Large Time Behaviors Of The3D Generalized Navier-Stokes Equations In Half Spaces

The mathematical models of fluid dynamics have been attracted more and more attention in the past half century due to the large applications to the fields such as me-teorology science, marine science, aeronautical science and biology medicine. It is well known that the classic Navier-Stokes equations are accepted as a right system modelling the above fluid flows. However, The nonlinear damping effect required in some complex fluids such as the porous media flow with friction effects. In this thesis, we consider one class generalized Navier-Stokes equations which exhibit the nonlinear damping Here b and π are unknown velocity fields and pressure fields,|b|β-1b is the nonlinear damping term. The purpose of this thesis is to investigate the large time behavior of solutions of the above generalized Navier-Stokes equations. More precisely, with the aid of the spectral decomposition technique and LP-Lq estimates of the linear heat equations, we will derive some decay results of weak solutions to the generalized Navier-Stokes equations(1). The thesis is organized as follows.In the chapter1, before introducing some fundamental main physical background of Navier-Stokes equations, we review the related previous work of the equations (1). Then we gave some preliminary which is necessary in our thesis.In the chapter2, firstly we apply the spectral decomposition approach to derive an estimation of the nonlinear operator:Theorem0.1Suppose that b is a weak solution of the equations(1)in the half spaces. Then So we obtain the time decay rate results of the weak solutions to the3D generalized Navier-Stokes equations in R+3by borrowing the spectral decomposition approach due to Borchers and Miyakawa[6]and the modified LP-Lq estimates of Bae and Choe[5]. Theorem0.2Suppose that b is a weak solution to the equations (1). Then we have:Theorem0.3Suppose that b is a weak solution of (1), bo∈Lσ2∩LTforl≤r＜2. Then fort≥1, we have:In the chapter3, we begin to consider the asymptotic stability for the nontrivial solution of the equations (1) with non-zero force. More precisely, the following result will be studied.Theorem0.4Suppose b0∈H1(R+3)∩Lβ+1(R+3),b(x,t) is a solution of (1) with β≥7/2. then for any initial perturbation zo(x)∈L2(R+3), there exists a weak solution v(x,t) of the perturbed problem which is written as follow: Then the weak solution v(x, t) converges asymptotically to b(x, t) as...