Some Studies On The Multi-dimensional Compressible Fluid Flow

In this thesis we study the multi-dimensional compressible fluid flow,including two-dimensional Riemann problem for the compressible Euler Equations and the L~1-stability of weak solutions together with the global existence of spherically symmetric weak solutions for the compressible Navier-Stokes-Poisson(NSP) equations.There are four chapters in this thesis.In Chapter 1,we introduce some related physical and mathematical backgrounds about the equations and problems we studied and our main results obtained in this thesis.In Chapter 2,we construct patches of solutions,in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves,to the two-dimensional Euler equations.The semi-hyperbolic patch of solutions is different from the planar rarefaction,shock,simple,or pseudo-subsonic waves.This type of solutions appears in the transonic flow over an airfoil and Guderley reflection,and is common in the numerical solutions of Riemann problems.In Chapter 3,we consider the isentropic compressible NSP equations arising from transport of charged particles or motion of gaseous stars in astrophysics.We are interested in the case that the viscosity coefficients depend on the density and shall degenerate in the appearance of(density) vacuum.We prove the L~1-stability of weak solutions for arbitrarily large data on multi-dimensional bounded or periodic domain or whole space.In Chapter 4,we prove the existence of global weak solutions to the compressible NSP equations with density-dependent viscosity coefficients when the initial data are large and spherically symmetric by constructing suitable aproximate solutions. The solutions are obtained as limits of solutions in annular regions between two balls.