可压缩欧拉方程在不变子空间中的精确解∗

摘要

The Euler equations are a very important model in fluid mechanics, which have been wide used in many areas. Constructing their explicit solutions is a very significant part in mathematical physics. Explicit solutions can provide the concrete examples to understand their nonlinear phenomena and physical implications. This paper is devoted to construct the explicit solutions to compressible Euler equations by using the invariant subspace method. In the sense of variable changes, the invariant conditions yield the invariant subspaces related to compressible Euler equations. On these invariant subspaces, they are reduced to systems of first-order ordinary differential equations. Then some explicit solutions of compressible Euler equations are obtained by solving these systems.%欧拉方程是流体力学中非常重要的模型，被广泛应用于许多领域。构造它的精确解是数学物理中非常有意义的工作。精确解可以为理解它的非线性现象和物理意义提供具体的例子。本文旨在通过不变子空间方法构造可压缩欧拉方程的精确解。在变量变换意义下，由不变条件给出与可压缩方程相关的不变子空间；在这些不变子空间中，它被约化为一阶常微分方程组；通过求解这些常微分方程组，最终得到可压缩欧拉方程的一些精确解。