几类水波方程的相关研究

发布时间：2018-05-14 22:31

本文选题：深度变化的浅水波方程 + 旋转双组份Camassa-Holm系统　；参考：《南京师范大学》2016年博士论文

【摘要】：本文主要对几类水波方程进行了一些研究.我们主要研究了四个问题：深度变化的浅水波方程,旋转双组份Camassa-Holm系统,分层流中赤道边缘波的不稳定性及带负表面张力的稳定周期有旋重力波.论文由五部分构成,结构如下：第一章给出了本文的研究背景及所要用到的一些基本知识.第二章,我们考虑在不平坦底部的浅水中模拟表面波的一个非线性发展方程.首先我们建立了在Sobolev空间Hs(s),其中s3/2的适定性.此外,我们同样得到了在Besov空间B2,13/2拚的适定性结果.接着,我们在空间Hs(S),s3/2中得到了爆破准则,并且得到了关于此模型一个简化模型的一些爆破结果.最后我们研究了关于强解的持久性.第三章,我们推导了一个在有旋流中考虑Coriolis力的修正双组份Camassa-Holm系统,其是在赤道水波中的一个模型.我们研究了地球旋转所引起的非局部及非线性对爆破所带来的影响.我们所做的分析是基于特征线法、守恒量和此动力系统的局部结构,具体过程是讨论Riccati型微分不等式.此外,利用李雅普诺夫函数我们得到了保证波整体存在的的条件.接下来,我们对此系统的局部孤立波解进行了分类.根据旋转参数及流体对流中的平衡指数,我们得出除了光滑孤立波解外,还存在'peakon'及'cusped'这样的奇异孤立波解.最后,我们证明了此模型的水平对称弱解必为行波解.第四章,我们首先给出了在有沿岸方向赤道潜流的分层流中赤道边缘波的一个显式精确解.接着,我们分析了对正压的不可压流体的短波扰动法.最后,我们应用此方法于分层流的赤道边缘波中,得到了当此波的坡度超过一个具体的临界值时其是不稳定的.第五章,我们考虑带有负表面张力的二维稳定周期有旋重力水波.所研究问题的小振幅解的局部曲线是通过利用Crandall-Rabinowitz局部分岔理论得到的.利用整体分岔理论结合带有Venttsel边界条件的椭圆方程的Schauder理论,我们将小振幅解的局部曲线延拓到解的整体连续体.此外,我们也证明了对于波剖面从波峰到波谷是单调的且在波谷处为严格单调的这些波而言,其关于波峰必是对称的.
[Abstract]:In this paper, several kinds of water wave equations are studied. We mainly study four problems: the shallow water wave equation with depth variation, the rotating two-component Camassa-Holm system, the instability of the equatorial edge waves in stratified flows and the steady period of gravity waves with negative surface tension. The paper is composed of five parts. The structure is as follows: the first chapter gives the research background and some basic knowledge to be used. In chapter 2, we consider a nonlinear evolution equation for simulating surface waves in shallow water with uneven bottom. First, we establish the possitivity of S 3 / 2 in Sobolev space. In addition, we also obtain an appropriate qualitative result for B _ 2 ~ (13) / 2 in Besov space. Then, we obtain the blow-up criterion in the space HsSX / 3 / 2, and obtain some blasting results for a simplified model of this model. Finally, we study the persistence of strong solutions. In chapter 3, we derive a modified two-component Camassa-Holm system considering Coriolis forces in a swirling flow, which is a model in equatorial water waves. The nonlocal and nonlinear effects of earth rotation on blasting are studied. Our analysis is based on the method of characteristic line, the conserved quantity and the local structure of the dynamical system. The concrete process is to discuss the differential inequality of Riccati type. In addition, by using Lyapunov function, we obtain the conditions to guarantee the existence of the whole wave. Next, we classify the local solitary wave solutions of the system. Based on the rotational parameters and the equilibrium exponents in the fluid convection, we obtain the existence of singular solitary wave solutions such as Peakon 'and Cincus ped' in addition to the smooth solitary wave solutions. Finally, we prove that the horizontal symmetric weak solution of the model must be a traveling wave solution. In chapter 4, we first give an explicit and exact solution of the equatorial edge wave in a stratified flow with a coastal equatorial submersible current. Then, we analyze the short wave perturbation method for barotropic incompressible fluids. Finally, we apply this method to the equatorial edge wave of stratified flow and obtain that the wave is unstable when the slope of the wave exceeds a specific critical value. In chapter 5, we consider the two-dimensional steady period with negative surface tension. The local curve of the small amplitude solution is obtained by using Crandall-Rabinowitz 's local bifurcation theory. By using the global bifurcation theory and the Schauder theory of elliptic equations with Venttsel boundary conditions, we extend the local curve of the small amplitude solution to the global continuum of the solution. In addition, we also prove that for these waves whose wave profile is monotone from peak to trough and strictly monotone at the trough, the wave crest must be symmetric.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.29

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