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

考虑可压缩Navier-Stokes方程的初边值问题，给出了在平衡态附近小初值条件下解的渐近性态。在有界区域上，讨论了可压缩Navier-Stokes-Poisson方程的初边值问题，证明了整体弱解的存在性定理。同时，我们考虑了定常麦克斯韦或Stokes方程组解的分量正则性，并考虑了可压缩Navier-Stokes方程在半个定常平衡点附近解的渐近行为。导出了线性化问题的解公式，并得到了线性化问题解的一些估计。它示出，在柯西问题的情况下，线性化问题的解决方案的领导部分被分解成两部分，一个行为像扩散波和其他一个行为像纯扩散。然而， ...更多信息 在空间导数的考虑上，出现了一些不同于Cauchy问题的地方。同时证明了线性化问题的解在某些空间中大范围地逼近非定常Stokes问题的解，从而得到了非定常Stokes问题的解公式，并应用线性化分析和加权能量法的结果，研究了非线性问题解的大范围性态.结果表明，在柯西问题中可能存在一些非线性相互作用现象。考虑描述可压缩粘性等熵气体流在自引力作用下运动的Navier-Stokes-Poisson方程。证明了三维有界区域上有限能量弱解的存在性，讨论了平衡点的稳定性，并考虑了定常麦克斯韦或Stokes系统解的分量正则性.我们假设有一个表面，被认为是一个接口，其中一个系统的解决方案是光滑的，除了这个接口。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，在麦克斯韦体系中，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，只有其切向分量的法向导数才能产生穿过界面的解的奇异性。此外，这些结果被证明是最佳的。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