Stability of the 3D MHD equations without vertical dissipation near an equilibrium

1.   Introduction

The MHD equations describe the evolution in time of velocity field $u$ and magnetic field $b$ of some electrically conducting fluids such as plasmas, liquid metals and salt water or electrolytes^[1,2]. The set of equations that describe MHD is a combination of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electro-magnetism. The field of MHD was initiated by Hannes Alfvén^[3], for which he received the Nobel Prize in physics in 1970.

The MHD equations are also of great interest in mathematics. In recent years, the stability of the MHD equations has attracted considerable interest, and one focus has been on the MHD equations with partial or fractional dissipation and diffusion. Elegant works have been made (see, e.g., ^{[4,5,6,7,8,9,10,11,12]}). For the 3D incompressible generalized MHD equations with fractional dissipation and diffusion,

where $\alpha, \beta > 0$ are real parameters, $u = u(x, t)\in\mathbb{R}^3$ represents the velocity field, $B = B(x, t)\in\mathbb{R}^3$ represents the magnetic field, $P = P(x, t)\in\mathbb{R}$ represents the pressure, $\nu > 0$ denotes the kinematic viscosity, and $\eta > 0$ denotes the magnetic diffusivity. For notational convenience, we write $\partial_i$ for the partial derivatives $\partial_{x_i}$($i$ = 1, 2, 3). The fractional Laplacian operator $(-\Delta)^{\alpha}$ is defined via the Fourier transform,

for

The MHD equations with fractional dissipation given by (1.1) have recently attracted considerable interest, due to their mathematical importance and physical applications. The justification for the study of this fractionally dissipated system can be made from several different perspectives. First, (1.1) represents a two-parameter family of systems and contains the MHD systems with standard Laplacian dissipation as special cases. (1.1) allows us to simultaneously examine a whole family of equations and potentially reveals how the properties of its solutions are related to the sizes of $\alpha$ and $\beta$. Second, the fractional diffusion operators can model the so-called anomalous diffusion, a much studied topic in physics, probability and finance (see, e.g., ^[13,14]). Third, fractional dissipation has been widely used in turbulence modeling to control the effective range of the non-local dissipation and to make numerical resolutions more efficient (see, e.g., ^[15]).

A range of global well-posedness results on (1.1) have been obtained. When $\alpha = \beta = 1$, $(1.1)$ reduces to the standard MHD equations, which is well-known possessing global $L^2$ weak solutions; in two dimensions, it is also unique^[7,16]. When combined with the already established global weak solutions^[8], these bounds allow us to conclude that MHD equations possess a global classical solution if $\alpha$ and $\beta$ satisfy

Wu^[9] was able to sharpen this result by replacing the fractional Laplacian operators by general Fourier multiplier operators. In particular, $(1.1)$ with a $\frac{(-\Delta)^{\alpha}}{\log (I-\Delta)} u$ and $\frac{(-\Delta)^{\beta}}{\log (I-\Delta)} b$ for $\alpha$ and $\beta$ satisfying the bounds above is also globally well posed^[6]. Yamazaki obtained the global regularity for the case when $\alpha = 2$ and $\beta = 0$ and for a logarithmically reduced fractional dissipation ^[17]. Logarithmic refinement of these fractional powers is contained in ^[18]. Ye and Xu^[19] considered the global existence of the 2D generalized incompressible MHD system with velocity dissipation exponent $\alpha > \frac{1}{4}$ and magnetic diffusion exponent $\beta = 1$. After that, Ye established the global regularity solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space ^[20]. Recently, Dai and Ji^[5] established the local existence and uniqueness in inhomogeneous Besov spaces when

Since (1.1) was proposed in ^[8], there have been considerable activities, and the global well-posedness problem on (1.1) is now much better understood (see, e.g., ^{[21,22,23,24,25,26]}).

The nonlinear stability for the ideal MHD equations was established in several beautiful papers ^{[27,28,29,30]}. Nevertheless, the stability of (1.1) remains unknown. The focus of this paper is the stability of perturbation near a background magnetic field which is

The perturbation $(u, b)$ with

solves the MHD system

We now address the problem as to whether the steady weak solution of the MHD equations (1.1) does in fact depend continuously on the perturbation of $(u^{(0)}, B^{(0)})$ given in the problem. In this paper, we consider (1.1) with only horizontal fractional dissipation and lacking vertical dissipation

with $\alpha, \beta\in(\frac{1}{2}, 1]$. The concept of horizontal dissipation comes from geophysical fluid dynamics (see ^[31]), and meteorologists model the turbulent diffusion with anisotropic viscosity $-\nu_h \Delta_h-\nu_3\partial_3^2$, where the horizontal kinetic viscosity coefficient $\nu_h$ and the vertical kinetic viscosity coefficient $\nu_3$ are empirical constants and satisfy $0 < \nu_3 \ll \nu_h$. In this paper, we take the limit case $\nu_h = \nu$ and $\nu_3 = 0$. To give a complete view of current studies on the stability problem concerning the MHD equations with partial dissipation, we mention some of the encouraging results in ^{[4,10,16,22,32,33,34,35,36,37,38,39,40]} and the references therein.

