A Regularity Result for the Incompressible Magnetohydrodynamics Equations with Free Surface Boundary

TL;DR

This paper proves a regularity result for three-dimensional incompressible free-boundary MHD equations, showing that magnetic fields help control vorticity even with minimal initial regularity, in a small bounded domain.

Contribution

It establishes the first low-regularity a priori estimates for free-boundary incompressible MHD equations, highlighting the magnetic field's regularizing effect.

Findings

Magnetic field controls vorticity in low-regularity solutions.

A priori estimates are obtained under minimal initial regularity.

The results apply to small bounded domains with free surfaces.

Abstract

We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and that of the magnetic field. To the best of our knowledge this is the first result that focuses on the low regularity solution for incompressible free-boundary MHD equations.

Equations

$$\{\begin{array}{ll}{\partial }_{t}u+u\cdot \nabla u-B\cdot \nabla B+\nabla (p+\frac{1}{2}|B{|}^2)=0,& \text{in}\mathcal{D};\\ {\partial }_{t}B+u\cdot \nabla B-B\cdot \nabla u=0,& \text{in}\mathcal{D};\\ \text{div }u=0,\text{div }B=0,& \text{in}\mathcal{D},\end{array}$$

$$\Omega (0)=\{x:(0,x)\in \mathcal{D}\},(u,B)=(u_0,B_0),\text{in}\{0\}\times {\Omega }_0.$$

$$\{\begin{array}{ll}({\partial }_t+u\cdot \nabla ){|}_{\partial \mathcal{D}}\in \mathcal{T}(\partial \mathcal{D})& \\ p=0& \text{on}\partial \mathcal{D},\\ |B|=c,B\cdot \mathcal{N}=0& \text{on}\partial \mathcal{D},\end{array}$$

$$\displaystyle~{}~{}~{}~{}~{}\frac{d}{dt}\Big{[}\frac{1}{2}\int_{\Omega(t)}|u|^{2}+\frac{1}{2}\int_{\Omega(t)}|B|^{2}\Big{]}$$

$$={\int }_{\Omega (t)}u\cdot D_{t}u+{\int }_{\Omega (t)}B\cdot D_{t}B$$

$$=-{\int }_{\Omega (t)}u\cdot \nabla (p+\frac{1}{2}|B{|}^2)+{\int }_{\Omega (t)}u\cdot (B\cdot \nabla B)+{\int }_{\Omega (t)}B\cdot (B\cdot \nabla u)$$

$$=-{\int }_{\partial \Omega (t)}(u\cdot \mathcal{N})p-\underset{=0\text{by Gauss theorem}}{\underbrace{{\int }_{\partial \Omega (t)}\frac{1}{2}(u\cdot \mathcal{N})c^2}}+{\int }_{\Omega (t)}u\cdot (B\cdot \nabla B)-{\int }_{\Omega (t)}u\cdot (B\cdot \nabla B)=0.$$

$$-{\nabla }_{\mathcal{N}}(p+\frac{1}{2}|B{|}^2)\ge {ϵ}_0>0\text{on}\partial \Omega (t).$$

