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

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.

💡 Research Summary

This paper addresses the initial‑boundary‑value problem (IBVP) for the three‑dimensional compressible magnetohydrodynamic (MHD) system posed in an exterior domain Ω = ℝ³ \ ¯D, where D is a bounded simply‑connected region. The fluid is viscous with density‑dependent viscosities of the form μ(ρ)=μ ρ^δ, λ(ρ)=λ ρ^δ, where the exponent satisfies 0 < δ < 1, i.e., the viscosities degenerate as the density approaches vacuum. The initial state is assumed to contain far‑field vacuum: (ρ, u, H) → (0, 0, 0) as |x|→∞. The velocity satisfies Navier‑slip boundary conditions (u·n=0, curl u×n=−A(x)u) on ∂Ω, while the magnetic field obeys perfect‑conductor conditions (H·n=0, curl H×n=0).

The main goal is to prove the local existence and uniqueness of strong solutions for large, regular initial data that satisfy a compatibility condition involving the weighted momentum L(u₀)=ρ₀^{1−δ/2}g. The authors introduce two auxiliary variables: ψ = ∇log ρ, which captures the logarithmic gradient of the density, and J = H / ρ^{1+δ/2}, a density‑weighted average magnetic field. This change of variables rewrites the most singular terms—those containing ρ^{−1+δ} ∇ρ and ρ^{−1+δ} H·∇H—into expressions involving ψ and J that are more amenable to energy estimates. In particular, J satisfies a standard parabolic equation with source terms that are at most first‑order in spatial derivatives, while ψ appears only linearly in the boundary condition for J.

The analysis proceeds by constructing smooth approximate problems on truncated annular domains Ω_R = Ω ∩ B_R, with strictly positive initial density ρ₀^ε > 0 to avoid complete degeneracy. For each ε and R, a classical solution is obtained via Galerkin approximation and fixed‑point arguments. Uniform a priori estimates independent of ε and R are then derived, which allow passage to the limit ε→0, R→∞ and yield a solution of the original IBVP.

Four layers of a priori estimates are established:

1. Basic L²‑energy and mass conservation: Using the continuity equation and the momentum equation multiplied by u, the authors obtain uniform bounds for ρ, u, and H in L². The degenerate viscosity term μ ρ^δ|∇u|² is controlled by exploiting ψ∈L⁶ and ρ^{δ−½}∈L⁶, guaranteeing integrability even where ρ→0.

2. First‑order Sobolev bounds for ψ and J: The equation for ψ follows from differentiating the continuity equation, while J satisfies a parabolic equation (1.20). Multiplying these equations by ψ and J respectively yields L²‑in‑time, H¹‑in‑space bounds for both variables. The boundary condition curl J×n = −(1+δ/2)(ψ·n)J couples ψ and J; the authors show that ψ·n remains bounded on ∂Ω because ρ retains a positive lower bound near the boundary (Lemma 2.5). This enables the use of trace theorems to control the boundary contributions.

3. Higher‑order Sobolev estimates: By applying div‑curl decomposition and the Helmholtz‑Leray projection, the authors obtain H²‑bounds for u and H. The viscous operator L u = μΔu+(μ+λ)∇div u together with Korn’s inequality yields control of ∇²u. For J, the second‑order terms involving div ψ = Δ log ρ are handled using the already established ψ‑bounds and careful commutator estimates. The magnetic diffusion term ηΔH provides direct H²‑regularity for H, which in turn assists in estimating J_t via the relation J_t = H_t ρ^{−1−δ/2} − H u·∇ρ^{−1−δ/2} + (1+δ/2)H ρ^{−1−δ/2} div u.

4. Time‑derivative estimates and continuity: The authors differentiate the momentum, induction, and J‑equations in time, multiply by the respective time derivatives, and integrate. The most delicate term is the boundary integral ∫_{∂Ω}(curl J×n)·J_t dS, which is transformed using the expression for J_t in terms of H_t and the density‑weighted factors. Because ρ stays bounded away from zero near ∂Ω, the trace theorem applies, and the integral is absorbed into the energy inequality. Grönwall’s lemma then yields uniform bounds for u_t, H_t, J_t, and the weighted quantities ρ^{1−δ/2}u_t, ρ^{1−δ/2}H_t.

Collecting all estimates, the authors prove that for any initial data satisfying (1.12)–(1.13) there exists a time T₀ > 0 such that a unique strong solution (ρ, u, H) exists on