Multi-dimensional compressible Navier-Stokes equations with free boundary and symmetry.

摘要

The motivation of this work has come from the fluid thermodynamics. Dynamics of a viscous polytropic gas can be described by the compressible Navier-Stokes equations of conservation of mass, momentum, and energy. We study and prove the global existence of a weak solution in a domain outside of a solid core, assuming spherical symmetry and a free boundary. The gas dynamics close to the free boundary with the surrounding vacuum is described by the stress free condition and the zero heat flux requirement. Initially, the gas is assumed to continuously fill finite volume with bounded positive density between the static core and the free boundary. The absolute temperature of the gas is assumed to be initially bounded and strictly positive.; The proof of the global existence is carried out in several important steps. The Lagrangian transformation and the symmetry assumption enable us to transform the equations into one-dimensional form with geometrical source terms over a fixed mass domain. A sequence of solutions to approximate problems is constructed by using the space discretization method. With the help of a priori energy estimates, we show their global time existence, prove their weak convergence, and show that their limit is a solution to the original problem. The key milestones in the process of obtaining energy estimates include the property of finite free boundary expansion, the dissipation of the generalized total energy, the uniform point-wise boundedness of the density and the velocity, the uniform relative point-wise boundedness of the density from below, the uniform boundedness of the temperature away from the absolute zero, and the weak form of the temperature boundedness from above. These results are accompanied by quite a few higher order energy estimates that are necessary for the compactness framework and form the base for our convergence arguments. At the end we discuss classical solutions, external forces, and other extensions of our main results.