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

Abstract

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.