带旋度二维行波的自由面及流函数的正则性研究

In this project, we cosider regularity of the free surface and stream function of a two-dimensional gravity- or capillary-gravity-driven steady flow of water over a flat bed. The flow is allowed to be rotational and characterized by vorticity. Under the assumption that the density of the flow is constant, and there are no stagnation points, that is, the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow, we intend to study the real analyticity of all the streamlines, including the free surface, of the steady flow, with only H?lder continuous vorticity function; furthermore, if the vorticity possesses some more regularity property rather than H?lder continuity, namely Gevrey regularity of index s, we intend to show the stream function will admit the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals to 1, we will obtain that the stream function is real-analytic. Our analysis will rely on an appropriate hodograph change of variable that transforms the free boundary value problem (corresponding in a frame moving at the constant wave speed to the governing equations for water waves with vorticity) into a nonlinear boundary problem for a quasi-linear elliptic equation in a fixed rectangular domain, and some a priori Schauder estimates for general nonlinear elliptic equations with nonlinear oblique boundary conditions. We will show the real analyticity of all the streamlines and stream function by giving successively a quantitative bound for each derivative of the streamlines in the H?lder norm. The above regularity results hold for both periodic and solitary water waves propogating on the free surface，and imply that we will obtain the maximal regularity of free surface for a vorticity with a finite regularity only...The two-dimensional stratified flow, that is, flow with varying density, will also be considered. We study real analyticity of all the streamlines of the steady stratified flow in the absence of stagnation points, with a H?lder continuous Bernoulli function and a H?lder continuously differentiable density function; furthermore, if the Bernoulli function and the density function possess some Gevrey regularity of index s, we show the stream function will admit the same Gevrey regularity.The corresponding regularity of all the streamlines and stream function for flows with infinite depth or with stagnation points will also be investigated...Results concerning the regularity of the streamlines and stream function of a fluid is important in our understanding of nonlinear wave motion, since we can approximate them to any level of accuracy by using power series. This contributes to derive monotonicity results on derivatives of the stream function along streamlines which are fundamental to establishing exact qualitative results concerning the pressure distribution function and particle trajectories throughout the fluid domain.

本项目拟研究重力及表面张力作用下的带旋度的二维稳态水流的自由面和流函数的正则性问题。对密度为常值且不存在相对静止点(即表面波速大于每个粒子运动的水平速度)的水流，当旋度函数为 H?lder 连续时，通过利用部分速端变换，带非线性边值的非线性椭圆方程的Schauder先验估计，及归纳地对流线函数各阶导数的H?lder范数的相对精确控制,考查该稳态流的包括自由面在内的所有流线的实解析性；也利用Gevrey正则性在速端变换下的不变性，证明当旋度函数具有指标为s的Gevrey正则性时，流函数具有相同的Gevrey正则性，特别地当s为1时将得到流函数的实解析性。相应的正则性研究也推广到密度变化的层流和存在相对静止点的水流。本项目中结果与表面波形（周期波，固波等）无关，在旋度函数有有限正则性时给出了自由面的最大正则性。自由面及流函数的正则性结果有助于研究诸如压力函数的性质及粒子的运动轨迹等波动问题。

带旋度二维行波是近几年来一个比较活跃的研究课题, 所得结果包括小振幅及大振幅水波的存在性,波面的正则性,对称性等性质..本项目主要研究了位于平底面上方的整个流体区域中不含相对静止点的 非粘性不可压缩水流的自由波面及流函数的解析正则性. 结果主要包含以下三个方面: 1流体的密度为常值时旋转二维行波自由面及流函数的解析性, 这里假设流体只受重力作用或者也受表面张力作用。我们在旋度函数为Hölder连续的情况下，得出了包含自由面在内的所有流线的最大正则性，即解析正则性. 这改进了发表在期刊Ann. of Math 上的旋度函数为Hölder连续时自由面为光滑的, 旋度函数为解析时自由面才具有解析正则性的结果. 另外我们还得到当旋度函数具有指数为 s, 0<s<1, 的Gevrey正则性时，流函数在整个流体区域中具有与之相同的正则性. 特别地,当旋度函数为解析时, 即 s=1时, 则流函数也是解析的. 2流体的密度依赖于伪流函数值的 旋转二维层流上自由行波面及流函数的正则性. 我们在伯努利函数为Hölder连续及旋度函数为Hölder连续可微的情况下，得出了包含自由面在内的所有流线的解析正则性；另外我们还得到当伯努利函数和旋度函数具有Gevrey正则性时，流函数具有与之相同的正则性. 这里的正则性结果对只受重力, 只受表面张力, 同时受重力及表面张力的水波都是适用的. 3 带负旋度的小振幅孤波下粒子的运动轨迹, 这里假设旋度是随流体深度的增加而减小的. 我们得到上述满足上述条件的二维孤波下的粒子具有如下性质: (a)每一粒子会在某时刻位于波峰下方, 在此时刻前位于波峰前方, 此时刻后位于波峰后方; (b) 如果平凡流是同向的，则位于底面上方的粒子沿一单峰路径向右运动；底面的粒子沿直线向右运动; (c) 当旋度函数的二阶导数为非正且孤波的振幅足够小时, 根据平凡流是混合的或是逆向的, 位于底面上方的粒子或按单峰路径向右运动, 或按单圈路径向左运动, 或沿单峰路径向左运动; 位于底面的粒子一直位于底面， 或者粒子先向后再向前运动，最后沿直线向左运动，或者粒子一直向左运动. .流体中粒子的运动轨迹问题与流线的正则性相关. 自由面的解析性有很多应用. 比如在波面的解析性条件下,可得出波面的严格单调性及具有有限个最小值及最大值点等结论.