Existence of a Time Periodic Solution for the Compressible Euler Equation with a Time Periodic Outer Force in a Bounded Interval
Existence of a Time Periodic Solution for the Compressible Euler Equation with a Time Periodic Outer Force in a Bounded Interval
复制标题
有界区间内具有时间周期外力的可压缩欧拉方程时间周期解的存在性
DOI：
10.1007/s00205-023-01874-9
复制
发表时间：
2023
影响因子：
2.5
通讯作者：
Naoki
中科院分区：
文献类型：
--
作者：
Tsuge;Naoki
本文研究有界区间内的等熵气体流动，并施加一个时间周期外力。这种运动由具有外力的可压缩欧拉方程描述。本文的目的是证明时间周期解的存在性。在证明时间周期解的存在性时，我们面临两个困难的问题。一个问题是要证明初始数据和相应的解决方案，在该时间段包含在同一个有界集。为了克服这一点，我们采用了一个不变的区域推导出的质量和能量。这使我们能够详细研究解决方案的行为。此外，该方法还提供了一种衰减估计来抑制外力引起的解的增长。第二个问题是构造一个从初始数据到该时间段对应解的连续映射。我们需要映射来应用不动点定理。为了构造这个，我们引入了一种新型的Lax-Friedrichs格式，它具有由离散近似解组成的递归关系。利用不动点定理，我们可以证明一个不动点的存在性，它代表一个时间周期解。
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.