2D接口数学
Mathematics for 2d Interfaces
摘要：
我们在这里介绍了有关两种不可压缩、非粘性、密度恒定且涡度集中在界面上的流体之间界面不稳定性数学分析的最新结果的调查。这种配置包括所谓的开尔文-亥姆霍兹（两个密度相等），瑞利-泰勒（两个不同的，非零的密度）和水波（其中一个密度为零）问题。在简要回顾了欧拉方程的强解和弱解的结果之后，我们导出了描述界面运动的界面方程（如Birkhoff-Rott方程）。线性分析使我们能够展示这些方程的主要特征（如椭圆率特性）;然后描述完整的非线性方程的后果。特别地，如果Kelvin-Helmholtz和Rayleigh-Taylor问题的解超过一定的正则性阈值，则它们必然是解析的（结果是非解析框架中的初值问题的不适定性）。我们也说几句话的现象，可能会发生低于这个规律性阈值。最后，特别注意的水波问题，这是更稳定的Kelvin-Helmholtz和Rayleigh-Taylor配置。这里介绍的大多数结果都是二维的（界面具有一维），但我们简要描述了三维情况下的相似性和差异性。
英文摘要：
We present here a survey of recent results concerning the mathematical analysis of instabilities of the interface between two incompressible, non viscous, fluids of constant density and vorticity concentrated on the interface. This configuration includes the so-called Kelvin-Helmholtz (the two densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and the water waves (one of the densities is zero) problems. After a brief review of results concerning strong and weak solutions of the Euler equation, we derive interface equations (such as the Birkhoff-Rott equation) that describe the motion of the interface. A linear analysis allows us to exhibit the main features of these equations (such as ellipticity properties); the consequences for the full, non linear, equations are then described. In particular, the solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily analytic if they are above a certain threshold of regularity (a consequence is the illposedness of the initial value problem in a non analytic framework). We also say a few words on the phenomena that may occur below this regularity threshold. Finally, special attention is given to the water waves problem, which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor configurations. Most of the results presented here are in 2d (the interface has dimension one), but we give a brief description of similarities and differences in the 3d case.
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