with yieldswhereTaking advantage of Lemma 1,By Young’s inequality,Using , we haveSimilarly, by Lemma 1, and are bounded byBy Lemma 1,Combining the above bounds with (38), we infer thatWe bound by Lemma 3. By Lemma 6, (10), and (17), we obtainThen, the desired global -bound follows from these bounds and Osgood’s inequality. This completes the proof of Proposition 1.

Appendix

We recall the definitions of the Littlewood–Paley decomposition and Besov spaces in this appendix (see, e.g., [13, 16–19]).

Let be the usual Schwartz class and be its dual. Write for each ,

The Littlewood–Paley decomposition means that there exist functions such that

Thus, for , we have

Choose such that

Then, for all ,and hencein for any . In addition, set

Definition A.1. The inhomogeneous Besov space with and consists of satisfyingwhere is as defined in (A.2).

An important tool in dealing with Fourier localized functions is the following Bernstein’s inequality.

Proposition A.1. Let . Let .(1)Ifsatisfies for some integer and a constant , then(2)Ifsatisfies for some integer and constants , then where and are constants depending on , and only.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This study was supported by the Fundamental Research Funds for Universities of Henan Province (grant no. NSFRF180317), the Foundation of the Education Department of Henan Province (grant no. 14B110014), and the RFDP of Henan Polytechnic University (grant no. B2013-054).