Global regularity to the 3D Cauchy problem of inhomogeneous magnetic Bénard equations with vacuum

1.   Introduction

In this paper, we consider the following inhomogeneous incompressible magnetic Bénard equations in $\Bbb R^3$:

which is equipped with the following initial conditions:

Here, the unknown functions $\rho$, $u$, $\theta$, $b$ and $p$ are the density, absolute temperature, magnetic field and pressure of the fluid, respectively. $\mu > 0$ stands for the viscosity constant and $\kappa > 0$ is the heat conductivity coefficient. $\eta > 0$ is the magnetic diffusive coefficient. $e_3 = (0, 0, 1)^T$.

The magnetic Bénard equations (1.1) illuminates the heat convection phenomenon under the dynamics of the velocity and magnetic fields in electrically conducting fluids such as plasmas (see ^[12] for details). If we ignore the Rayleigh-Bénard convection term $u\cdot e_2$, system (1.1) recovers the inhomogeneous incompressible MHD equations (i.e., $\theta\equiv 0$), which have been discussed in numerous studies on the existence, uniqueness, and regularity of the solutions, please see ^{[1,2,3,4,6,7,9,11,14,15,16,22]} and references therein.

When $b\equiv0$, system (1.1) reduces to inhomogeneous incompressible Bénard system. For the initial density allowing vacuum states, imposing a compatibility condition on the initial data, Cho-Kim ^[5] showed the local existence of strong solution in bounded domains $\Omega\subset\Bbb R^N(N = 2, 3)$. Later on, Zhong ^[17] removed the compatibility condition, and extended the Cho-Kim's results to the whole space $\Bbb R^2$. Meanwhile, Zhong ^[18,19] showed the global existence of strong solutions to the 2D Cauchy problem and 2D initial boundary value problem with general large data, respectively. Recently, using time-weighted estimates, Zhong ^[20] investigated the global existence and exponential decay of strong solutions without a compatibility condition, provided that some initial conditions are suitably small.

Similar to the results achieved for inhomogeneous Bénard system, the authors ^[8,21] respectively showed the local and global existence of strong solutions to the Cauchy problem of (1.1) and (1.2) in $\Bbb R^2$. However, the global existence of strong solution to the 3D Cauchy problem of (1.1) and (1.2) with vacuum is not addressed. In fact, this is the main aim of this paper.

Before formulating our main results, we first explain the notations and conventions used throughout this paper. For simplicity, we set

For $1\le r\le \infty$, $k\ge 1$, the Sobolev spaces are defined in a standard way.

The main result of this paper is formulated in the following theorem.

Theorem 1.1. For constants $\bar{\rho} > 0$, assume that the initial data $(\rho_0, u_0, \theta_0, b_0)$ satisfies

Then, there is a positive constant $\epsilon_0$, which depends only on $\bar{\rho}$ and the initial data, such that if

then the problems (1.1) and (1.2) has a unique global solution $(\rho, u, \theta, b)$ on $\Bbb R^3\times(0, T)$ satisfying

Moreover, it holds that

and there exists some positive constant $C$ depending only on $\bar{\rho}$ and initial data, such that for all $t\ge 1$,

Remark 1.1. Very recently, Zhong ^[20] establish the unique global strong solutions with vacuum to the Cauchy problem of the inhomogeneous incompressible Bénard equations. However, the corresponding strong solutions admits the exponential decay-in-time property which is quite different from Theorem 1.1 for magnetic Bénard system. This means that the magnetic field acts some significant roles on the large time behaviors of solutions. In particular, this decay rates of solutions are new for the magnetic Bénard system.

Remark 1.2. When $b = 0$, Theorem 1.1 is different from Zhong's ^[20] result, since the initial velocity $u_0$ and initial temperature $\theta_0$ need not to be small in our result.

2.   Preliminaries

We begin with the local existence and uniqueness of strong solutions whose proof can be performed by strategies as those in ^[5,21].

Lemma 2.1. Assume that $(\rho_0, u_0, \theta_0, b_0)$ satisfies (1.3). Then there exists a small time $T_* > 0$ and a unique strong solution $(\rho, u, \theta, b, p)$ to the problems (1.1) and (1.2) in $\Bbb R^3\times (0, T)$ satisfying for some constant $M_* > 1$ depending on $T_*$ and the initial data

Next, the following well-known Gagliardo-Nirenberg inequality (see ^[13]) will be used later.