A natural consideration is how the parameters $\alpha$ and $\beta$ are determined, and this is what we choose to do in our tentative estimation work, based primarily on energy estimate method. When we bound the ${\dot{H}}^{3}$-norm of $(u, b)$, we plan to utilize a series of anisotropic inequalities derived from a Sobolev embedding inequality and Gagliardo-Nirenberg (G-N) interpolation inequality^[41,42,43]. Based on the relationship between the parameters of these inequalities, we ended up choosing $\alpha, \beta\in(\frac{1}{2}, 1]$ in (1.3) to study the stability of the MHD equations with fractional horizontal dissipation.

To construct a steady solution of (1.3), we make use of a bootstrap argument by anisotropic estimates

Here $\Lambda_h = (-\Delta_h)^{\frac{1}{2}}$ denotes the Zygmund operator. Our precise result is stated in the following theorem.

Theorem 1.1. Consider (1.3) with initial data $(u_0, b_0)\in H^{3}(\mathbb{R}^3)$ satisfying $\nabla\cdot u_{0} = \nabla\cdot b_{0} = 0$ and $\alpha, \beta\in(\frac{1}{2}, 1]$. Then, there exists a constant $\delta = \delta(\nu, \eta) > 0$ such that, if

then (1.3) has a unique global classical solution satisfying

for any $t > 0$, and $C = C(\nu, \eta)$ is a constant.

A natural starting point is to bound $\|u(t)\|_{H^{3}}+\|b(t)\|_{H^{3}}$ via energy estimate. We are able to derive the following energy inequality:

Combined with the bootstrapping argument (see^[44]), we can prove Theorem 1.1. However, the proof of Theorem 1.1 is not superficial. Due to the lack of the vertical dissipation and vertical magnetic diffusion, some nonlinear terms are not easy to control in terms of $\|u(t)\|_{H^3}+\|b(t)\|_{H^3}$ or the dissipation parts $\|\Lambda_{h}^\alpha u\|_{H^3}$ and $\|\Lambda_{h}^\beta b\|_{H^3}$. One of the most difficult terms is

Clearly, it does not appear possible to bound the subterms

directly in terms of $\|\Lambda_{h}^\alpha u\|_{H^3}^2\|\Lambda_{h}^\beta b\|_{H^3}^2,$ but in terms of $\|u(t)\|_{H^3}^2\|b(t)\|_{H^3}^2.$ Therefore, we hope the sum of the corresponding exponents of the two subterms to be less than or equal to 1 for all given $\alpha$ and $\beta$, which is

To establish the inequality of (1.6), we choose $\alpha, \beta\in(\frac{1}{2}, 1]$. In the case of

the subterms of (1.7) can be estimated directly by

The other case is

Our strategy is to extract part from the rest subterms

to fill the subterms of (1.7) by G-N interpolation inequality. One reason which cannot be ignored is that $\| \partial_{3}\Lambda_{h}^{\alpha}u_{h}\|_{L^2}$ could be bounded by either $\|u\|_{H^3}$ or $\|\Lambda_{h}^\alpha u\|_{H^3}$, and $\|\nabla_{h} \partial_{3}^{2} b\|_{L^2}$ could be bounded by either $\|b\|_{H^3}$ or $\|\Lambda_{h}^\alpha b\|_{H^3}$. In the last section of our paper, we have successfully used this method to solve all similar difficulties in proving stability and obtain inequality (1.6).

Lemma 1.2. Assume that $\alpha, \beta, \gamma \in(\frac{1}{2}, 1]$, $f, g, h, \Lambda_{h}^\alpha f, \Lambda_{h}^\beta g, \Lambda_{h}^\gamma h\ and\ \partial_{3}h$ are all in $L^2(\mathbb{R}^3)$. Then,

Here, we write $A \lesssim B$ to mean that $A \leq C B$ for some constant $C$ and $\Lambda_{3} = (- \partial_{33})^\frac{1}{2}$.

These anisotropic inequalities are greatly powerful in the study of global regularity and stability problems on partial differential equations with only partial dissipation. Similar inequalities have previously been used in the investigation of partially dissipated MHD systems and related equations (see, e.g., ^[45,46])

The rest of this paper is divided into two sections. Section 2 provides the proofs of Theorem 1.1 and Lemma 1.2. Section 3 derives the energy inequality (1.6).

