Compressible flow with vacuum and physical singularity.
The first part is devoted to the study of Euler equations in one space dimension, in the compressible, isentropic, vacuum case with damping. The vacuum boundary is said to be physical if in a small neighborhood of the boundary \(0 <| dc^2/dx| < +\infty\), where \(c\) is the sound speed. That leads to the fact that the pressure has a non zero-bounded effect on the evolution of the vacuum boundary and exhibits a singular behaviour. In particular, for smooth initial data there is some ``waiting time'' before which the boundary does not move, and after that time it starts to move. Similar effects can be seen for Euler-Poisson system which is the goal of the second part of the article. For Euler equations, the authors consider the case when the vacuum boundary is the particle path through~\( x = a\). The characteristics degenerate through that vacuum boundary, and the idea is to introduce a change of variable to symmetrize the system while having non-degenerate propagation speeds (that is possible under the ``physical singularity'' condition). The authors are then able to prove a short-time existence result for smooth initial data (in~\( H^3\)), and to prove that the physical boundary condition holds for a short time. For the Euler-Poisson system, the authors perform the study in three space dimensions with physical symmetry (the idea is to model gaseous stars), and prove similar results, using again a coordinate transformation.
scientific article; zbMATH DE number 1757317