Local and global well-posedness of compressible Navier-Stokes equations

The well-posedness theory of the full compressible Navier-Stokes equations (CNS for short) has been an important subject in the development of mathematical physics. For the spherically symmetric flow, even though many favourable regularity properties may be expected for its similarity to the one-dimensional equations, there are still a large number of open questions remaining unsolved. Two central difficulties of spherically symmetric CNS are the coordinate singularity at the centre of symmetry, and the upper/lower bounds of density. In Chapter 1, the Cauchy problems of CNS for the general multidimensional flows, both with and without spherical symmetry, are introduced. Then a heuristic analysis on the Lagrangian coordinate transformation is conducted, which gives insight to the strategies used in the later chapters. Finally, several previous progresses on the well-posedness theory of CNS are highlighted at the end of this chapter. In Chapter 2, the global-in-time existence of a spherically symmetric weak solution to the Cauchy problem in Eulerian coordinate is obtained. The main strategy is to construct a sequence of approximation problems posed in finite annular domains. This circumvents the problem of coordinate singularity at the origin. Moreover, they are also formulated in the Lagrangian coordinates, so that one can obtain a-priori estimates which are uniform with respect to the dimension of annular domains. The weak solution is then constructed as the limit of approximate solutions in the original Eulerian coordinate. The main result of this chapter is produced in collaboration with Gui-Qiang G. Chen and Shengguo Zhu. In Chapter 3, for the Lagrangian approximation problem introduced in Chapter 2, the existence and uniqueness of a local-in-time strong solution are shown. This is done via the Picard iteration scheme, however some major difficulties arise due to the fact that second order differential operators in momentum and energy equations is quasilinear under the Lagrangian formulation. To resolve this, a set of a-priori bounds on each iterative solutions is established so that the quasilinear operators are elliptic at each iterative step. Moreover, the time dependent high regularity estimates for linear iterative solution are carefully derived so that Banach Fixed-point theorem can be applied to obtain the desired solution. In Chapter 4, it is proved that the local-in-time strong solution in Chapter 3 can be extended to all time by bootstrap argument. Moreover, global-in-time strong solutions in the unbounded Lagrangian domain are also obtained by taking the limit of outer and inner radii of annular domains. The main difference of this chapter compared to Chapter 2 is that the limit is taken in the Lagrangian coordinates, and the initial data are assumed to have higher regularities.