LINEAR WAVES THAT EXPRESS THE SIMPLEST POSSIBLE PERIODIC STRUCTURE OF THE COMPRESSIBLE EULER EQUATIONS

• Published in: Acta mathematica scientia 2009-11, Vol.29 (6), p.1749-1766
• Main Authors: Temple, Blake, Young, Robin
• Format: Article
• Language: English
• Subjects: 35L65; 76N10; compressible Euler; conservation laws; Euler方程组; periodic solutions; 可压缩Euler方程; 周期结构; 扩展方程; 欧拉方程组; 波浪结构; 线性波; 非线性解
• ISSN: 0252-9602; 1572-9087
• DOI: 10.1016/S0252-9602(10)60015-X

In this paper we show how the simplest wave structure that balances compression and rarefaction in the nonlinear compressible Euler equations can be represented in a solution of the linearized compressible Euler equations. Such waves are exact solutions of the equations obtained by linearizing the compressible Euler equations about the periodic extension of two constant states separated by entropy jumps. Conditions on the states and the periods are derived which allow for the existence of solutions in the Fourier 1-mode. In [3, 4, 5] it is shown that these are the simplest linearized waves such that, for almost every period, they are isolated in the kernel of the linearized operator that imposes periodicity, and such that they perturb to nearby nonlinear solutions of the compressible Euler equations that balance compression and rarefaction along characteristics in the formal sense described in [3]. Their fundamental nature thus makes them of interest in their own right.