Global Well-Posedness of Free Interface Problems for the Incompressible Inviscid Resistive MHD

Abstract

We consider the plasma-vacuum interface problem in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field. The lower plasma region is governed by the incompressible inviscid and resistive MHD, the upper vacuum region is governed by the pre-Maxwell equations, and the effect of surface tension is taken into account on the free interface. The global well-posedness of the problem, supplemented with physical boundary conditions, around the equilibrium is established, and the solution is shown to decay to the equilibrium almost exponentially. Our results reveal the strong stabilizing effect of the magnetic field as the global well-posedness of the free-boundary incompressible Euler equations, without the irrotational assumption, around the equilibrium is unknown. One of the key observations here is an induced damping structure for the fluid vorticity due to the resistivity and transversal magnetic field. A similar global well-posedness for the plasma-plasma interface problem is obtained, where the vacuum is replaced by another plasma.

Appendix A. Analytic tools

In this appendix we will collect the analytic tools which are used throughout the paper.

1.1 Harmonic extension

Define the specialized Poisson sum in $\mathbb {T}^2\times \mathbb {R}$ by (see [53])

$$\begin{aligned} \mathcal {P} f(x) {:}{=} \left\{ \begin{array}{lll} \displaystyle \sum _{\xi \in \mathbb {Z} ^2 } e^{2\pi i \xi \cdot x_h} \sum _{j=0}^m\alpha _j e^{- |\xi |\lambda _jx_3} \hat{f}(\xi ), &{} x_3> 0 \\ \displaystyle \sum _{\xi \in \mathbb {Z} ^2 } e^{2\pi i \xi \cdot x_h} e^{2\pi \left| \xi \right| x_3} \hat{f}(\xi ),&{} x_3\le 0, \end{array}\right. \end{aligned}$$

(A.1)

where

$$\begin{aligned} \hat{f}(\xi ) = \int _{\mathbb {T}^2} f(x_h) e^{-2\pi i \xi \cdot x_h},\quad \xi \in \mathbb {Z}^2 . \end{aligned}$$

(A.2)

Here $0<\lambda _0<\lambda _1<\cdots<\lambda _m<\infty$ for $m\in \mathbb {N}$, and $\alpha =(\alpha _0,\alpha _1,\dots ,\alpha _m)^T$ is the solution to

$$\begin{aligned} V(\lambda _0,\lambda _1,\dots ,\lambda _m)\,\alpha = (1,1,\dots ,1)^T, \end{aligned}$$

(A.3)

where V is the $(m+1) \times (m+1)$ Vandermonde matrix. The Poisson sum (A.1) is specialized in that $\mathcal {P} f$ is differentially continuous across $\Sigma$ up to any order as needed provided that m is sufficiently large. Moreover, the following estimate holds.

Lemma A.1

It holds that for all $s\in \mathbb {R}$,

$$\begin{aligned} \left\| \mathcal {P}f \right\| _{s} \lesssim \left| f\right| _{s-1/2}. \end{aligned}$$

(A.4)

Proof

One may refer to Lemma A.9 of [24] for instance. $\quad \square$

1.2 Time extension

The following lemmas allow one to extend the initial data to be time-dependent functions, which are “hyperbolic” versions of the “parabolic” ones in [24] with some minor modifications.

Lemma A.2

Suppose that $\partial _t^j \eta (0) \in H^{2N-j+3/2}(\Sigma )$ for $j=0,\dotsc ,2N-1$. There exists an extension $\eta ^0$ defined on $[0,\infty )$, achieving the initial data, so that

$$\begin{aligned} \sum _{j=0}^{2N+1}\sup _{[0,\infty ]}\left| \partial _t^j \eta ^0\right| ^2_{ {2N-j+3/2}}+\sum _{j=0}^{2N+2} \int _0^\infty \left| \partial _t^j \eta ^0\right| ^2_{ {2N-j +2}} \lesssim \sum _{j=0}^{2N-1} \left| \partial _t^j \eta (0)\right| ^2_{ {2N-j+3/2 }}. \end{aligned}$$

(A.5)

Proof

For each $j=0,\dotsc ,2N-1$, we denote $f_j = \partial _t^j \eta (0)$ and let $\varphi _j \in C_0^\infty (\mathbb {R}^{})$ be such that $\varphi _j^{(k)}(0) = \delta _{j,k}$ for $k=0,\dotsc ,2N-1$. Then $\eta ^0$ is constructed as a sum $\eta ^0 = \sum _{j=0}^{2N-1} F_j$, where $F_j$ is defined via its Fourier coefficients:

$$\begin{aligned} \hat{F}_j(\xi ,t) = \varphi _j(t \left\langle \xi \right\rangle ) \hat{f}_j(\xi )\left\langle \xi \right\rangle ^{-j},\ j=0,\dots ,2N-1, \end{aligned}$$

where $\left\langle \xi \right\rangle = \sqrt{1+ \left| \xi \right| ^2}$. It follows by modifying the proof of Lemma A.5 of [24] suitably that $\eta ^0$ satisfies the conclusion. $\quad \square$

Lemma A.3

Suppose that $\partial _t^j u(0) \in H^{2N-j}(\Omega _-)$ for $j=0,\dotsc ,2N-1$. There exists an extension $u^0$ defined on $[0,\infty )$, achieving the initial data, so that

$$\begin{aligned} \sum _{j=0}^{2N}\sup _{[0,\infty ]}\left\| \partial _t^j u^0\right\| ^2_{ {2N-j}}+\sum _{j=0}^{2N} \int _0^\infty \left\| \partial _t^j u^0\right\| ^2_{ {2N-j +1/2}} \lesssim \sum _{j=0}^{2N-1} \left\| \partial _t^j u(0)\right\| ^2_{ {2N-j }}. \end{aligned}$$

(A.6)

Proof

It follows similarly as Lemma A.2, by using additionally the usual theory of extensions and restrictions in Sobolev spaces between $H^k(\Omega _-)$ and $H^k(\mathbb {R}^3)$ for $k\ge 0$.

