On an analysis globally in time of solutions for surface waves

KdV方程和川原方程是由长波近似的面波基本方程形式导出的。在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，它们分别是水深与波长的比率和自由面的振幅与水深的比率。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，而极限δ=ε^4→0对应于川原极限。井口对KdV极限和川原极限给出了严格的数学证明，并证明了这些近似方程的解在适当的长时间间隔内实际上近似于原始基本方程的解。他还分析了不平坦底部的存在对这些长波近似的影响。T.Miyakawa研究了二维压缩…的纳维斯托克斯方程在流体占据整个空间或单元盘外部的情况下，更易产生粘性流体。他发现了解的群对称性和解的时空衰变性质之间的关系。他还研究了占据外部区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和流动对障碍物的影响之间的关系。西田研究了球外部区域内可压缩Navier-Stokes方程的球对称解的时间渐近行为。他证明了在对初始数据和外力作适当的假设下，定常解是渐近稳定的。Y.Kagei研究了半空间中可压缩Navier-Stokes方程定常解的渐近稳定性。他发现了一个很好的解决线性化问题的公式。通过使用该公式并对振荡积分进行非常仔细的分析，他得到了可能的最佳衰减估计，并证明了渐近稳定性。较少

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