$$-{\nabla }_{\mathcal{N}}p\ge {ϵ}_0>0\text{on}\partial \Omega (t).$$

$$({\partial }_t+{\nabla }_u)\text{curl }u\sim \nabla (\text{curl }B).$$

$${\partial }_t\eta (t,y)=u(t,\eta (t,y)),\eta (0,y)=y.$$

$$v(t,y)=u(t,\eta (t,y)),b(t,y)=B(t,\eta (t,y)),q(t,y)=p(t,\eta (t,y)).$$

$${\partial }_{\mu }{a}^{\mu \alpha }=0.$$

$$\{\begin{array}{ll}{\partial }_t{v}_{\alpha }-b_{\beta }{a}^{\mu \beta }{\partial }_{\mu }{b}_{\alpha }+a_{\alpha }^{\mu }{\partial }_{\mu }Q=0& \text{in}[0,T]\times \Omega ;\\ {\partial }_t{b}_{\alpha }-b_{\beta }{a}^{\mu \beta }{\partial }_{\mu }{v}_{\alpha }=0& \text{in}[0,T]\times \Omega ;\\ a_{\alpha }^{\mu }{\partial }_{\mu }{v}^{\alpha }=0,a_{\alpha }^{\mu }{\partial }_{\mu }{b}^{\alpha }=0& \text{in}[0,T]\times \Omega ;\\ v_3=0& \text{on}{\Gamma }_0;\\ a^{\mu \nu }{b}_{\mu }{b}_{\nu }=c^2,Q=\frac{1}{2}{c}^2,a_{\nu }^{\mu }{b}^{\nu }{N}_{\mu }=0& \text{on}{\Gamma }_{;}\\ \frac{\partial Q}{\partial N}={\partial }_{3}Q\le -{ϵ}_0& \text{on}{\Gamma }_1.\end{array}$$

$$\Omega ={\mathbb{T}}^2\times (0,\overline{ϵ}),$$

$$N(t):=\Vert \eta (t){\Vert }_{H^{3+\delta }}^2+\Vert v(t){\Vert }_{H^{2.5+\delta }}^2+\Vert b(t){\Vert }_{H^{2.5+\delta }}^2.$$

$$-\frac{\partial Q}{\partial N}\big{|}_{t=0}=-\partial_{3}Q\big{|}_{t=0}\geq\epsilon_{0}>0,\quad\text{on}\,\,\Gamma_{1}$$

Full text

1. 1 Introduction

2. 1.1 History and background

3. 1.2 MHD system in Lagrangian coordinates and Main result

1 Introduction

The goal of this manuscript is to investigate the solutions in low regularity Sobolev spaces for the following incompressible inviscid MHD equations in a moving domain:

[TABLE]

describing the motion of conducting fluids in an electromagnetic field, where $\mathcal{D}={\cup}_{0\leq t\leq T}\{t\}\times\Omega(t)$ and $\Omega(t)\subset\mathbb{R}^{3}$ is the domain occupied by the fluid with small volume whose boundary $\partial\Omega(t)$ moves with the velocity of the fluid. Under this setting, the fluid velocity $u=(u_{1},u_{2},u_{3})$, the magnetic field $B=(B_{1},B_{2},B_{3})$, the fluid pressure $p$ and the domain $\mathcal{D}$ are to be determined; in other words, given a simply connected bounded domain $\Omega(0)\subset\mathbb{R}^{3}$ and the initial data $u_{0}$ and $B_{0}$ satisfying the constraints $\text{div}\,u_{0}=0$ and $\text{div}\,B_{0}=0$, we want to find a set $\mathcal{D}$ and the vector fields $u$ and $B$ solving (1.1) satisfying the initial conditions:

[TABLE]

We also require the following boundary conditions on the free boundary $\partial\mathcal{D}={\cup}_{0\leq t\leq T}\{t\}\times\partial\Omega(t)$:

[TABLE]

where $\mathcal{T}(\partial\mathcal{D})$ is the tangent bundle of $\partial\mathcal{D}$, $\mathcal{N}$ is the exterior unit normal to $\partial\Omega_{t}$ and $c\geq 0$ is a constant. The first condition of (1.3) means that the boundary moves with the velocity of the fluid, the second condition of (1.3) means that the region outside $\Omega_{t}$ is vacuum, where $B\cdot\mathcal{N}=0$ on $\partial\Omega_{t}$ implies that the fluid is a perfect conductor; in other words, the induced electric field $\mathcal{E}$ satisfies $\mathcal{E}\times\mathcal{N}=0$ on $\partial\Omega_{t}$. Also, the condition $|B|=c$ on $\partial\Omega_{t}$ yields that the physical energy is conserved, i.e., denoting $D_{t}=\partial_{t}+u\cdot\nabla$, and invoking the divergence free condition for both $u$ and $B$, we have:

