Global Smooth Solutions For The Compressible Viscous And Heat-conductive Gas

Navier-Stokes equations is a model describing a motion of fluid flows. Due to its impor-tance in the fields of mathematics and physics, many mathematicians and physicists have deeply studied the Navier-Stokes equations and obtained a number of results, so the study on the mathematical theory of the Navier-Stokes equations has drawn exten-sive attention by wordwide mathematicians. In this paper, we introduce the progress and current status of the research on the viscous, heat-conductive gas of the Navier-Stokes equations. Based on the previous results, we study the global well-posedness of the Navier-Stokes equations related models with deep physicalsense. Using some new approaches, techniques and tools, we have overcome some mathematical difficulties of the physical model to study the global smooth solution. This paper is concerned with the global existence of the smooth solutions to a system of equations describing one-dimensional motion of a self-gravitating, radiative and chemically reactive gas in Lagrangian mass coordinates. The main innovation of this paper is that (ⅰ) We have proved the problem under consideration admits a unique global smooth solutions when q≥2, improving Umehara and Tani's results. (ⅱ) Using embedding theorems and the delicate interpolation inequalities, we solved the complex estimation problems for the higher order of partial derivatives in the proof of the global solution existence.