An application of real aualytical wethad to nonlinear evolution eguations

We studied the initial-boundary value problem for Navier-Stokes@equation in3dimensional exterior domain which describes the motion of incompressible viscous fluids.In1930，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。In1950，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 y D2∞I D2 at infinity.After Finn，Heywood proved the stability of prs in the L I D 22 I D 2 framework，which now become one of the most important base of numerical investigation of the fluid motion.But，prs does not belong to L I D 22个D 2 space，and therefore we have to study the stability of prs in the class to which prs belongs.This problem remained more than30years.In our present study，w…More e solved this problem。We used the following argument。We took the Oseen approcimation in the 3dim。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 I D 2 p I D 2-L I D 2 q I D 2 estimate in the whole space by cut-off technique，we proved the optimal L I D 2 p I D 2-L I D 2 q D 2 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 I D2∞ⅡD2。By using this estimate，we could solve the stability problem of Finn‘s prs.(2)In2dimensinal 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 with3dim。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 i D2p i D2-L I D2q i D2 estimate(1&lt；q≤p≤∞)of Stokes semigroup in a2 dim。Exterior domain by using the similar argument to the 3dim。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 I D2p I D2-L I D2q ii 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：Less

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