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

可压缩流动方程组解的稳定性

基本信息

批准号：
17540167
负责人：
KOBAYASHI Takayuki
金额：
$ 2.18万
依托单位：
Saga University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

项目摘要

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

我们考虑可压缩的Navier-Stokes方程的初边值问题，给出了平衡态附近小初值问题解的渐近行为。对于有界域上可压缩的Navier-Stokes-Poisson方程的初边值问题，我们证明了弱解的整体存在定理。此外，我们还考虑了定常Maxwell或Stokes系统解的按分量正则性，并考虑了可压缩的N-S方程在给定的常值平衡点附近解的渐近行为。给出了线性化问题的解公式，并给出了线性化问题解的估计。结果表明，与柯西问题的情形一样，线性化问题解的前导部分被分解为两部分，一部分表现为扩散波，另一部分表现为纯扩散。然而，在那里，…此外，在考虑空间导数时，出现了一些与柯西问题不同的方面。在某些空间中，线性化问题的解在大时间内逼近于非定常Stokes问题的解，并由此得到了非定常Stokes问题的解公式，然后应用线性化分析和加权能量方法研究了非线性问题解的大时间性态。结果表明，在柯西问题中可能存在一些没有出现的非线性相互作用现象。我们考虑了描述可压缩粘性等熵气体在自引力作用下运动的Navier-Stokes-Poisson方程。证明了三维有界域上有限能量弱解的存在性，讨论了平衡点的稳定性，讨论了定常Maxwell或Stokes系统解的按分量正则性。我们假设有一个曲面，被认为是一个界面，并且除了这个界面外，其中一个系统的解是光滑的。然后，只有在这些假设下，我们才能证明解的某些组成部分在界面上是光滑的。也就是说，在麦克斯韦系统中，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，界面上的解的奇异性只会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。较少