An application of real aualytical wethad to nonlinear evolution eguations

我们studied the initial-boundary value problem for Navier-Stokes equation in 3 dimensional exteriordomain which describes the motion of incompressible viscous fluids. In 1930J. Leray proved the存在于its weak solutions without uniqueness. But，我们不能得到任何qualitative property of the motion of fluid from Leray's solutions. In 1950R. Finn started to study the qualitative property of soltuions to the stationary Navier-Stokesequation,prs (prs) by him. He proved the unique existencetheorem of prs for small external force and the velocity at infinity. After Finn，Heywood proved the stability of prs in the L - 22 - D2 framework，which now become one of the最重要的基础numerical investigation of the fluid motion. But，prs does not belong to L - 22 - D2 space，我们已经研究了这个问题的稳定性，在这类prs belongs.这个问题remained more than 30 years. In our present studyMore e solved this problem. We used the following argument. We took the Oseen approcimationin the 3 dim. Exterior domain，then we investigated the local energy estimate near the boundary of optimal order of the OseenCombinig . equation by the fractional differentiablity of Oseen resolvent near the origin.联合this and L D2p D2-L D2q D2 estimate in the whole space by ct -off technique，we proved the optimal L i D2p ii -L i D2q ii - D2 estimate of the Oseen semigroup in 3dim . Exteriordomain. The most important point is that all The constant appearing in The estimate is independentof u - D2∞- D2. By using this estimate我们可以解决Finn's prs的稳定性问题(2)In 2 dimensinal case，我们知道Leray's weak solutions. But，since the Stokes fundamental solution has logarihmic singularity我们不知道如何解决property of solution in the exterior domain or even the whole space compared with3 dim. Case. We could obtain the asymptotic expansion of Stokes resolvent near the origin and Wefound the logarithmic singularity is canceled out by the反射phenomenon near thebounday,and therefore we obtained the optimal L D2p D2-L D2q D2 estimate (1< q≤p≤∞)of Stokessemigroup in a 2 dim. Exterior domain by using the similar argument to the 3 dim. Case. Applyingthis estimate,we could show the best convergent rate of weak solutions to NS when time goes to infinity.(3) weconsidered the motion of the compressible viscous fluid也是. Extending the method developed in study(1),we obtained the optimal L - D2p d -L D2q D2 estimate of solutions to linearized eqautions in the 3dim. Exterior domainwhich is applied to obtain the optimal convergent rate of solutions to the original nonlinearprobolem at time infinity. 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