Lemma 2.2. (Gagliardo-Nirenberg inequality) Assume $f\in W^{1, m}\cap L^r$, it holds that

where $\vartheta = (\frac{1}{r}-\frac{1}{q})/(\frac{1}{r}-\frac{1}{m}+\frac{1}{2})$, and if $m < 2$, then $q$ is between $r$ and $\frac{2m}{2-m}$, if $m = 2$, then $q\in [2, \infty)$, if $m > 2$, then $q\in [2, \infty]$ and constant $C$ depending on $q$, $m$ and $r$.

Next, we have some regularity results for the Stokes equations, which have been proven in ^[9].

Lemma 2.3. Suppose that $F\in L^r$ with $1 < r < \infty$. Let $(u, p)\in H^1\times L^2$ be the unique weak solution to the following Stokes problem

then $(\nabla^2u, \nabla p)\in L^r$ and there exists a constant $C$ depending only on $\mu$ and $r$ such that

3.   A priori estimates

In this section, we will establish some necessary a priori estimates, which together with the local existence (cf. Lemma 2.1) will complete the proof of Theorem 1.1. To this end, we let $(\rho, u, \theta, b, p)$ be a strong solutions of (1.1) and (1.2) in $\Bbb R^3\times[0, T]$. For simplicity, we use the letters $C$ and $C_i(i = 1, 2, \ldots)$ to denote some positive constant which dependent on $\bar{\rho}$ and the initial data, and write $C(\alpha)$ to emphasize that $C$ dependent on $\alpha$.

We first aim to get the following key a priori estimates on $(\rho, u, \theta, b, p)$. Set

Proposition 3.1. Assume that

there exists some small positive constant $\epsilon_0$ depending only on $\bar{\rho}$ and the initial data, such that if $(\rho, u, \theta, b, p)$ is a smooth solution of (1.1) and (1.2) on $\Bbb R^3\times (0, T]$ satisfying

then the following estimate holds

Moreover, we have

Proof of Proposition 3.1. 1) It following from transport Equation $(1.1)_1$, and making use of $(1.1)_5$ (see Lions [10, Theorem 2.1]) that

2) Multiplying $(1.1)_{2, 3, 4}$ by $u$, $\theta$ and $b$, respectively, and integrating by parts over $\Bbb R^3$, one obtains by using ${\rm div}u = {\rm div}b = 0$,

which leads

provided $\chi_0\le \epsilon_1: = \min \Big\{1, \frac{1}{64C_1^6\|\rho_0\|_{L^1}^4}\Big\}$. Integrating (3.23) over $[0, T]$ yields that

where

3) Multiplying $(1.1)_{2, 3}$ by $u_t$ and $\theta_t$ respectively, and integrating the resulting equality by parts. We infer from Gagliardo-Nirenberg inequality and Young's inequality, (3.4) that

which yields

4) Multiplying $(1.1)_4$ by $b_t$ in $L^2$ and integrating by parts, we obtain

5) According to Lemma 2.2 and $F = \rho u_{t}+\rho u \cdot \nabla u+b\cdot\nabla b+\rho \theta e_{3}$, we derive

which directly leads that

It follows from $(1.1)_4$, Hölder's and Gagliardo-Nirenberg inequalities, we get

which together with (3.10) implies that

which provided $\chi_0\le \epsilon_2 = \min\{\epsilon_1, \frac{1}{4C_2^2}\}$.

Similarly, by using the following $L^{2}$-estimate of elliptic system, we have

that is

6) Combining (3.8), (3.9), (3.12) and (3.13), it is easy to deduce that

Hence, choosing $\chi_0$ suitably small, we have

Integrating (3.14) with respect to $t$, and using (3.1), one obtains

As a consequence, we have

provided $\chi_0\le \epsilon_3: = \min\{\epsilon_2, \frac{1}{8C_3}, \frac{1}{16\tilde{C}}\}$.

7) Mmultiplying $(1.1)_4$ by $3|b|b$ and integrating by parts, we derive

Consequently,

To deal with the right-hand side of (3.16), we need to use the following variant of the Kato inequality

which combined with Hölder's inequality and Galiardo-Nirenberg inequality leads to

Thus, substituting (3.17) into (3.16), we obtain from Young's inequality that

This together with (3.7), (3.15) and Gronwall's inequality yields

provided $\chi_0\le \epsilon_4 = \min\Big\{\epsilon_3, \frac{1}{4C_4^2}\Big\}$. Thus, choosing $\epsilon_0 = \min\{\epsilon_4, \epsilon_7\}$ ($\epsilon_7$ can be chosen in the following lemmas), one obtains

Finally, combining (3.15) with (3.7) and (3.12) imply the desired (3.3). We completed the proof of Proposition 3.1. □

Lemma 3.1. Under the conditions of Proposition 3.1, it holds that

Proof. 1) Using $(1.1)_4$, Hölder's and Gagliardo-Nirenberg inequalities, we have

which implies

This along with Gronwall's inequality, (3.7) and (3.3) yields the desired (3.19).

