Complexity
Volume 2021 (2021), Article ID 9943271, 8 pages
https://doi.org/10.1155/2021/9943271

Research Article

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

Yinhong Cao

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454100, China

Correspondence should be addressed to Yinhong Cao; caoyh@hpu.edu.cn

Received 19 March 2021; Accepted 11 May 2021; Published 4 June 2021

Academic Editor: Heng Liu

Copyright © 2021 Yinhong Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 problem

where , , , 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 that

Then, (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 , and

Then, 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 and

where 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, for

and 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 obtain

Then, (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 have

Applying Lemma 1, we arrive at

Note that

Then, 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 have

with . Write (18) into the integral form

We further localize it by applying with :

For , taking the -norm to this equation, we derive that

Choose such that

By Sobolev’s inequality,

Similarly, we have

We multiply the second equation in (1) by and integrate the result in space domain to obtain

which yields

Note that ; then, inserting (23)–(26) into (21), we derive that

Integrating this in time, together with (12), we obtain

Next, we apply to the first equation of (1) to obtain

Using ,

where we have chosen satisfying

Similarly, we have

Inserting (30)–(33) into (29),

Then, Gronwall’s inequality and (10)–(12) yields

This together with (26) yields (17). Substituting (35) into (28), we obtain

Thus, 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 yields

where

Taking advantage of Lemma 1,

By Young’s inequality,

Using , we have

Similarly, by Lemma 1, and are bounded by

By Lemma 1,

Combining the above bounds with (38), we infer that

We bound by Lemma 3. By Lemma 6, (10), and (17), we obtain

Then, 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 hence

in for any . In addition, set

Definition A.1. The inhomogeneous Besov space with and consists of satisfying

where 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).

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