An application of real aualytical wethad to nonlinear evolution eguations

我们研究了Navier-Stokes equation在3个维度外部域中的初始边界值问题，该域描述了不可压缩的可见流体的运动。在1930年， J. Leray提出了没有独特性的脆弱解决方案的存在。然而，我们无法从勒雷的解决方案中获得流体运动的任何定性属性。1950年， R。Finn开始研究固定海军-斯托克斯等式的质量特性的解决方案，他的定义是一个物理上合理的解决方案（prs）。He proved the unique existence theorem of prs for small external force and the velocity uy D2马伊D2 at infinity。在Finn之后， Heywood在Ly D22和D2框架中提供了prs的稳定性，现在成为流体运动中最重要的数字调查基础之一。但是， prs不属于L-D22和D2空间，因此我们必须研究prs在阶级中的稳定性，使其成为prs所属的。这个问题已经超过30年了。在我们目前的研究中， W ... More 我已经解决了这个问题。我们使用了以下的论点。我们在3dim的外部域中研究了Oseen方法，然后我们在Oseen equation附近研究了当地能量估计，通过展示Oseen resolvent附近的边界的最佳秩序。Combinig this and L-D2p-L-D2q-D2 estimate in the whole space by cut-off technique， we proved the optimal L-D2p-L-D2q-D2 estimate of the Oseen semigroup in 3 dim。Exterior domain。The most important point is that all the constant appearing in the estimate is independent of uイD2マイエD2。根据这一估计，我们可以解决芬恩的稳定性问题。（2）在两个维度的情况下，我们知道勒雷脆弱解决方案的独特存在。但是，自从斯托克斯的基本解决方案具有逻辑的单一性以来，我们不知道NS在外部域或甚至整个空间与3个尺寸比较的解决方案的特性。We could obtain the asymptotic expansion of Stokes resolvent near the origin and we found that the logarithmic singularity is canceled out by the reflection phenomenon near the bounday， and therefore we obtained the optimal L D2-L D2 q（1< q？p？） of Stokes semigroup in a2 dim。Exterior domain by using the similar argument to the 3 dim。Case。根据这一估计，我们可以向NS展示最佳的疲软解决方案转换率，当时间达到无限时。（3）我们考虑到了复杂的可见流体的运动。Extended the method developed in study （1）， we obtained the optimal LイD2 pイD2-LイイD2 estimate of solutions to linearized eqautions in the 3 dim. Exterior domain， which is applied to obtain the optimal convergent rate of solutions to the original nonlinear probolem at time infinity。Less（低）

We studied the initial-boundary value problem for Navier-Stokes equation in 3 dimensional exterior domain which describes the motion of incompressible viscous fluids. In 1930, J. Leray proved the existence of its weak solutions without uniqueness. But, we can not get any qualitative property of the motion of fluid from Leray's solutions. In 1950, R. Finn started to study the qualitative property of soltuions to the stationary Navier-Stokes equation, which was termed physically reasonable solutions (prs) by him. He proved the unique existence theorem of prs for small external force and the velocity uィイD2∞ィエD2 at infinity. After Finn, Heywood proved the stability of prs in the LィイD22ィエD2 framework, which now become one of the most important base of numerical investigation of the fluid motion. But, prs does not belong to LィイD22ィエD2 space, and therefore we have to study the stability of prs in the class to which prs belongs. This problem remained more than 30 years. In our present study, w … More e solved this problem. We used the following argument. We took the Oseen approcimation in the 3 dim. Exterior domain, and then we investigated the local energy estimate near the boundary of optimal order of the Oseen equation by showing the fractional differentiablity of Oseen resolvent near the origin. Combinig this and LィイD2pィエD2-LィイD2qィエD2 estimate in the whole space by cut-off technique, we proved the optimal LィイD2pィエD2-LィイD2qィエD2 estimate of the Oseen semigroup in 3 dim. Exterior domain. The most important point is that all the constant appearing in the estimate is independent of uィイD2∞ィエD2. By using this estimate, we could solve the stability problem of Finn's prs.(2) In 2 dimensinal case, we know the unique existence of Leray's weak solutions. But, since the Stokes fundamental solution has logarihmic singularity, we know less property of solution to NS in the exterior domain or even the whole space compared with 3 dim. Case. We could obtain the asymptotic expansion of Stokes resolvent near the origin and we found that the logarithmic singularity is canceled out by the reflection phenomenon near the bounday, and therefore we obtained the optimal LィイD2pィエD2-LィイD2qィエD2 estimate (1< q≦ p≦ ∞) of Stokes semigroup in a 2 dim. Exterior domain by using the similar argument to the 3 dim. Case. Applying this estimate, we could show the best convergent rate of weak solutions to NS when time goes to infinity.(3) We considered the motion of the compressible viscous fluid too. Extending the method developed in study (1), we obtained the optimal LィイD2pィエD2-LィイD2qィエD2 estimate of solutions to linearized eqautions in the 3 dim. Exterior domain, which is applied to obtain the optimal convergent rate of solutions to the original nonlinear probolem at time infinity. Less