2.   Proof of theorem

This section proves Theorem 1.1 and Lemma 1.2.

2.1.   Proof of Theorem 1.1

Roughly speaking, the bootstrap argument starts with an ansatz that $E(t)$ is bounded, say,

and shows that $E(t)$ actually admits a smaller bound, say,

when the initial condition is sufficiently small. A rigorous statement of the abstract bootstrap principle can be found in T. Tao's book^[44].

It follows that

for some pure constants $C$. To initiate the bootstrapping argument, we make the ansatz

We then show that (2.1) allows us to conclude that $E(t)$ actually admits an even smaller bound by taking the initial $H^{3}$-norm $E(0)$ sufficiently small. In fact, when (2.2) holds, (2.1) implies

or

Therefore, if we choose $\delta > 0$ sufficiently small such that

then

$E(t)$ actually admits a smaller bound in (2.3) than the one in the ansatz (2.2). The bootstrapping argument then assesses that (2.2) holds for all times when $E(0)$ obeys (2.4). This completes the proof.

2.2.   Proof of Lemma 1.2

The proof makes use of the version of Minkowski's inequality

for any $1\leq q\leq p\leq \infty$, where $f = f(x, y)$ with $x\in\mathbb{R}^m$, and $y\in\mathbb{R}^n$ is a measurable function on $\mathbb{R}^m\times \mathbb{R}^n$, and the following basic one-dimensional Sobolev embedding inequality ^[12], for $f\in H^s(\mathbb{R})$,

where $s > \frac{1}{2}$. By the above inequality and Hölder's inequality,

Let $s_1 = \alpha, s_2 = \beta, s_3 = 1$, and we obtain

Let $s_1 = \alpha, s_2 = \beta, s_3 = \gamma$, and we obtain

Here, $\|f\|_{L^\infty_{x_1}L^2_{x_2}L^2_{x_3}}$ represents the $L^\infty$-norm in the $x_1$-variable, followed by the $L^2$-norm in $x_2$ and the $L^2$-norm in $x_3$. This finishes the proof of Lemma 1.2.

3.   The $H^3$-stability

Due to the equivalence of $\|(u, b)\|_{H^{3}}$ with $\|(u, b)\|_{L^2}+\|(u, b)\|_{{\dot{H}}^{3}}$, it suffices to bound the $L^{2}$-norm and the ${\dot{H}}^{3}$-norm of $(u, b)$. By a simple energy estimate and $\nabla\cdot u = \nabla\cdot b = 0$, we find that the $L^{2}$-norm of $(u, b)$ obeys

The rest of the proof focuses on the ${\dot{H}}^{3}$-norm. Applying $\partial_{i}^{3}$ to (1.3) and then dotting by ($\partial_{i}^{3}u$, $\partial_{i}^{3}b$), we obtain

where

Note that, by integration by parts,

and

To bound $I_{2}$, we decompose it into three pieces,

where we have used the fact that $\int_{\mathbb{R}^3}u\cdot\nabla \partial_{3}^{3} u\cdot \partial_{3}^{3} u \, dx = 0.$ $I_{21}$ is naturally split into three parts,

By Lemma 1.2 and G-N interpolation inequality,

where we have applied inequality

We now turn to $I_{212}$, by Lemma 1.2,

In fact, we take $\|\nabla_{h} u\|_{L^2}^{1-\frac{1}{2\alpha}}\|\nabla_{h} \partial_{3} u\|_{L^2}^{1-\frac{1}{2\alpha}}$ from $\|\nabla_{h} u\|_{L^2}^\frac{1}{2}\|\nabla_{h} \partial_{3} u\|_{L^2}^\frac{1}{2}$ and combine it with $\| \partial_{3}^{3}\Lambda_{h}^\alpha u_{h}\|_{L^2}^\frac{1}{2\alpha}\| \partial_{3}^{3}\Lambda_{h}^\alpha u\|_{L^2}^\frac{1}{2\alpha}$ to reach our desired bound. In addition, by G-N interpolation inequality, we get

We next consider the term $I_{213}$, and we have

where we used

and $\nabla\cdot u = 0$. To deal with $I_{22}$, this term is split into three parts,

Similarly to (3.4),

Applying Lemma 1.2 and G-N interpolation inequality, we obtain

Note that we separate out a part of $\|\nabla_{h} \partial_{3} u\|_{L^2}^\frac{1}{2}\|\nabla_{h} \partial_{3}^{2} u\|_{L^2}^\frac{1}{2}$ and make it controlled by $\|\Lambda_{h} u\|_{H^3}$. Similarly,