$\square$

Lemma A.4

Suppose that $\partial _t^j b(0) \in H^{2N-j+1}(\Omega _-)$ for $j=0,\dotsc ,2N-1$. There exists an extension $b^0$ defined on $[0,\infty )$, achieving the initial data, so that

$$\begin{aligned} \sum _{j=0}^{2N+1}\sup _{[0,\infty ]}\left\| \partial _t^j b^0\right\| ^2_{ {2N-j+1}}+\sum _{j=0}^{2N+1} \int _0^\infty \left\| \partial _t^j b^0\right\| ^2_{ {2N-j +3/2}} \lesssim \sum _{j=0}^{2N-1} \left\| \partial _t^j b(0)\right\| ^2_{ {2N-j+1 }}. \end{aligned}$$

(A.7)

Proof

It follows in the same way as Lemma A.3. $\quad \square$

1.3 Product estimates

The following standard estimates in Sobolev spaces are needed.

Lemma A.5

Let the domains below be either $\Omega _\pm$, $\Sigma$ or $\Sigma _\pm$, and d be the dimension.

• (1) Let $0\le r \le s_1 \le s_2$ be such that $s_1 > d/2$. Then

  $$\begin{aligned} \left\| fg\right\| _{H^r} \lesssim \left\| f\right\| _{H^{s_1}} \left\| g\right\| _{H^{s_2}}. \end{aligned}$$

  (A.8)

• (2) Let $0\le r \le s_1 \le s_2$ be such that $s_2 >r+ d/2$. Then

  $$\begin{aligned} \left\| fg\right\| _{H^r} \lesssim \left\| f\right\| _{H^{s_1}} \left\| g\right\| _{H^{s_2}}. \end{aligned}$$

  (A.9)

Lemma A.6

It holds that for $s>5/2$,

$$\begin{aligned} \left\| fg\right\| _{-1} \lesssim \left\| f\right\| _{-1} \left\| g\right\| _{{s}}. \end{aligned}$$

(A.10)

1.4 Poincaré-type inequality

The following Poincaré-type inequality related to ${\bar{B}}\cdot \nabla$ holds.

Lemma A.7

For any constant vector ${\bar{B}} \in \mathbb {R}^3$ with ${\bar{B}}_3\ne 0$, it holds that

$$\begin{aligned} \left\| f\right\| ^2_0 \le \frac{1}{{\bar{B}}_3^2}\left\| ({\bar{B}}\cdot \nabla )f\right\| ^2_0+\left| f\right| ^2_0 \end{aligned}$$

(A.11)

and

$$\begin{aligned} \left| f\right| ^2_0 \le \frac{1}{{\bar{B}}_3^2}\left\| ({\bar{B}}\cdot \nabla )f\right\| ^2_0+\left\| f\right\| ^2_0 . \end{aligned}$$

(A.12)

Proof

It follows by the fundamental theorem of calculus, see Lemma A.4 in [52]. $\square$

1.5 Normal trace estimates

The following $H^{-1/2}$ boundary estimate holds for functions satisfying $v\in L^2$ and ${{\,\mathrm{div}\,}}^{\varphi }v \in L^2$.

Lemma A.8

Assume that $\left\| \nabla \varphi \right\| _{L^\infty } \le C$, then

$$\begin{aligned} \left| v \cdot \mathcal {N}\right| _{-1/2}\lesssim \left\| v\right\| _0+\left\| {{\,\mathrm{div}\,}}^{\varphi }v\right\| _0 . \end{aligned}$$

(A.13)

Proof

One may refer to Lemma 3.3 in [24]. $\quad \square$

1.6 Hodge-type estimates

The following Hodge-type estimate holds, when the boundary conditions are not specified. Let the domain be either $\Omega _-$ or $\Omega +$ or $\Omega$.

Lemma A.9

Let $r\ge 1$ be an integer. Then it holds that

$$\begin{aligned} \left\| v\right\| _r\lesssim \left\| v\right\| _{0,r}+\left\| ({\text {curl}}v)_h \right\| _{r-1}+\left\| {\text {div}}v\right\| _{r-1} . \end{aligned}$$

(A.14)

Proof

Notice that (A.14) follows easily for $r=1$. Now for $r\ge 2$, applying the previous estimate of $r=1$ gives that for $\ell =1,\dots , r$,

$$\begin{aligned} \left\| v\right\| _{\ell ,r-\ell }&\lesssim \left\| v\right\| _{\ell -1,r-\ell +1}+\left\| ({\text {curl}}v)_h\right\| _{\ell -1,r-\ell }+\left\| {\text {div}}v\right\| _{\ell -1,r-\ell } \nonumber \\&\lesssim \left\| v\right\| _{\ell -1,r-\ell +1}+\left\| ({\text {curl}}v)_h\right\| _{r-1}+\left\| {\text {div}}v\right\| _{r-1} . \end{aligned}$$

(A.15)

By an induction argument on $\ell =1,\dots , r$, one gets (A.14). $\quad \square$