Global solutions to the two-dimensional compressible MHD equations with density-dependent bulk viscosity
Abstract
We study the two-dimensional periodic resistive compressible magnetohydrodynamic equations with constant shear viscosity and magnetic diffusivity, bulk viscosity ρβ, and pressure ργ. For every β>1 and γ>1, we construct global finite-energy renormalized weak solutions from strictly positive bounded W1,q densities, q>2, and arbitrary H1 velocity and magnetic data. The construction combines approximation-independent finite density moments, generalized effective-flux identities, and a joint logarithmic renormalization of the density and its entropy defect. An augmented relative energy containing the β-power density potential yields weak--strong uniqueness against positive H2 strong solutions for the same full pressure range. We also establish global strong existence for a large-background family when γ>max{5−2β,β+1}. Writing m=∫ρ0, the density perturbation is bounded in the scaled norm m(γ−3)/2|ρ0−m|H2, while the fixed H2 velocity and magnetic norms need not be small. For sufficiently large m, frequency-dependent acoustic dissipation, staged vorticity weights, and a separate initial-layer flux estimate close a simultaneous bootstrap. The admissible pressure exponents have infimum 7/3, with the endpoint excluded. For β>2 the relevant damping rate is the minimum of the viscous and pressure-relaxation rates; keeping both rates extends the argument without an upper restriction on β. The strong theorem has no upper restriction on γ under this scaled density hypothesis; for fixed additive density perturbations it applies in the stated lower range up to γ=3.
Related
- Singular viscosity-exponent limit for one-dimensional compressible Navier-Stokes equations with far field vacuum
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BibTeX
@misc{li2026global,
title = {{Global solutions to the two-dimensional compressible MHD equations with density-dependent bulk viscosity}},
author = {Yachun Li and Peng Lu and Zhaoyang Shang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05312},
eprint = {2610.05312},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05312},
note = {CC0},
}
RIS
TY - UNPB
TI - Global solutions to the two-dimensional compressible MHD equations with density-dependent bulk viscosity
AU - Yachun Li
AU - Peng Lu
AU - Zhaoyang Shang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05312
UR - https://arxiv.org/abs/2610.05312
AB - We study the two-dimensional periodic resistive compressible magnetohydrodynamic equations with constant shear viscosity and magnetic diffusivity, bulk viscosity ρ^β, and pressure ρ^γ. For every β>1 and γ>1, we construct global finite-energy renormalized weak solutions from strictly positive bounded W^1,q densities, q>2, and arbitrary H^1 velocity and magnetic data. The construction combines approximation-independent finite density moments, generalized effective-flux identities, and a joint logarithmic renormalization of the density and its entropy defect. An augmented relative energy containing the β-power density potential yields weak--strong uniqueness against positive H^2 strong solutions for the same full pressure range. We also establish global strong existence for a large-background family when γ>max{5−2β,β+1}. Writing m=∫ρ_0, the density perturbation is bounded in the scaled norm m^(γ−3)/2|ρ_0−m|_H^2, while the fixed H^2 velocity and magnetic norms need not be small. For sufficiently large m, frequency-dependent acoustic dissipation, staged vorticity weights, and a separate initial-layer flux estimate close a simultaneous bootstrap. The admissible pressure exponents have infimum 7/3, with the endpoint excluded. For β>2 the relevant damping rate is the minimum of the viscous and pressure-relaxation rates; keeping both rates extends the argument without an upper restriction on β. The strong theorem has no upper restriction on γ under this scaled density hypothesis; for fixed additive density perturbations it applies in the stated lower range up to γ=3.
ER -