We deal with $I_{23}$ in the same method, as $I_{23}$ is naturally split into three parts,

By Lemma 1.2 and G-N interpolation inequality, we have

We estimate $I_{232}$ similarly as $I_{213}$, which yields

We next consider the term $I_{233}$, and utilizing the incompressible condition again, we have

Combined with (3.3)–(3.12), we obtain

To bound $I_3$, we can refer to the way which is used in $I_2$ and then divide it into three terms,

$I_{31}$ can be further decomposed into three parts,

By Lemma 1.2 and G-N interpolation inequality,

where we have applied inequality

By divergence-free condition $\nabla\cdot b = 0$ and Lemma 1.2, we have

Significantly, we have used G-N interpolation inequality

We simplify $I_{313}$ by the same way as $I_{213}$, that is,

where we have used

and

To deal with $I_{32}$, we also split it into three parts,

By Lemma 1.2 and G-N interpolation inequality,

Naturally,

and

In the same way, $I_{33}$ is split into three parts,

Lemma 1.2 and G-N interpolation inequality imply

We estimate $I_{332}$ similarly as $I_{313}$, which yields

Now we turn to the next term $I_{333}$, by $\nabla\cdot b = 0$,

Utilizing Young's inequality, combining with (3.13)–(3.22), we have

Now, we try to bound $I_4$, and we split it into three parts,

Similarly as $I_{31}$, $I_{41}$ can be divided directly into three parts,

and each term can be bounded by Lemma 1.2 and G-N interpolation inequality. Same as (3.14)–(3.15), we have

and

By $\nabla\cdot u = 0$ and $\| \partial_{3}\Lambda_{h}^\beta b\|_{L^2}^{\frac{1}{\beta}-1}\leq \|b\|_{H^3}^{\frac{1}{\beta}-1}$,

Similarity, $I_{42}$ can also be divided directly into three terms,

Then, Lemma 1.2 and (3.17)–(3.18) imply

and

$I_{423}$ can also be bounded via $\nabla\cdot u = 0$, Lemma 1.2 and G-N interpolation inequality,

To deal with $I_{43}$, we rewrite it as

Again, by Lemma 1.2 and G-N interpolation inequality,

The estimate for $I_{432}$ is more complex, and utilizing Lemma 1.2 and G-N interpolation inequality, we have

where $\theta = \frac{\frac{1}{2\alpha}+\frac{1}{2\beta}-1}{\frac{1}{2\alpha}-\frac{1}{2\beta}+1}, 1-\theta = \frac{2-\frac{1}{\beta}}{\frac{1}{2\alpha}-\frac{1}{2\beta}+1}$. It is worth noting that

allows us to extract part of

which can be bounded by $\|u\|_{H^3}^{\theta\frac{1}{2\alpha}}$ and $\|b\|_{H^3}^{\theta(1-\frac{1}{2\beta})}$, and this brings us the hope of controlling $I_{432}$ suitably. We estimate $I_{433}$ by the same way as $I_{423}$, which is

Combining with (3.23)–(3.32), we obtain

It remains to estimate $I_5$, and since the estimation in $I_{5}$ is similar to what is done in $I_{2}-I_{4}$, we will omit the specific calculation process for $I_5$.

We turn to estimate $I_{51}$,

By Lemma 1.2 and G-N interpolation inequality,

and

where $\theta = \frac{\frac{1}{2\alpha}+\frac{1}{2\beta}-1}{\frac{1}{2\alpha}-\frac{1}{2\beta}+1}, 1-\theta = \frac{2-\frac{1}{\beta}}{\frac{1}{2\alpha}-\frac{1}{2\beta}+1}$. Now, we focus on $I_{52}$ and set

By Lemma 1.2 and G-N interpolation inequality,

and

We try to bound $I_{53}$,

Again, by Lemma 1.2 and G-N interpolation inequality,

and

Combined with (3.33)–(3.42), we obtain

Adding (3.1), (3.2) and integrating in time,

and inserting all the bounds obtained above for $I_{2}$ through $I_{5}$, we obtain (1.6). For example, the bounds for $I_{2}$ yield

The time integrals of $I_{3}-I_{5}$ are similarly bounded, which completes the proof of (1.6).

Acknowledgments

Ji is supported by the National Natural Science Foundation of China (NNSFC) under grant number 12001065 and Creative Research Groups of the Natural Science Foundation of Sichuan under grant number 2023NSFSC1984. The authors are grateful to the referee for careful reading of the paper and valuable suggestions and comments.

Conflict of interest

The authors declare no conflict of interest.