Institute of Applied Mathematics
Shenzhen Polytechnic University
Shenzhen 518055, China
email: yszheng@szpu.edu.cn
Yasi Zheng
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
\(0\leq \rho_0\in L^\infty(\mathbb T^3),
u_0,b_0\in H^1(\mathbb T^3), \text{div}b_0=0\).
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.
Submitted June 18, 2026. Published September 3, 2026.
Math Subject Classifications: 35D35, 76N10, 76W05.
Key Words: Compressible magnetohydrodynamic equations; existence and uniqueness;
effective viscous flux; logarithmic Gronwall inequality
DOI: 10.58997/ejde.2026.67
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Yasi Zheng Institute of Applied Mathematics Shenzhen Polytechnic University Shenzhen 518055, China email: yszheng@szpu.edu.cn |
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