Researches Of Global Well-posedness On The Compressible Navier-Stokes Equations And Related Models

Navier-Stokes system is a model describing a motion of fluid flows. Due to its sig-nificance in the fields of mathematics and physics, many mathematicians and physi-cists have deeply investigated the Navier-Stokes equations and obtained a number of results. However, some physical problems, though they seem simple, are not well ex-plained in mathematics until now, the study on the mathematical theory of the Navier-Stokes equations has drawn the extensive attention from the worldwide mathemati-cal society. The progress and current status of the research on the viscous gas, radia-tive fluid of the Navier-Stokes equations and 1D liquid crystal model coupling with a equation of a heat flow are reviewed. Based on the previous results, we investigate the global well-posedness of the Navier-Stokes equations, which are of deep physical sense, and their related models. By means of some new approaches, techniques and tools, we have overcome some mathematical difficulties of these physical models to study the global existence, regularity and exponential stability of weak solutions to the Navier-Stokes equations under some reasonable assumptions on the initial data.In this dissertation, we study the following problems:(1) When the viscous gas is connected to vacuum states with jump discontinu-ities, and the initial density of the gas has the compact support, on the basis of the known results in Hl, we employ the energy method to establish the regularity of so-lutions to a free boundary problem of 1D compressible isentropic Navier-Stokes equa-tions. The novelties for this problem are as follows:(ⅰ) We first study the regularity of solutions in H2[0,1] and H4[0,1]. (ⅱ) By the embedding theorem and the delicate inter-polation techniques, we overcome the mathematical difficulties caused by the higher order of partial derivatives in the proof of the regularity of solutions.(2) When the viscous gas is connected to vacuum states with continuous density, i.e., density function and viscosity coefficient are degenerate at free boundaries, we prove the interior regularity of 1D compressible isentropic Navier-Stokes equations. The novelty is as follows:By introducing a suitable weighted power a of pa (i.e., a factor in the termsρα((?)xβu)2 andρα[(?)xβ(ρθ)]2 with some positive integerβ), we overcome the difficulty that the density function of the gas has no positive lower bound at the free boundaries and obtain the interior regularity of solutions to the system.(3) We establish the global existence and exponential stability of solutions for a Stefan-Boltzmann model of a viscous, reactive and radiative fluid. We correct some defects about the global existence and asymptotic behavior of solutions to the system in [1], and improve the results in [1]. The novelties for this problem include:(ⅰ) We correct some defects in [1] and obtain the larger range of the temperature growth ex-ponents (q,β) than that in [1], which is a result of having a great physical sense. (ⅱ) We bound the norms of (u,v,θ, Z) and their derivatives in terms of expression of the form (1+sup0≤s≤t‖θ‖L∞)ΛwithΛbeing a positive constant only depending on q andβ. (ⅲ) We first establish the global existence and exponential stability of global solu-tions in H2[0,1] and H4[0,1] for this model. (ⅳ) We first establish the global existence of classical solutions.(4) We study the global existence and regularity of 1D liquid crystal system, which consists of the compressible Navier-Stokes equations coupling with a heat flow equation for harmonic maps. The novelty is that we prove regularity of solutions in H4 by using the embedding theorem and the delicate interpolation techniques to over-come the complexity caused by the higher order estimates on partial derivatives.