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Nonlinear Stability of MHD Contact Discontinuities with Surface Tension

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Abstract

We consider the motion of two inviscid, compressible, and electrically conducting fluids separated by an interface across which there is no fluid flow in the presence of surface tension. The magnetic field is supposed to be nowhere tangential to the interface. This leads to the characteristic free boundary problem for contact discontinuities with surface tension in three-dimensional ideal compressible magnetohydrodynamics (MHD). We prove the nonlinear structural stability of MHD contact discontinuities with surface tension in Sobolev spaces by a modified Nash–Moser iteration scheme. The main ingredient of our proof is deriving the resolution and tame estimate of the linearized problem in usual Sobolev spaces of sufficiently large regularity. In particular, for solving the linearized problem, we introduce a suitable regularization that preserves the transport-type structure for the linearized entropy and divergence of the magnetic field.

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Acknowledgements

The authors would like to thank the anonymous referees for comments and suggestions that improved the quality of the paper.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Koptyug av. 4, Novosibirsk, Russia, 630090

    Yuri Trakhinin

  2. Novosibirsk State University, Pirogova str. 1, Novosibirsk, Russia, 630090

    Yuri Trakhinin

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Tao Wang

  4. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

    Tao Wang

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Communicated by N. Masmoudi.

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The research of Yuri Trakhinin was supported by the Russian Science Foundation under Grant No. 20-11-20036. The research of Tao Wang was supported by the National Natural Science Foundation of China under Grants 11971359 and 11731008.

A Jump Conditions with or without Surface Tension

A Jump Conditions with or without Surface Tension

We assume that the surface \(\Sigma (t)\) is smooth with a well-defined unit normal \(\varvec{n}(t,x)\) and moves with the normal speed \(\mathcal {V}(t,x)\) at point \(x\in \Sigma (t)\) and time \(t\ge 0\). Let \(\Omega ^+(t)\) and \(\Omega ^-(t)\) denote the space domains occupied by the two conducting fluids at time t, respectively. Without loss of generality we assume that the unit normal \(\varvec{n}\) points into \(\Omega ^+(t)\). Piecewise smooth weak solutions of the compressible MHD equations (1.1)–(1.2) must satisfy the following MHD Rankine–Hugoniot conditions on the surface of discontinuity \(\Sigma (t)\) (see Landau–Lifshitz [18, §70]):

$$\begin{aligned}&-\mathcal {V}[\rho ]+\varvec{n}\cdot [\rho v]=0, \end{aligned}$$
(A.1a)
$$\begin{aligned}&-\mathcal {V}[\rho v]+\varvec{n}\cdot [\rho v\otimes v-H\otimes H ]+\varvec{n}[q]=0, \end{aligned}$$
(A.1b)
$$\begin{aligned}&-\mathcal {V}[H]-\varvec{n}\times [ v\times H]=0, \end{aligned}$$
(A.1c)
$$\begin{aligned}&-\mathcal {V}\big [\rho E+\tfrac{1}{2}|H|^2\big ]+\varvec{n}\cdot [ v(\rho E+p)+ H\times (v\times H)]=0, \end{aligned}$$
(A.1d)
$$\begin{aligned}&\,\varvec{n}\cdot [H]=0. \end{aligned}$$
(A.1e)

Here \([g]:=g^+-g^-\) denotes the jump in the quantity g across \(\Sigma (t)\) with

$$\begin{aligned} g^{\pm }(t,x):=\lim _{\epsilon \rightarrow 0^+}g(t,x\pm \epsilon \varvec{n}(t,x)) \quad \text {for }x\in \Sigma (t). \end{aligned}$$

The condition (A.1a) means that the mass transfer flux \(\mathfrak {j}:=\rho (v\cdot \varvec{n}-\mathcal {V})\) is continuous through \(\Sigma (t)\). We can rewrite (A.1) in terms of \(\mathfrak {j}\) as

$$\begin{aligned} \left\{ \begin{aligned}&{[}{\mathfrak {j}}{]}=0,\quad \mathfrak {j} [v_{\varvec{n}}]+[q]=0,\quad \mathfrak {j} [v_{\tau }]=H_{\varvec{n}}[H_{\tau }],\quad [H_{\varvec{n}}]=0,\\&\mathfrak {j} \left[ \tfrac{1}{\rho }H_{\tau }\right] =H_{\varvec{n}}[v_{\tau }],\quad \mathfrak {j}\left[ E+\tfrac{1}{2\rho }|H|^2\right] +[qv_{\varvec{n}}-(H\cdot v)H_{\varvec{n}}]=0, \end{aligned}\right. \end{aligned}$$
(A.2)

