On an analysis globally in time of solutions for surface waves

KdV方程和Kawahara方程是由表面波的基本方程作为长波近似形式导出的。将方程改写成适当的无量纲形式后，我们得到了两个无量纲参数δ和ε，分别是水深与波长之比和自由水面振幅与水深之比。极限δ→0对应于长波近似。更准确地说，极限δ=ε^2→0对应于KdV极限，极限δ=ε^4→0对应于河原极限。井口在数学上对KdV和Kawahara极限给出了严格的证明，并证明了这些近似方程的解实际上在适当的长时间间隔内近似于原始基本方程的解。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研究了二维不可压缩流体的Navier-Stokes方程，在流体占据整个空间或单位圆盘外部的情况下，粘性流体更容易发生。他发现了解的群对称性和解的时空衰减特性之间的关系。他还研究了占据外域的二维不可压缩理想流体的欧拉方程，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。nishibata研究了可压缩Navier-Stokes方程的球对称解在球外域的渐近时间行为。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。kagei研究了半空间中可压缩Navier-Stokes方程的平稳解的渐近稳定性。他发现了一个很好的解线性化问题的公式。通过使用该公式并对振荡积分进行仔细的分析，他得到了可能的最佳衰减估计，并证明了渐近稳定性。少

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