Initial Trace For Degenerate Parabolic Equations And Global Solutions For Compressible Navier-Stokes Equations

This paper consists of two parts. Part 1 is the second chapter in the paper, and to study the existence of an initial trace of nonnegative solutions for the following problemwe prove that the initial trace is an outer regular Borel measure which may not be locally bounded for some values of parameters p,q and m. We study also the corresponding Cauchy problems with a given generalized Borel measure as initial data.The second part is devided into two chapters, it's main aim is to study the global solutions to 1-d compressible Navier-Stokes equations, and it is organized as follows:In Chapter 2, we introduce the second boundary value problem for the compressible Navier-Stokes equations in the half space. We also prove the existence of global solutions when u_+≥0, and the method we used is the basic energy eatimate and the continuity argument. The difficulties we haved overcomed are that how to estimate the boundaries for the energy esimate and the upper bound and the lower bound for 6. The method used here is similar to [1].In Chapter 3, we come to the Cauchy problem. Under the assumptions that u_≤u_+ and s_ = s_+, we prove the global existence for the solution to the compressible Navier-Stokes equations. Here we pull in the viscous rarefaction waves, and make use of the smallness ofεwhich is to control the a priori estimates for v andθ.