Existence, Stability, and Qualitative Theory of Traveling Water Waves

水波是大气和海洋科学研究的基本课题。它们已经被研究了几百年，但它们的许多最基本的特征在数学上仍然知之甚少。例如，直到最近，才在发展稳定的旋转水波的严格存在理论方面取得了重大进展，即流体主体中的涡度波。这些类型的波浪在自然界中无处不在：涡度是由流入的洋流、吹过海洋的风或盐度造成的密度分层产生的。对这些现象背后的数学有更深的理解将有许多科学和实际应用。例如，密度分层水中的行波(内波)是海洋沿岸流的一般特征；众所周知，它们在推动海洋循环方面发挥着重要作用。空气中的涡度同样与风产生水波的过程密切相关，这是一个对海洋和气候模拟以及海洋工程非常重要的主题。研究人员研究了密度分层水中的行波、具有局域涡度的行波以及几类重要定常波的稳定性。他还研究了分层波是否连续依赖于它们的密度分布的问题。这个问题对于分层流体模型在多大程度上能够可靠地表示海洋中的流动具有重要的实际意义。这个项目的具体目标是：(I)扩展定常分层波和旋转行波的存在理论；(Ii)发展研究内波定性性质的新工具；以及(Iii)确定几类重要定常波的稳定性/不稳定性。研究人员和同事改进了目前的大振幅内波存在理论，允许在远场中有一个大致的速度分布。同时，他们还研究了分层波是否持续依赖于其密度分布的问题。证实或否认这一事实具有严重的实际意义：如果这不是真的，那么连续分层流体与通常用于模拟它们的分层流体系统之间存在着不可逾越的鸿沟。相反，如果相关性是平滑的，则连续模给出了分层近似产生的误差的界限。研究人员研究具有局域涡度的行波。在早期的工作中，他和他的合作者证明了具有紧支撑涡度(包括点涡旋和涡斑)的行波类的存在，但完全没有被探索的是涡度局部化但不紧凑的中间区域。利用无界区域上的分叉理论和半线性椭圆理论的技巧来攻击这一重要的区域。他们还利用新开发的确定非正则哈密顿系统束缚态稳定性/不稳定性的工具，研究了具有点涡旋的行波的轨道稳定性。最后讨论了风对水波的影响问题。这需要研究双流体欧拉系统中小振幅定常波的稳定性，这有助于揭示海-气界面在将能量从风转移到水中所起的作用。

Water waves are a fundamental subject of study in atmosphere and ocean science. They have been investigated for hundreds of years, yet many of their most basic features remain poorly understood mathematically. For instance, only very recently has significant progress been achieved in developing a rigorous existence theory for steady rotational water waves, that is, waves with vorticity in the bulk of the fluid. These types of waves are ubiquitous in nature: vorticity is generated by incoming currents, the wind blowing over the ocean, or the density stratification due to salinity. A deeper understanding of the mathematics underlying these phenomena would have many scientific and practical applications. For example, traveling waves in density-stratified water (internal waves) are a generic feature of coastal flows in the ocean; they are known to play a major role in driving the circulation of ocean. Vorticity in air is likewise strongly connected to the process of wind generation of water waves, a topic of great importance to ocean and climate modeling as well as ocean engineering. The investigator studies traveling waves in density-stratified water, traveling waves with localized vorticity, and the stability of several important classes of steady waves. He also studies the question of whether stratified waves depend continuously on their density distribution. This question has serious practical implications for how well layered-fluid models can reliably represent flows in the ocean. The specific goals of this project are to (i) expand the existence theory for steady stratified waves and rotational traveling waves; (ii) develop new tools for studying the qualitative properties of internal waves; and (iii) ascertain the stability/instability of several important classes of steady waves. The investigators and colleagues improve on the current existence theory for large-amplitude internal waves by allowing for a general velocity profile in the far field. Concurrently, they study the question of whether stratified waves depend continuously on their density distribution. Confirmation or denial of this fact has serious practical implications: if it is not true, then there is an unbridgeable gap between continuously stratified fluids and the layered-fluid systems that are commonly used to model them. Conversely, if the dependence is smooth, then the modulus of continuity gives bounds on the error arising from the layered approximation. The investigator studies traveling waves with localized vorticity. In earlier work, he and collaborators proved the existence of classes of traveling wave with compactly supported vorticity (including point vortices and vortex patches), but completely unexplored is the intermediate regime where the vorticity is localized but not compact. This important regime is attacked using techniques from bifurcation theory and semi-linear elliptic theory on unbounded domains. Using newly developed tools for determining stability/instability of bound states for non-canonical Hamiltonian systems, they also study the orbital stability of traveling waves with point vortices. Lastly, the problem of wind generation of water waves is considered. This entails studying the stability of small-amplitude steady waves in a two-fluid Euler system, which sheds light on the role of the air-sea interface in the transfer of energy from wind to water.