[TABLE]

We will establish a priori bounds for the MHD equations (1.1)-(1.3) when $u_{0},B_{0}\in H^{2.5+\delta}(\Omega(0))$ for $\delta\in(0,0.5)$ under the physical sign condition

[TABLE]

We recall here that for the free-boundary problem of the motion of a incompressible fluid without magnetic field (i.e., the incompressible free-boundary Euler equations), the physical sign condition reads

[TABLE]

Condition (1.4) was first discovered by Hao and Luo [hao2014priori] when proving the a priori energy estimate for the free boundary incompressible MHD equations with $H^{4}$ initial data. Very recently, they proved that (1.1)-(1.3) is ill-posed when (1.4) is violated [hao2018ill]. The quantity $p+\frac{1}{2}|B|^{2}$ (i.e., the total pressure) plays an important role here in our analysis. In fact, it determines the acceleration of the moving surface boundary.

1.1 History and background

In the absence of the magnetic field $B$, the system (1.1) is reduced to the free-boundary Euler equations which has attracted much attention in the past two decades. Important progress has been made for both incompressible and compressible flows, with or without surface tension, and with or without vorticity. Without attempting to be exhaustive, we refer [alazard2014cauchy, christodoulou2000motion, coutand2007LWP, disconzi2017prioriI, ignatova2016local, kukavica2017local, lindblad2002, lindblad2005well, lindblad2018priori, luo2017motion, nalimov1974cauchy, SchweizerFreeEuler, shatah2008geometry, shatah2008priori, shatah2011local, wu1997LWPww, wu1999LWPww, zhang2008free] for more details.

On the other hand, the MHD equations describe the behavior of an electrically conducting fluid (e.g., a plasma) acted on by a magnetic field. In particular, the free-boundary MHD equations (also known as the plasma-interface problem) describes the phenomenon when the plasma is separated from the outside wall by a vacuum, whose motion can be formulated as the incompressible free-boundary MHD equations.

Although the MHD equations in a fixed domain have been the focus of a great deal of activities, e.g., [cao2010two, chen2008existence, friedlander2011global, friedlander1990nonlinear, friedlander1995stability, jiang2010incompressible, wu2011global], much less is known for the free-boundary case. The main difficulty is the strong coupling between $u$ and $B$ (i.e., the appearance of $B\cdot\nabla B$ and $B\cdot\nabla u$ terms in the first and second equations of (1.1), respectively). In fact, the appearance of the Lorentzian force term $B\cdot\nabla B$ destories the Cauchy invariance, which provides good estimates for $\text{curl }v$ when $B$ is absent; indeed, one can see this by commuting the curl operator through the first equation of (1.1), which implies111We refer (LABEL:cauchy_invariance_fail1)-(LABEL:cauchy_invariance_fail_2) for the detailed computation.

[TABLE]

Nevertheless, it is remarkable that the magnetic field $B$ yields certain regularizing effect (cf. [wangyanjin2012]), which can be derived from the transport equation of $B$ (i.e., the second equation of (1.1)). Such regularizing effect plays an important role to control the full Sobolev norms of $\text{curl }B$ and $\text{curl }u$ and hence the full Sobolev norm of $B$ and $u$ via the div-curl estimate. We will provide more details on this in Section LABEL:section_1.3.

For the free-boundary MHD equations, the local (in time) well-posedness (LWP) of the linearized equations was studied by Morando-Trakhinin-Trebeschi [morando2014well], Secchi-Trakhinin [secchi2011well] and Trakhinin [trakhinin2010well]. For the nonlinear equations, Hao-Luo [hao2014priori] proved the a priori energy estimate with $H^{4}$ initial data and the LWP was established by Secchi-Trakhinin [secchi2013well] and Gu-Wang [gu2016construction]. Also, we mention here that in Hao [hao2016motion] and Sun-Wang-Zhang [sun2017well], the authors studied the a priori energy estimate and LWP, respectively, for the free-boundary MHD equations with nontrivial vacuum magnetic field.

