Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum
The authors study the mathematical model to the 3D compressible isentropic MHD flows. These flows are governed by the Cauchy problem NEWLINE\[NEWLINE\begin{aligned} & \frac{\partial \rho}{\partial t}+\text{div}\,(\rho u)=0, \\ & \frac{\partial }{\partial t}(\rho u)+\text{div}\,(\rho u\otimes u)+\nabla P(\rho)= (\nabla \times H)\times H+\mu\Delta u+(\mu+\lambda)\nabla\text{div}\,u, \\ & \frac{\partial H}{\partial t}-\nabla\times(u\times H)=-\nabla\times(\nu\nabla\times H),\quad \text{div}\,H=0 \end{aligned}NEWLINE\]NEWLINE for \(t>0\) and \(x\in \mathbb{R}^3\), NEWLINE\[NEWLINE (\rho,u,H)\rightarrow (\bar{\rho},0,0)\quad \text{as}\;|x|\rightarrow \infty,\;t>0, \tag{infc}NEWLINE\]NEWLINE NEWLINE\[NEWLINE (\rho,u,H)(x,0)=(\rho_0,u_0,H_0)(x),\quad x\in \mathbb{R}^3. \tag{ic}NEWLINE\]NEWLINE Here the density \(\rho\), the velocity \(u\), the pressure \(P\) and the magnetic field \(H\) are unknown unctions. The viscosity coefficients \(\lambda,\mu\) and the resistivity coefficient \(\nu\) are given constants. \(\rho_0,u_0,H_0\) are given functions and \(\bar{\rho}\geq 0\) is the given constant. The pressure \(P\) satisfies to the power law NEWLINE\[NEWLINE P(\rho)=A\rho^\gamma \quad \text{with}\;A>0,\;\gamma>1. NEWLINE\]NEWLINENEWLINENEWLINEIt is proved that if the initial data \((\rho_0,u_0,H_0)\) are small in the sense of energy then the Cauchy problem ({mhd1}), ({infc}), ({ic}) has the unique global solution. The large-time behavior of the global solution is investigated in \(L^q\) spaces. The proof is based on a priory estimates as usual.
scientific article; zbMATH DE number 6200933