On The Stationary Solutions Of The Compressible Navier-Stokes Equations

In this thesis, we are concerned with the existence, uniqueness and nonlinear stability of stationary solutions to the compressible Navier-Stokes equations effected by the external force of general form in R³, and the large time behavior of the non-stationary solutions is also studied.In Chapter two, we prove the existence and uniqueness of stationary solutions of the compressible Navier-Stokes equations For general external force (G, F, H), which is small in suitable norm, we prove that the above problem admits a unique stationary solution (P^*,ν^*,θ^*) = ((?)＋q^*,ν^*,(?)＋(?)*) and the regularity and Spatial decay of (q^*,ν^*, (?)^*) is also obtained. Our argument in this part is based on the analysis of the linearized problem by using the weighted L₂ and L∞estimates. Compared with the isentropic case [25], we find that, due to the occurrence of the term qθ▽θ, to get certain weighted L₂ estimates for the linearized equation which are sufficient to solve the corresponding nonlinear problem, it is necessity for us to choose (P,ν,θ) rather than (p,ν,θ) to be the independent variables. It is worth pointing out that for the non-isentropic case, the estimates we obtained onσ^*(here,ρ^* =ρ(P^*,σ^*) =(?)＋σ^*) is not so good as for the isentropic case obtained in [25].In Chapter three, we study the stability of the stationary solution (ρ^*,ν^*,θ^*) with respect to the initial disturbance, i.e., the solvability of the non-stationary solution of N-S equation with the initial data: What we can obtain is that if H³-norm of the initial disturbance is small enough, then the solution to the above initial value problem exists uniquely and globally in time. Although the estimates onσ^* are not so good as for the isentropic case, we can also deduce the desired priori estimate.In Chapter four, we investigate the almost optimal convergence rate of non-stationary solution to the corresponding stationary solution when the initial data are small in H³ and bounded in L_6/5: (σ,ν,θ)(t) enjoy the decay of (1＋t)^{-1/2＋η}, where＞0 is any small number. Due to the not so good spatial decay of stationary solution, we can't get the optimal decay rate. Our main trick here is to combine the energy estimate deduced in Chapter three with the Stricker type estimates which generalizes the L_p - L_q estimate on the linearized equation to the Lorentz space by interpolation.