Some Mathematical Results On The Compressible Navier-Stokes Equations

The thesis is composed of two parts. The first part concerns the isentropic Navier-Stokes equations while the second one studies the nonisentropic Navier-Stokes equations. The first part has five chapters. In chapter 1, we study the isentropic compressible Navier-Stokes equations with radially symmetric data in anannular domain or exterior domain and the initial density need not be positive. The existence and uniqueness of global strong solutions have been proved very recently by H. J. Choe and H. Kim (Math. Meth. Appl. Sci., 2005) under the technical restrictionγ≥2 and their estimates of the solutions depend on the time. In this chapter, we assumeγ≥1 and prove the same results. Furthermore, we prove the uniform-in-time L~∞upper bounds of the density and the H~1-bounds of the velocity.Chapter 2 gives some blow-up criteria of strong solutions to t he isentropic Navier-Stokes equations in Lorentz space and Morrey space. Especially, our similar results on the dnesity-dependent Navier-Stokes equations improve the very recent results due to H. Kim (SIAM J. Math. Anal., 2006).Chapter 3 proves that the weak solutions to the isentropic Navier-Stokes equations satisfy the energy equality under some conditions on the regularity of weak solutions.Chapter 4 studies the problem of the limit process when the shear viscosity goes to zero for global solutions to the isentropic Navier-Stokes equations for the one-dimensional spiral flows between two circular cylinders. This improves the results obtained by H. Frid and V. Shelukhin (Comm. Math. Phys., 1999).Chapter 5 makes use of the method in chapter 1 to study the one-dimensional isentropic Navier-Stokes equations for non-Newton fluid and obtain the existence and uniqueness of global strong solutions.The second part also consists of five chapters which considers the full Navier-Stokes equations for viscous polytropic fluids. In chapter 6, we prove the existence and uniqueness of local strong solutions for all initial data satisfying some compatibility conditions. The initial density need not be positive and may vanish in an open set. Especially, we prove the following generalized Poincare-type inequality:Chapter 7 uses this inequality to prove that the nonexistence of nontrivial time-periodic variational solutions to the full Navier-Stokes equations for viscous polytropic fluids.In chapter 8, we prove the Lipschitz continuous dependence on the initial data of spherically symmetric global weak solutions to equations of a viscous polytropic ideal gas in bounded annular domains with the initial data in Lebesgue space.Chapter 9 presents a uniqueness result of weak slutions under some conditions on the regularity of solutions.In the final chapter 10, we analyze the question of the limit process when the shear viscosity goes to zero for global solutions to the Navier-Stokes equations for compressible heat conductive fluids for the flows which are invariant over cylindrical sheets. We improve the results obtained by H. Frid and V. Shelukhin (SIAM J. Math. Anal., 2000).