In this manuscript, we establish the local a priori energy estimate with $u,B\in H^{2.5+\delta}$ with $\delta>0$ is arbitrary. This agrees with the minimal regularity assumption (i.e., $H^{\frac{d}{2}+1+\delta}$, where $d$ is the spatial dimension) that one may expect for the velocity field in the theory of the free-boundary incompressible Euler equations (see, e.g., [DKT, kukavica2014regularity, kukavica2017local]). In fact, Bourgain-Li [bourgain2015strong] proved that the incompressible Euler equations with $H^{\frac{d}{2}+1}$ initial data are ill-posed even in the free space $\mathbb{R}^{d}$.

1.2 MHD system in Lagrangian coordinates and Main result

We reformulate the MHD equations in Lagrangian coordinates, in which the free domain becomes fixed. Let $\Omega$ be a bounded domain in $\mathbb{R}^{3}$. Denoting coordinates on $\Omega$ by $y=(y_{1},y_{2},y_{3})$, we define $\eta:[0,T]\times\Omega\to\mathcal{D}$ to be the flow map of the velocity $u$, i.e.,

[TABLE]

We introduce the Lagrangian velocity, magnetic field and fluid pressure, respectively, by

[TABLE]

Let $\partial$ be the spatial derivative with respect to $y$ variable. We introduce the cofactor matrix $a=[\partial\eta]^{-1}$, which is well-defined since $\eta(t,\cdot)$ is almost the identity map when $t$ is sufficiently small. It’s worth noting that $a$ verifies the Piola’s identity, i.e.,

[TABLE]

Here, the summation convention is used for repeated upper and lower indices, and in above and throughout, all indices (e.g., Greek and Latin) range over $1,2,3$.

Denote the total pressure $p_{\text{total}}=p+\frac{1}{2}|B|^{2}$ and let $Q=p_{\text{total}}(t,\eta(t,y))$. Then (1.1)-(1.3) can be reformulated as:

[TABLE]

Remark**.**

In above and throughout, the upper index of $a$ represents the number of the rows whereas the lower index represents the number of the columns, i.e., $a^{\text{row}}_{\text{column}}$.

For the sake of simplicity and clean notation, here we consider the model case when

[TABLE]

where $\overline{\epsilon}\ll 1$ and $\partial\Omega=\Gamma_{0}\cup\Gamma_{1}$ and $\Gamma_{1}=\mathbb{T}^{2}\times\{\overline{\epsilon}\}$ is the top (moving) boundary, $\Gamma_{0}=\mathbb{T}^{2}\times\{0\}$ is the fixed bottom. We shall treat the general bounded domains with small volume in Section 6 by adapting what has been done in [DKT]. However, choosing $\Omega$ as above allows us to focus on the real issues of the problem without being distracted by the cumbersomeness of the partition of unity. Let $N$ stands for the outward unit normal of $\partial\Omega$. In particular, we have $N=(0,0,-1)$ on $\Gamma_{0}$ and $N=(0,0,1)$ on $\Gamma_{1}$.

In this paper, we prove:

Theorem 1.1**.**

Let $\Omega$ be defined as in (1.9). Let $(\eta,v,b)$ be the solution of (1.8) and $\delta\in(0,0.5)$. Assume that $v(0,\cdot)=v_{0}\in H^{2.5+\delta}(\Omega)$ and $b(0,\cdot)=b_{0}\in H^{2.5+\delta}(\Omega)$ be divergence free vector fields and $b_{0}\cdot N=0$ on $\partial\Omega$. Let

[TABLE]

Then for sufficiently small $\overline{\epsilon}$, there exists a $T>0$, depending only on $N(0)$ and $\overline{\epsilon}$ such that $N(t)\leq P(N(0))$ for all $t\in[0,T]$, provided the physical sign condition

[TABLE]

holds. Here, $P$ is a polynomial of its arguments.