Mathematical Sciences: Analysis on Waves in Stratified Fluids of Infinite Depth

9623060太阳：这个项目的目的是对无限深的层状流体中的孤立波和周期波进行数学和数值研究，这些流体受到引力的作用。该提案由两部分组成。第一部分给出了一个悬而未决的问题的严密解答，即除了具有连续密度分层的层外，连续分层流体中是否存在无穷大处具有代数衰减的孤立内波。这项拟议的研究将严格证明孤立波的存在，该孤立波来自一个被称为本杰明-小野方程的形式模型方程。第二部分讨论了无边界两层流体中大振幅周期波的存在性。将引入一种新的问题形式，将控制方程转化为单一的积分方程式。然后，对任意密度比的积分方程解进行数值和理论研究，不受解的幅值限制。在这个项目中，主要的目的是证明层状流体的完全非线性控制方程有有限和大振幅的解，并用数值和理论方法给出解的各种性质。微分方程式和泛函分析理论的相互作用对于获得严格的证明是必不可少的，而数值计算则提供了一些关于解的关键信息。密度变化流体(又称层状流体)中的大振幅重力波在深度较大的流体中具有相当大的地球物理意义。大幅度内波扰动是海洋和低层大气中的常见特征。特别是，孤立波和周期波都与各种海洋和大气现象有关，其中孤立波的形式是局部单峰，周期波在一定距离后总是具有相同的形式。大幅度的孤立波与大气中龙卷风的形成以及海洋内巨大的动量和能量传输有关。这项工作的重点是获得孤立波或周期波存在的条件，并预测如果存在孤立波或周期波的幅度可以有多大。它还将尝试使用数值计算来捕捉这些波的定性特征。这项研究的结果可能会对这些波的形成提供一些理论上的解释，并对海洋和大气中的某些波动有更多的了解，以便在不同的物理应用中利用或避免这些波。***

9623060 Sun The purpose of this project is to present a mathematical and numerical study of solitary and periodic waves in stratified fluids of infinite depth subject to the gravitational force. The proposal consists of two parts. The first part intends to give a rigorous answer to an open question whether there exist solitary internal waves with algebraic decay at infinity in a continuously stratified fluid that is bounded only by a rigid bottom and has a constant density except for a layer with continuous density stratification. The proposed research will give a rigorous justification of the existence of solitary waves derived from a formal model equation, called the Benjamin-Ono equation. The second part deals with the existence of periodic waves of large amplitude in a two-layer fluid without boundaries. A new formulation of the problem will be introduced, which transforms the governing equations into a single integral equation. Then the solutions of the integral equation will be studied numerically and theoretically for any density ratio without restrictions on the amplitude of solutions. In this project, the main thrust is to show that the fully nonlinear governing equations for stratified fluids have solutions of finite and large amplitude and give the various properties of the solutions using numerical and theoretical approaches. An interplay of theories in differential equations and functional analysis will be essential to obtaining the rigorous justifications, while numerical computation gives some crucial information on the solutions. %%% The gravity waves of large amplitude in a fluid of density variation, also called a stratified fluid, with great depth are of considerable geophysical interest. Large amplitude internal wave disturbances are common features in the oceans as well as in the lower atmosphere. In particular, a solitary wave, whose form is a localized single hump, and a periodic wave, which always has a same form after a certain distance, are relevant to various oceanic and atmospherical phenomena. The solitary waves of large amplitude have been associated with the formation of tornados in the atmosphere and the enormous transport of momentum and energy within the oceans. This work focuses on obtaining the conditions under which the solitary or periodic waves can exist and predicting how large the amplitude of the waves can be if they exist. It will also try to capture the qualitative features of these waves using numerical computations. The results obtained from this research may provide some theoretical explanation of the formation of these waves and give more understanding of certain wave motions in the oceans and atmosphere so that these waves may be either utilized or avoided in different physical applications. ***