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

Technical Summary: Strong Solutions to the Initial-Boundary-Value Problem of Compressible MHD Equations with Degenerate Viscosities and Far-Field Vacuum in 3D Exterior Domains

Problem Statement
This paper addresses the local existence and uniqueness of strong solutions to the initial-boundary-value problem (IBVP) for the three-dimensional compressible Magnetohydrodynamic (MHD) equations in an exterior domain $\Omega = \mathbb{R}^3 \setminus \bar{D}$. The system is characterized by two significant physical complexities:

1. Far-field vacuum: The initial density $\rho_0$ approaches zero as $|x| \to \infty$.
2. Degenerate viscosities: The shear and bulk viscosity coefficients are power functions of the density, $\mu(\rho) = \mu\rho^\delta$ and $\lambda(\rho) = \lambda\rho^\delta$, with $0 < \delta < 1$.

The system is subject to Navier-slip boundary conditions for the velocity and perfect conducting conditions for the magnetic field. The primary mathematical challenge arises from the singularity of terms involving $\rho^{\delta-1}$ and $\nabla\rho^{\delta-1}$ as the density vanishes, which is exacerbated by the strong coupling between the fluid motion and the magnetic field.

Methodology
The authors employ a combination of approximation techniques, energy estimates, and a novel reformulation of the system to handle the singularities and boundary conditions.

1. Variable Transformation: To manage the singular terms arising from the density-dependent viscosities and the magnetic field, the authors introduce new variables:

   • $\psi = \nabla \log \rho$, which helps in handling the gradient of the density.
   • $J = \rho^{-(1-\delta)/2} H$, a density-weighted average magnetic field. This variable is crucial as it transforms the magnetic field equation into a standard parabolic equation, effectively decoupling the singularity from the magnetic field's evolution in the estimates.

2. Reformulated System: The original MHD equations are rewritten in terms of $(\rho, u, H, J)$. The momentum equation is multiplied by $\rho^{-\delta}$ to isolate the singular terms, and the magnetic field equation is transformed into a parabolic equation for $J$ with specific boundary conditions derived from the perfect conducting conditions.

3. Approximation and A Priori Estimates:

   • The problem is first solved on a bounded annular domain $\Omega_R$ with strictly positive initial density to establish local classical solutions.
   • A series of uniform a priori estimates are derived for the smooth solutions. Key estimates include bounds on $\nabla \rho^{(\delta-1)/2}$, $\nabla u$, $\nabla H$, and the time derivatives of these quantities.
   • The authors utilize cut-off functions and trace theorems to handle boundary terms, particularly the boundary condition for $J$ which depends on $\psi \cdot n$. A critical step involves transforming the boundary term involving $J_t$ back to $H_t$ to utilize the integrability of $\nabla H_t$.

4. Uniqueness Proof: To prove uniqueness, the authors consider the difference between two solutions. A specific difficulty arises from the second-order derivatives of the density in the difference equations. To circumvent this, they introduce a transformation involving $g_i = \rho_i^{-\delta}$ and utilize the fact that the initial difference $\bar{g}|_{t=0} = 0$, allowing them to close the estimates without requiring direct control of the second derivatives of the density difference.

Key Contributions and Results
The main result, Theorem 1.1, establishes the local existence and uniqueness of strong solutions for regular large initial data satisfying specific decay conditions near the vacuum. The solution $(\rho, u, H)$ exists on a time interval $[0, T_0]$ and satisfies specific regularity classes, including $u \in L^\infty(0, T_0; D^1_0 \cap D^2) \cap L^2(0, T_0; D^3)$ and $H \in L^\infty(0, T_0; H^2)$.

Specific Technical Insights:

• Role of the Magnetic Field: A significant finding is that the magnetic field $H$ maintains a decay rate faster than the density $\rho$ throughout the time evolution. Specifically, if the initial data satisfies $J_0 = H_0 / \rho^{(1+\delta)/2}_0 \in H^1$, then $J \in L^\infty(0, T_0; H^1)$. This property is essential for controlling the singular terms associated with the magnetic field in the momentum equation.
• Boundary Behavior: The analysis demonstrates that the density retains a positive lower bound near the boundary $\partial \Omega$ for a short time, despite the far-field vacuum. This is achieved through the derived estimate $\rho^{(\delta-1)/2} \in L^\infty(0, T_0; L^6(\Omega_0))$.
• Generalization: The results generalize the local theory for compressible Navier-Stokes equations with degenerate viscosities (specifically the work of Li-Lü-Yuan [24]) to the compressible MHD system.

Significance
The paper claims that for the case $0 < \delta < 1$, previous approaches used for the Navier-Stokes equations or MHD with $\delta \ge 1$ are insufficient due to the strong singularity of $\rho^{\delta-1}$ and the new difficulties introduced by the magnetic field. This work fills a gap in the literature regarding the IBVP for compressible MHD with degenerate viscosities and far-field vacuum in 3D exterior domains. The introduction of the density-weighted magnetic field $J$ and the demonstration of its decay properties provide a new mechanism for handling singularities in coupled fluid-magnetic systems with degenerate coefficients.

The authors note that their methods can be adapted to the Cauchy problem (without boundaries) and the case of zero magnetic dissipation ($\eta=0$), though the primary focus remains on the IBVP with non-zero dissipation.