On an analysis globally in time of solutions for surface waves

面波解的全局及时分析

基本信息

批准号：
15540200
负责人：
IGUCHI Tatsuo
金额：
$ 2.24万
依托单位：
Tokyo Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540200/
关键词：
surface wave KdV equation Kawahara equation forced KdV equation Navier-Stokes equations hyperbolic-parabolic system long wave approximation group symmetry Benjamin-Ono方程式表面張力 forced kdV方程式 kawahara方程式漸近挙動熱対流

项目摘要

The KdV equation and the Kawahara equation were derived formally from the basic equations for surface waves as long wave approximations. After rewriting the equations in an appropriate non-dimensional form, we have two non-dimensional parameters δ and ε the ratio of the water depth to the wave length and the ratio of the amplitude of the free surface to the water depth, respectively. The limit δ→0 corresponds to the long wave approximation. More precisely, the limit δ=ε^2→0 corresponds to the KdV limit and the limit δ=ε^4→0 corresponds to the Kawahara limit. T.Iguchi gave mathematically rigorous justifications for the KdV and the Kawahara limits and proved that the solutions of these approximate equations in fact approximate the solutions of the original basic equations for an appropriate long time interval. He also analyzed an effect of the presence of an uneven bottom to these long wave approximations.T.Miyakawa investigated the Navier-Stokes equations for a two-dimensional incompres … More sible viscous fluid in the cases where the fluid is occupied the entire space or outside of the unit disc. He discovered the relation between the group symmetries of the solution and the space-time decay properties of the solution. He also investigated the Euler equation for a two-dimensional incompressible ideal fluid occupied an exterior domain and discovered a relation between the square integrability of the pressure and the effect of the flow to the obstacle.S.Nishibata investigated the asymptotic behavior in time of the spherically symmetric solution for compressible Navier-Stokes equations in the exterior domain of the sphere. He proved that a stationary solution is asymptotically stable under suitable assumptions for the initial data and the external forces. He did not suppose any smallness conditions for the data.Y.Kagei investigated the asymptotic stability of a stationary solution for the compressible Navier-Stokes equations in the half-space. He discovered a nice solution formula to the linearized problem. By using the formula and carrying out the analysis very carefully for the oscillatory integrals, he obtained the best possible decay estimates and proved the asymptotic stability. Less

KdV 方程和 Kawahara 方程是从作为长波近似的表面波基本方程形式上导出的，将方程重写为适当的无量纲形式后，我们得到两个无量纲参数 δ 和 ε（水深之比）。分别为波长和自由表面振幅与水深之比。极限 δ→0 对应于长波近似。更准确地说，极限 δ=ε^2→0 对应。到KdV极限，极限δ=ε^4→0对应于Kawahara极限。T.Iguchi对KdV和Kawahara极限给出了严格的数学证明，并证明了这些近似方程的解实际上是近似方程的解。他还分析了不均匀底部的存在对这些长波近似的影响。T.Miyakawa 研究了二维的纳维-斯托克斯方程。当流体占据整个空间或单位圆盘外部时，他发现了溶液的群对称性与溶液的时空衰变性质之间的关系。二维不可压缩理想流体的欧拉方程占据了一个外域，并发现了压力的平方可积性与流动对障碍物的影响之间的关系。S.Nishibata 研究了时间上的渐近行为球外域可压缩纳维-斯托克斯方程的球对称解他证明了在初始数据和外力的适当假设下平稳解是渐近稳定的。他没有假设数据的任何小条件。 Y.Kagei 研究了半空间中可压缩纳维-斯托克斯方程的平稳解的渐近稳定性，通过使用该公式并进位，他发现了一个很好的线性化问题的解公式。通过对振荡积分衰减进行非常仔细的分析，他获得了最好的估计并证明了渐近稳定性。