Some Studies On The Mathematical Theory Of Compressible Fluid Mechanics

There are important physical background and application value on the study of fluid mechanics equations. The research is one of the most important research fields on the theory of nonlinear partial differential equation. It can make us understand the motion of fluid particles by the theory or numeri-cal computation method clearly. Navier-Stokes equation is one of the most basic equations of the fluid mechanics equations. In this paper, we study the existence of weak solutions, as well as dynamics of vacuum states for Navier-Stokes equation and the existence of weak solutions reduced gravity two-and-a-half layer model. The main result are as follows:· Global existence of spherically symmetric weak solutions to the free boundary value problem of 3D isentropic compressible Navier-Stokes equa-tions(CNS). we prove the global existence of spherically symmetric weak so-lutions to the free boundary problem for the CNS with vacuum and free boundary separating fluids and vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time.· The dynamics of vacuum states for 1D compressible Navier-Stokes equations are considered. For any global entropy weak solution, we show that the flow density is continuous on both space and time, and is positive everywhere for all the time, if no vacuum state exists initially, (i.e., non-formation of vacuum states happens). Furthermore, we prove that there is a global weak solution which contains one piece of discontinuous finite vacuum for some time period, meanwhile the vacuum is shown to be compressed at an algebraic rate and then vanishes within finite time.·Global existence of weak solutions to a reduced gravity two-and-a-half layer model, We obtain the existence of global weak solution to a reduced gravity two and a half layer model in one dimensional bounded spatial domain ?= (0,1) or periodic domain ? = T1. Also, we show that any possible vacuum state has to vanish within finite time, then the weak solution becomes a unique strong one.