Existence of a Time Periodic Solution for the Compressible Euler Equation with a Time Periodic Outer Force in a Bounded Interval

In this paper, we study isentropic gas flow in a bounded interval and apply a time periodic outer force. This motion is described by the compressible Euler equation with the outer force. Our purpose in this paper is to prove the existence of a time periodic solution. When we prove the existence of the time periodic solution, we are faced with two difficult problems. One problem is to prove that initial data and the corresponding solutions at the time period are contained in the same bounded set. To overcome this, we employ an invariant region deduced from the mass and energy. This enable us to investigate the behavior of solutions in detail. In addition, this method provide us a decay estimate to suppresses the growth of solutions caused by the outer force. The second problem is to construct a continuous map from initial data to the corresponding solutions at the time period. We need the map to apply a fixed point theorem. To construct this, we introduce a new type Lax-Friedrichs scheme, which has a recurrence relation consisting of discretized approximate solutions. By virtue of the fixed point theorem, we can prove a existence of a fixed point, which represents a time periodic solution.