Stability of Non-constant Equilibrium Solutions for Compressible Viscous and Diffusive MHD Equations with the Coulomb Force

Abstract

We consider stability problems for the compressible viscous and diffusive magnetohydrodynamic (MHD) equations arising in the modeling of magnetic field confinement nuclear fusion. In the first part, we investigate the Cauchy problem to the barotropic MHD system. With the help of the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy estimates, we prove that smooth solutions exist globally in time near the non-constant equilibrium solutions. We also obtain the asymptotic behavior of solutions when the time goes to infinity. The result shows that gradients of both the velocity and the magnetic field converge to the equilibrium solutions with the same norm $$ \Vert \cdot \Vert _{H^{s-3}}$$, while the density converge with stronger norm $$ \Vert \cdot \Vert _{H^{s-1}}$$. In the second part, the initial value problem to the full MHD system is studied. By means of the techniques of choosing a non-diagonal symmetrizer and elaborate energy estimates, we prove the existence and uniqueness of global solutions to the system when the initial data are close to the non-constant equilibrium states. We find that both the density and temperature converge to the equilibrium states with the same norm $$ \Vert \cdot \Vert _{H^{s-1}}$$. These phenomena on the charge transport show the essential relationship of the equations between the barotropic and the full MHD systems.