A Study Of Several Mathematical Problems In Ocean Waves

About 70% of the earth's surface is covered by water,the motion of ocean water waves as one of the most common motions in nature,is widely used in fluid mechanics,marine science,atmospheric science and natural disaster prediction.Therefore,it is of great significance to study the dynamic characteristics of ocean water waves systematically,such as the Indonesian tsunami in 2004.As one of the important models to promote the development of Applied Mathematics,the study of ocean water waves has become one of the core problems of Applied Mathematics.In this thesis,we study the models of ocean water waves,and a variety of theories and techniques of differential equations are comprehensively applied,including qualitative theory,stability theory,bifurcation theory,Schauder estimate,maximum principle,etc.The research is mainly focused on the exact solution,instability,stratified water waves and steady periodic solution.This thesis is divided into the following five chapters.In Chapter 1,we introduce the research background and progress of ocean water waves,preliminary and main results.The preliminary mainly includes the introduction of basic physical quantities such as equations of ocean water waves,vorticity,stream function,flow force function and so on.In Chapter 2,we show an exact solution of internal ocean water waves with thermocline at arbitrary latitude,considering the influence of Coriolis force,and based on the short-wavelength instability approach,we demonstrate the criteria for the hydrodynamical instability of such water waves.Moreover,we study the velocity and pressure distributions of internal waves near the thermocline at arbitrary latitude with Coriolis force and underlying current.In addition,we present the range of underlying current in the northern and southern hemisphere.In Chapter 3,we discuss the model of ocean water waves at arbitrary latitude with windstress,considering the influence of Coriolis and centripetal forces,and mainly study dynamic characteristics of such stratified water waves.Moreover,in the two-layer flows,we prove some monotonicity results with respect to the strength of the wind speed near the ocean surface.In Chapter 4,we study two-dimensional steady periodic equatorial water waves which propagate on a free surface and with a specified fixed mean-depth.In particular,we focus on irrotational flows and develop an equivalent formulation using flow force function and modified height function to transform the nonlinear ocean water wave problem with unknown boundary to a quasilinear elliptic differential equation with fixed boundary.Based on the local and global bifurcation theory due to Crandall-Rabinowitz,we present the existence analysis of non-trivial local solution curve and the global continuum.In Chapter 5,we summarize the full thesis and look forward to the researches of water wave in the future.