2) It follows from (3.5) that

which implies

provided $\chi_0\le \epsilon_5 = \min\{\epsilon_4, \frac{1}{64C_5^6}\}$. Multiplying it by $t^\frac12$, we arrive at

3) In the view of (3.14), one obtains

which together with (3.3), (3.7) and (3.19) yields that

Thus, we directly obtain (3.21) from (3.12), (3.13) and (3.26). The proof of Lemma 3.1 is completed. □

Lemma 3.2. Under the conditions of Proposition 3.1, it holds that

Proof. 1) Differentiating $(1.1)_{2, 3}$ with respect to time variable $t$ give

Multiplying (3.30), (3.31) by $u_t$, $\theta_t$ respectively, and integrating it by parts, we arrive at

By using Hölder's, Gagliardo-Nirenberg inequalities, and (3.4), one gets

Putting all above estimates into (3.32), we show

which yields

provided $\chi_0\le \epsilon_6: = \min\{\epsilon_5, \Big(\frac{1}{2C_6}\Big)^6\}$. Hence, multiplying (3.33) by $t^\frac12$, and integrating by parts, we infer from Lemma 3.1 and (3.3) that

that is

provided $\chi_0\le \epsilon_7: = \min\{\epsilon_6, \Big(\frac{1}{2C_7}\Big)^6\}$.

2) Differentiating $(1.1)_4$ with respect to $t$, and multiplying the resulting equality with $b_t$ and then integrating by parts over $\Bbb R^3$, we arrive at

which leads

Multiplying (3.36) by $t$, and using Gronwall's inequality, we infer from (3.35) that

Then, the desired (3.27) follows from (3.12) and (3.13). We completed the proof of lemma. □

Lemma 3.3. Under the assumption of Theorem 1.1, it holds that

Proof. 1) It follows from the Lemma 2.3, Hölder's and Gagliardo-Nirenberg inequalities that for $r \in$ $(3, \min \{q, 6\})$,

which yields

Then, one derives from the Gagliardo-Nirenberg inequality and (3.12) that

which together with Lemma 3.2, (3.7) and (3.3) implies

2) Differentiating the continuity equation $(1.1)_1$ with respect to $x_i$ gives rise to

Multiplying (3.41) by $s|\rho_{x_i}|^{s-2}\rho_{x_i}\ (s = \{2, 6\})$ and integrating the resulting equation over $\Bbb R^3$ indicate that

It follows from the Gronwall's inequality and (3.40) that

Noticing the following facts

This ends the proof of Lemma 3.3. □

Lemma 3.4. Under the assumption of Theorem 1.1, it holds that for

Proof. Taking $\nabla$ operator to $(1.1)_4$, we get

Using the $L^2$-estimates of elliptic system, we derive

which yields to

We complete the proof of this lemma. □

4.   The proof of Theorem 1.1

By Lemma 2.1, there exists a $T_*$ such that the problems (1.1) and (1.2) has a unique local strong solution $(\rho, u, \theta, b)$ on $\Bbb R^3\times (0, T_*]$. In what follows, we shall extend the local solution to all the time.

Set

First, for any $0 < \tau < T_* < T < T^*$ with $T$ finite, it follows from Proposition 3.1, and Lemmas 3.1–3.4 that for any $p\ge 2$,

where we used the following standard Sobolev embedding

Moreover, one deduces from (3.4) and (3.38) that

Now, we claim that

Otherwise, if $T^* < \infty$, in the view of Lemmas 3.1–3.4, we have

satisfies (1.3) at $t = T^*$. Thus, we can take $(\rho, u, \theta, b)(T^*, x)$ as the initial data, and Lemma 2.1 implies that one can extend the local solutions beyond $T^*$. This contradicts the assumption of $T^*$ in (4.4). The proof of Theorem 1.1 is completed.

5.   Conclusions

This paper deals with the Cauchy problem of 3D inhomogeneous incompressible magnetic Bénard equations. Through some time-weighted a priori estimates, we prove the global existence of strong solution provided that the upper boundedness of initial density and initial magnetic field satisfy some smallness condition. Furthermore, we also obtain large time decay rates of the solution.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the anonymous reviewers and the Editor for their constructive comments which helped to improve the quality of the paper. Funding: This research was supported by grants from the Key Project of Jilin Provincial Science and Technology Development Plan (Grant No. 20210203056SF); Project Name: Research on the Construction of the "Two Products and One Equipment" Supervision and Traceability System under Information Conditions.

Conflict of interest

The authors declare that they have no competing interests.