Influence Of Degenerate Damping On Classical Solutions To Multi-dimensional Compressible Euler Equations

In fluid dynamics, the Euler equations, named after Leonhard Euler, are a set of quasilinear hyperbolic equations governing the motion of a perfect fluid. The equations represent conservation of mass, momentum and energy, and can be viewed as particular Navier-Stokes equations with zero viscosity and zero thermal conductivity. The Euler equations are the most important and fundamental equations in describing the motion of the inviscous flow. The equations are widely used in oceanography, aerography, aerodynamics and so on.The smooth solution of compressible Euler equations will generally blow up in finite time and may accompany with the formation of shocks, rarefaction waves and so on. The formation of shocks is one of the most important phenomena of a fluid, the history of its study can be found in and the references therein. In multi-dimensions, for a special class of initial data, Sideris [55] has proved that the smooth solution in three space dimensions can develop singularities in finite time, and Rammaha in [53] has proved a blowup result in two space dimensions. For more extensive literature on the blowup results and the blowup mechanism, see [1-4,6,8-10,17,33,40,56,57,59,68,69] and the references therein.The motion of compressible flow through a porous medium can be modeled as the following compressible Euler equations with frictional damping where the frictional coefficient v> 0 is a constant. The system (0.0.3) admits a global smooth solution when the initial data is a small perturbation of the equilibrium state, and the solution to the Cauchy problem is expected to tend to diffusion waves governed by Darcy's law. In some sense, the damping can prevent the development of singularities in small amplitude smooth solutions. The global existence of the Cauchy problem or the initial-boundary value problem and the large time behavior of the solutions have been established in [7,15,24,26,27,29,32,34,35,38,41-49,51,58,60,61, 63-65,70]. Also see [16,25,28,36,37,50] for other results on non-smooth solutions.In this doctoral thesis, we will consider the global existence or blowup of the clas-sical solutions to the following compressible Euler equations with degenerate dampingwhere x ? Rd (or R+d), the factional coefficient ?(t)=?/ (1+t)? with ?> 0 and ?? 0 are constants, the amplitude e> 0 is sufficiently small. Since only the classical so-lutions are considered in this thesis, we may assume that the initial data (p0,u0) is smooth enough and has compact support. Here we also point that, when we study the initial-boundary value problem in the half space R+d, (0.0.4) should be supplied with the slip-boundary condition.At first, we divide ??0, ?> 0 into four cases:Case 1:0??< 1, ?> 0 for d= 2,3;Case 2:?= 1, ?> 3-d, for d= 2,3;Case 3:?= 1,?? 3-d for d= 2;Case 4:?> 1,?> 0 for d= 2,3.The main results of this thesis can be briefly summarized as follows:? In Case 1, there exists global solutions to the Cauchy problem in the whole space Rd or the initial-boundary value problem in the half space R+d.? In Case 2, the smooth solution to the Cauchy problem (0.0.4) exists globally when the initial data satisfies curl u0=0.? In Case 3 and Case 4, the smooth solution to the Cauchy problem (0.0.4) will blow up in finite time.In Chapter 2, we study the Cauchy problem of the 3-D irrotational flow.In Chapter 3, we consider the Cauchy problem of the multi-dimensional compress-ible Euler equations in the whole space Rd.In Chapter 4, we focus on the initial-boundary value problem of the multi-dimensional compressible Euler equations in the half space R_+~d.