where \(v_{\varvec{n}}:=v\cdot \varvec{n}\) (resp. \(H_{\varvec{n}}:=H\cdot \varvec{n}\)) is the normal component of v (resp. H) and \(v_{\tau }\) (resp. \(H_{\tau }\)) is the tangential part of v (resp. H). If there is no flow across the discontinuity, that is, \(\mathfrak {j}=0\) on \(\Sigma (t)\), then compressible MHD permits two distinct types of characteristic discontinuities [18, §71]: tangential discontinuities (\(H_{\varvec{n}}|_{\Sigma (t)}= 0\)) and contact discontinuities (\(H_{\varvec{n}}|_{\Sigma (t)}\ne 0\)). For tangential discontinuities (or called current-vortex sheets), the jump conditions (A.2) become

$$\begin{aligned} H^{\pm }\cdot {\varvec{n}}=0,\quad [q]=0,\quad \mathcal {V}=v^+\cdot {\varvec{n}}=v ^-\cdot {\varvec{n}}\quad \text {on }\Sigma (t). \end{aligned}$$
(A.3)

Moreover, from (A.2), we obtain the following boundary conditions for MHD contact discontinuities:

$$\begin{aligned} H^{\pm }\cdot {\varvec{n}}\ne 0,\quad [p]=0,\quad [v]=[H]=0,\quad \mathcal {V}=v^+\cdot {\varvec{n}} \quad \text {on }\Sigma (t). \end{aligned}$$
(A.4)

With surface tension present on the interface \(\Sigma (t)\), we must take into account the corresponding surface force produced, so that the conditions (A.1b) and (A.1d) have to be modified respectively into (see Delhaye [13] or Ishii–Hibiki [16, Chapter 2])

$$\begin{aligned}&-\mathcal {V}[\rho v]+\varvec{n}\cdot [\rho v\otimes v-H\otimes H ]+\varvec{n}[q]=\mathfrak {s}\mathcal {H}\varvec{n},\\&-\mathcal {V}\big [\rho E+\tfrac{1}{2}|H|^2\big ]+\varvec{n}\cdot [ v(\rho E+p)+ H\times (v\times H)]=\mathfrak {s}\mathcal {H}\mathcal {V}, \end{aligned}$$

where \(\mathfrak {s}>0\) denotes the constant coefficient of surface tension and \(\mathcal {H}\) twice the mean curvature of \(\Sigma (t)\). Hence, for any interface with surface tension, the boundary conditions (A.2) should be replaced by

$$\begin{aligned} \left\{ \begin{aligned}&{[}{\mathfrak {j}}{]}=0,\quad \mathfrak {j} [v_{\varvec{n}}]+[q]=\mathfrak {s}\mathcal {H},\quad \mathfrak {j} [v_{\tau }]=H_{\varvec{n}}[H_{\tau }],\quad [H_{\varvec{n}}]=0,\\&\mathfrak {j}\left[ \tfrac{1}{\rho }H_{\tau }\right] =H_{\varvec{n}}[v_{\tau }],\quad \mathfrak {j}\left[ E+\tfrac{1}{2\rho }|H|^2\right] +[qv_{\varvec{n}}-(H\cdot v)H_{\varvec{n}}]=\mathfrak {s}\mathcal {H}\mathcal {V}. \end{aligned}\right. \end{aligned}$$

Considering \(\mathfrak {j}=0\) on \(\Sigma (t)\), we get two different possibilities of interfaces, viz.

  1. (a)

    current-vortex sheets with surface tension, for which the boundary conditions read as

    $$\begin{aligned} H^{\pm }\cdot {\varvec{n}}=0,\quad [q]=\mathfrak {s}\mathcal {H},\quad \mathcal {V}=v^+\cdot {\varvec{n}}=v ^-\cdot {\varvec{n}}\quad \text {on }\Sigma (t). \end{aligned}$$
    (A.5)
  2. (b)

    MHD contact discontinuities with surface tension, for which the boundary conditions read

    $$\begin{aligned} H^{\pm }\cdot {\varvec{n}}\ne 0,\quad [p]=\mathfrak {s}\mathcal {H},\quad [v]=[H]=0,\quad \mathcal {V}=v^+\cdot {\varvec{n}} \quad \text {on }\Sigma (t). \end{aligned}$$
    (A.6)

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Trakhinin, Y., Wang, T. Nonlinear Stability of MHD Contact Discontinuities with Surface Tension. Arch Rational Mech Anal 243, 1091–1149 (2022). https://doi.org/10.1007/s00205-021-01740-6

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