Existence and uniqueness of solutions for compressible magnetohydrodynamic equations with bounded density
- 1Shenzhen Polytechnic Univ., Shenzhen, China
Abstract
This article concerns the three-dimensional compressible magnetohydrodynamic equations on the torus in the presence of vacuum. We establish the existence and uniqueness of strong local solutions for low-regularity initial data without imposing an initial compatibility condition. More precisely, the initial data are assumed to satisfy 0≤ ρ0∈ L∞(𝕋3), u0,b0∈ H1(𝕋3), divb0=0. The main difficulties arise from the strong nonlinear coupling between the velocity and magnetic field and from the low regularity of the density. By introducing the effective viscous flux and deriving suitable time-weighted energy estimates, we establish a uniform local existence theory independent of the initial compatibility condition. For uniqueness, we estimate the density difference in a negative Sobolev space and introduce a coupled backward parabolic system to avoid derivatives of the density. The proof is completed by a logarithmic BMO estimate and an Osgood-type argument.
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BibTeX
@article{zheng2026existence,
title = {{Existence and uniqueness of solutions for compressible magnetohydrodynamic equations with bounded density}},
author = {Yasi Zheng},
journal = {Electronic Journal of Differential Equations},
year = {2026},
month = sep,
url = {http://ejde.math.txstate.edu/Volumes/2026/67/abstr.html},
note = {CC BY},
}
RIS
TY - JOUR
TI - Existence and uniqueness of solutions for compressible magnetohydrodynamic equations with bounded density
AU - Yasi Zheng
PY - 2026
DA - 2026/09/01
JO - Electronic Journal of Differential Equations
UR - http://ejde.math.txstate.edu/Volumes/2026/67/abstr.html
AB - This article concerns the three-dimensional compressible magnetohydrodynamic equations on the torus in the presence of vacuum. We establish the existence and uniqueness of strong local solutions for low-regularity initial data without imposing an initial compatibility condition. More precisely, the initial data are assumed to satisfy 0≤ ρ_0∈ L^∞(𝕋^3), u_0,b_0∈ H^1(𝕋^3), divb_0=0. The main difficulties arise from the strong nonlinear coupling between the velocity and magnetic field and from the low regularity of the density. By introducing the effective viscous flux and deriving suitable time-weighted energy estimates, we establish a uniform local existence theory independent of the initial compatibility condition. For uniqueness, we estimate the density difference in a negative Sobolev space and introduce a coupled backward parabolic system to avoid derivatives of the density. The proof is completed by a logarithmic BMO estimate and an Osgood-type argument.
SN - 1072-6691
ER -