Stability of non-constant equilibrium solutions for Euler–Maxwell equations

Abstract

We consider a periodic problem for compressible Euler–Maxwell equations arising in the modeling of magnetized plasmas. The equations are quasilinear hyperbolic and partially dissipative. It is proved that smooth solutions exist globally in time and converge toward non-constant equilibrium states as the time goes to infinity. This result is obtained for initial data close to the equilibrium states with zero velocity. The proof is based on an induction argument on the order of the derivatives of solutions in energy and time dissipation estimates. We also show the global stability with exponential decay in time of solutions near the equilibrium states for compressible Euler–Poisson equations.