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Weak solutions to a compressible viscous non-resistive MHD equations with general boundary data

Yang Li, Young-Sam Kwon, Yongzhong Sun

TL;DR

The authors establish global-in-time weak solutions for the 2D compressible viscous non-resistive MHD equations in domains with general inflow-outflow boundary data, under a 1D isentropic pressure law with exponent $\gamma>1$. They construct approximate solutions using Galerkin methods and diffusion-augmented transport, derive uniform energy and integrability estimates, and pass to the limit via renormalization, Bogovskii-type pressure estimates, and the effective viscous flux, proving strong convergence of density and magnetic field. They also prove a weak-strong uniqueness principle through a relative energy framework, showing that a weak solution coincides with any strong solution sharing the same initial and boundary data. The results extend prior Dirichlet-boundary analyses to open-fluid settings and provide a rigorous basis for convergence and error analysis of numerical schemes in open-boundary MHD problems.

Abstract

This paper is concerned with a compressible MHD equations describing the evolution of viscous non-resistive fluids in piecewise regular bounded Lipschitz domains. Under the general inflow-outflow boundary conditions, we prove existence of global-in-time weak solutions with finite energy initial data. The present result extends considerably the previous work by Li and Sun [\emph{J. Differential Equations.}, 267 (2019), pp. 3827-3851], where the homogeneous Dirichlet boundary condition for velocity field is treated. The proof leans on the specific mathematical structure of equations and the recently developed theory of open fluid systems. Furthermore, we establish the weak-strong uniqueness principle, namely a weak solution coincides with the strong solution on the lifespan of the latter provided they emanate from the same initial and boundary data. This basic property is expected to be useful in the study of convergence of numerical solutions.

Weak solutions to a compressible viscous non-resistive MHD equations with general boundary data

TL;DR

The authors establish global-in-time weak solutions for the 2D compressible viscous non-resistive MHD equations in domains with general inflow-outflow boundary data, under a 1D isentropic pressure law with exponent . They construct approximate solutions using Galerkin methods and diffusion-augmented transport, derive uniform energy and integrability estimates, and pass to the limit via renormalization, Bogovskii-type pressure estimates, and the effective viscous flux, proving strong convergence of density and magnetic field. They also prove a weak-strong uniqueness principle through a relative energy framework, showing that a weak solution coincides with any strong solution sharing the same initial and boundary data. The results extend prior Dirichlet-boundary analyses to open-fluid settings and provide a rigorous basis for convergence and error analysis of numerical schemes in open-boundary MHD problems.

Abstract

This paper is concerned with a compressible MHD equations describing the evolution of viscous non-resistive fluids in piecewise regular bounded Lipschitz domains. Under the general inflow-outflow boundary conditions, we prove existence of global-in-time weak solutions with finite energy initial data. The present result extends considerably the previous work by Li and Sun [\emph{J. Differential Equations.}, 267 (2019), pp. 3827-3851], where the homogeneous Dirichlet boundary condition for velocity field is treated. The proof leans on the specific mathematical structure of equations and the recently developed theory of open fluid systems. Furthermore, we establish the weak-strong uniqueness principle, namely a weak solution coincides with the strong solution on the lifespan of the latter provided they emanate from the same initial and boundary data. This basic property is expected to be useful in the study of convergence of numerical solutions.
Paper Structure (13 sections, 8 theorems, 85 equations)

This paper contains 13 sections, 8 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Approximating equations
  4. The approximating equations
  5. Solvability of the approximating equations
  6. Passing to the limits
  7. Uniform-in-$\delta$ estimates
  8. Pressure estimates
  9. Strong convergence of $\{\varrho_{\delta}\}_{\delta>0}$ and $\{b_{\delta}\}_{\delta>0}$
  10. Weak-strong uniqueness principle
  11. Relative energy inequality
  12. Gronwall-type argument
  13. Conclusion and remarks

Key Result

Theorem 2.1

Let $\Omega\subset \hbox{R}^2$ be a piecewise regular bounded Lipschitz domain and $\gamma>1$. Suppose that Then there exists a global weak solution to (in2)-(in4) in the sense of Definition def:1.

Theorems & Definitions (11)

  • Definition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Lemma 3.1
  • Proposition 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Corollary 4.1
  • ...and 1 more