Formation of Singularities and Existence of Global Continuous Solutions for the Compressible Euler Equations

• Published in: arXiv.org 2021-11
• Main Authors: Chen, Geng, Chen, Gui-Qiang G, Zhu, Shengguo
• Format: Article
• Language: English
• Subjects: Cauchy problems; Compressible fluids; Compressive strength; Data compression; Euler-Lagrange equation; Eulers equations; Far fields; Mathematical analysis; Rarefaction; Shock waves; Singularities
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1905.07758

We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional non-isentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum -- there exists a compression in the initial data. For the non-isentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum -- there exists a strong compression in the initial data. Furthermore, we identify two new phenomena -- decompression and de-rarefaction -- for the non-isentropic Euler flows, different from the isentropic flows, via constructing two respective solutions. For the decompression phenomenon, we construct a first global continuous non-isentropic solution, even though initial data contain a weak compression, by solving an inverse Goursat problem, so that the solution is smooth, except on several characteristic curves across which the solution has a weak discontinuity (i.e., only Lipschitz continuity). For the de-rarefaction phenomenon, we construct a continuous non-isentropic solution whose initial data contain isentropic rarefactions (i.e., without compression) and a locally stationary varying entropy profile, for which the solution still forms a shock wave in a finite time.