Global Existence Of Classical Solution Of A Dissipative Navier-stokes Equation

The thermal transpiration,phenomenon that the flow moves from the low temperature to the high one when the gas is very rarified,such as in high altitude localities or in vacuo.But nather the Euler equations nor the Navier-Stokes equations can't describe this phenomenon.So dissipative Navier-Stokes equation cannot be derived from the Navier-Stokes system of gas dynamics,while it can be derived from Kinetic theory using a Hilbert expansion method.Now we study two cases of the global existence of classical solutions of the dissipative Navier-Stokes equation:the dimension of space is one or the solution is spherically symmetric.In this two cases,the velocity u and the temperature ? can be decoupled and the dissipative Navier-Stokes equation becomes a quasilinear parabolic equation with nonlinear term |??|2.This paper consists of three chapters.In Chapter 1,We outlined the critical point theory,preparations,historical background and the main works of this paper.In Chapter 2,We establish the global existence of classical solution in dimen-sion one.We mainly prove the dissipative Navier-Stokes equation have a unique solution using two auxiliary theorems which derives from literature[10].Then we use standard Schauder estimates to lift the regularity of the solution.In Chapter 3,We study the global existence of spherically symmetric classi-cal solutions of the dissipative navier-Stokes equation.To solve above-mentioned quasilinear parabolic equation,the cyucial point is that we can reformulate this equation to an quasilinear parabolic equation in divergence form and remove the nonlinear tern |??|2 by variable transformation.The we transform quasilinear parabolic equation into linear system by constructing approximate solution.Fi-nally,we estimate(?)r??C?,?/2.by the maximum principle and the Holder continuity of weak solution of the quasilinear parabolic equation.