On a stability of the solutions to the equations of the Compressible flow

研究了可压缩Navier-Stokes方程的初边值问题，给出了在平衡态附近小初始数据解的渐近性。关于有界区域上可压缩Navier-Stokes-Poisson方程的初边值问题，证明了弱解在时间上全局的存在性定理。此外，我们还考虑了稳态Maxwell或Stokes系统解的分量正则性。研究了可压缩Navier-Stokes方程在给定常数平衡点上解的渐近性。导出了线性化问题的解公式，并给出了线性化问题解的一些估计。结果表明，与柯西问题一样，线性化问题的前导解被分解为两部分，一部分表现为扩散波，另一部分表现为纯粹扩散波。此外，还出现了一些与柯西问题不同的方面，特别是在考虑空间导数时。还证明了线性化问题的解在大时间内趋近于某些空间中的非平稳Stokes问题的解；并将所得结果应用于线性化分析和加权能量法，研究了非平稳Stokes问题的解的大时性。结果表明，柯西问题中可能存在一些未出现的非线性相互作用现象。考虑了在自引力作用下可压缩粘性等熵气体流动的Navier-Stokes-Poisson方程。证明了三维有界域上有限能量弱解的存在性，并讨论了平衡态的稳定性。我们考虑稳态Maxwell或Stokes系统解的分量正则性。我们假设有一个表面，作为一个界面，其中一个系统的解是光滑的，除了这个界面。然后，只有在这些假设下，我们才能证明解的某些分量在界面上是光滑的。也就是说，在麦克斯韦方程组中，溶液的法向分量在界面上总是规则的。另一方面，在Stokes系统中，解在界面上的奇异性只能出现在其切向分量的法向导数上。此外，这些结果被证明是最优的。少

We consider the initial boundary value problem of the Compressible Navier-Stokes equations and show the asymptotic behavior to the solutions for the small initial data near the equilibrium states. Concerning the initial boundary value problem of the Compressible Navier-Stokes-Poisson equations in a bounded domain, we prove the existence theorem of weak solutions globally in time. Also, we consider the component-wise regularity of the solution to stationary Maxwell or Stokes systems.Asymptotic behavior of solutions to the compressible Navier-Stokes equation on the half is considered around a given constant equilibrium. A solution formula for the linearized problem is derived, and some estimates for solutions of the linearized problem are obtained. It is shown that, as in the case of the Cauchy problem, the leading part of the solution of the linearized problem is decomposed into two parts, one behaves like diffusion waves and the other one behaves like purely diffusively. There, however … More , appear some aspects different from the Cauchy problem, especially in considering spatial derivatives. It is also shown that the solution of the linearized problem approaches in large times to the solution of the nonstationary Stokes problem in some spaces; and, as a result, a solution formula for the nonstationary Stokes problem is obtained Large time behavior of solutions of the nonlinear problem is then investigated in some by applying the results on the linearized analysis and the weighted energy method. The results indicate that there may be some nonlinear interaction phenomena not appearing in the Cauchy problem.We consider the Navier-Stokes-Poisson equation describing the motion of compressible viscous isentropic gas flow under the self-gravitational force. We prove the existence of finite energy weak solutions in three dimensional bounded domain and discuss the stability of equilibrium.We consider the component-wise regularity of the solution to stationary Maxwell or Stokes systems. We assume that there is a surface, regarded as an interface, and the solution to one of those systems is smooth except for this interface. Then, only under those assumptions, we can show that some components of solution are smooth across the interface. Namely, in the Maxwell system, the normal component of solution is always regular across the interface. In the Stokes system, on the other hand, the singularity of solution across the interface can arise only to the normal derivatives of its tangential components. Furthermore, those results are shown to be optimal. Less