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Strong solutions to the initial-boundary-value problem of compressible MHD equations with degenerate viscosities and far field vacuum in 3D exterior domains

Jiaxu Li, Boqiang Lü, Bing Yuan

TL;DR

This work addresses the local well-posedness of strong solutions to the IBVP for the 3D compressible MHD equations with density-dependent degenerate viscosities in exterior domains, allowing far-field vacuum. It introduces the density-weighted magnetic field J=ρ^{−(1+δ)/2}H and reformulates the system to derive robust a priori estimates, particularly handling singular terms from ρ^{δ−1}. The authors prove local existence and uniqueness for initial data with possible vacuum (0<δ<1, γ>1), and show that J remains in L∞(0,T0;H^1) and decays faster than ρ^{(1+δ)/2}, highlighting the magnetic field’s stabilizing influence. The results extend the theory of density-dependent viscous flows to MHD in exterior domains and offer insights into how magnetic effects can control singular behavior near vacuum regions.

Abstract

This paper concerns the initial-boundary-value problem (IBVP) of the compressible Magnetohydrodynamic (MHD) equations in 3D exterior domains with Navier-slip boundary conditions for the velocity and perfect conducting conditions for the magnetic field. For the case that the density approaches far-field vacuum initially and the viscosities are power functions of the density (ρ}δ with 0 < δ < 1), the local existence and uniqueness of strong solutions to the IBVP is established for regular large initial data. In particular, in contrast to the local theory of compressible Navier-Stokes equation Li-Lü-Yuan [24], we show that the magnetic field maintains the initial quality of decaying faster rate than density throughout the time evolution, which reveals the role of the magnetic field in handling singularities arising from density-dependent viscosities.

Strong solutions to the initial-boundary-value problem of compressible MHD equations with degenerate viscosities and far field vacuum in 3D exterior domains

TL;DR

This work addresses the local well-posedness of strong solutions to the IBVP for the 3D compressible MHD equations with density-dependent degenerate viscosities in exterior domains, allowing far-field vacuum. It introduces the density-weighted magnetic field J=ρ^{−(1+δ)/2}H and reformulates the system to derive robust a priori estimates, particularly handling singular terms from ρ^{δ−1}. The authors prove local existence and uniqueness for initial data with possible vacuum (0<δ<1, γ>1), and show that J remains in L∞(0,T0;H^1) and decays faster than ρ^{(1+δ)/2}, highlighting the magnetic field’s stabilizing influence. The results extend the theory of density-dependent viscous flows to MHD in exterior domains and offer insights into how magnetic effects can control singular behavior near vacuum regions.

Abstract

This paper concerns the initial-boundary-value problem (IBVP) of the compressible Magnetohydrodynamic (MHD) equations in 3D exterior domains with Navier-slip boundary conditions for the velocity and perfect conducting conditions for the magnetic field. For the case that the density approaches far-field vacuum initially and the viscosities are power functions of the density (ρ}δ with 0 < δ < 1), the local existence and uniqueness of strong solutions to the IBVP is established for regular large initial data. In particular, in contrast to the local theory of compressible Navier-Stokes equation Li-Lü-Yuan [24], we show that the magnetic field maintains the initial quality of decaying faster rate than density throughout the time evolution, which reveals the role of the magnetic field in handling singularities arising from density-dependent viscosities.
Paper Structure (6 sections, 13 theorems, 166 equations)

This paper contains 6 sections, 13 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A priori estimate
  4. Proof of Theorem 1.1
  5. Existence of the strong solution
  6. Uniqueness of the strong solution

Key Result

Theorem 1.1

For parameters $(\gamma,\delta)$ satisfy If the initial data $(\rho_0, u_0, H_0)$ satisfies and the compatibility condition: for some $g\in L^2$. Then there exist a positive time $T_0>0$ such that the IBVP 1-7 has a unique strong solution $(\rho, u, H)$ on $[0,T_0]\times \Omega$ satisfying

Theorems & Definitions (30)

  • Remark 1.1
  • Definition 1.1
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • ...and 20 more