Narrow Results

In this paper, we study the axially symmetric piston problem for compressible fluids when the velocity of the piston is a perturbation of a constant. Under the assumptions that both the velocity of the piston and the density of the gas outside the piston are small, we prove the global existence of a shock front solution by using a modified Glimm scheme.

Conical shock wave is generated when a sharp conical projectile flies supersonically in the air. We study the linear stability and existence of steady conical shock waves in supersonic flow for the equations of complete Euler system in 3D non-isentropic gas-dynamics.

We study the initial-boundary value problem for the general non-isentropic 3D Euler equations with data which are incompatible in the classical sense, but are “rarefaction-compatible”. We show that such data are also rarefaction-compatible of infinite order and the initial-boundary value problem has a piece-wise smooth solution containing a rarefaction wave.

We investigate the uniqueness of entropy solution to 2D Riemann problem of compressible isentropic Euler system with maximum density constraint. The constraint is imposed with a singular pressure. Given initial data for which the standard self-similar solution consists of one shock or one shock and one rarefaction wave, it turns out that there exist infinitely many admissible weak solutions. This extends the result of Markfelder and Klingenberg in [S. Markfelder and C. Klingenberg, The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock, Arch. Ration. Mech. Anal. 227(3) (2018) 967–994] for classical Euler system to the case with maximum density constraint. Also some estimates on the density of these solutions are given to describe the behavior of solutions near congestion.

The shock reflection problem is discussed, which is the basic phenomenon in fluid dynamics and related to quasilinear hyperbolic partial differential equations.