Mathematical aspects of surface water waves

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This project focuses on three aspects in the mathematical theory of surface water waves and related interfacial wave motions, (i) the existence of traveling waves and their properties, (ii) the Cauchy problem and dispersive properties due to surface tension, (iii) stability and instability of traveling waves. Solitary water waves of arbitrary amplitude are constructed for a general class of vorticity. A priori bounds will be obtained for Stokes-kind waves with vorticity. Long-time existence for small data will be established for the water wave problem and the vortex sheet problem with surface tension. The phase instability and blow-up will be investigated. Dispersive properties of the effect of surface tension and their consequences will be studied. The Benjamin-Feir instability will be analytically understood for Stokes waves on deep water. Stability and instability of generalized vortex patches will be investigated. Emphasis is taken on the large-scale dynamics and nonlinear behavior of the wave motions at interface; the studies ultimately hinge upon analytical proofs. Surface water waves are manifested in a variety of natural phenomena which may be observed on the surface of the ocean or the river; they range from ripples to tsunamis or rogue waves. The subject constantly attracts attention of mathematicians as well as physicists and engineers. Furthermore, a considerable part of the mathematical theory of wave motion has been pioneered on the basis of studies of water waves. A key objective of this project is to develop new methodologies and theories in the analytical studies of surface water waves and related interfacial waves. Results from this project will help to furnish underlying principles of numerical simulations and engineering designs for the surface water waves phenomena. Deep analysis of particular problems involving water waves will stimulate the development of new mathematical ideas and analytical techniques for solving other highly nonlinear problems.