Time-periodic and stationary solutions to the compressible Hall-magnetohydrodynamic system

Abstract

We are concerned with the 3-D compressible Hall-magnetohydrodynamic system with a time-periodic external force in a periodic domain, and establish the existence of a strong time-periodic solution under some smallness and symmetry assumptions by adapting a new approach. The basic idea of the proof is the following. First, we prove the existence of a time-periodic solution to the linearized system by applying the Tychonoff fixed point theorem combined with the energy method and the decay estimates. From the details of the proof, we see that the initial data of the time-periodic solution to the linearized system lies in some convex hull. Then, we construct a set-value function, such that the fixed point of this function is a time-periodic solution of the compressible Hall-magnetohydrodynamic system. The existence of the fixed point is obtained by the Kakutani fixed point theorem. Moreover, we establish the uniqueness of the time-periodic solution and the existence of the stationary solution.