Sharp time asymptotics for the quasi-geostrophic equation, the Boussinesq system and near plane waves of reaction-diffusion models

Abstract

Through this dissertation we present the sharp time decay rates for three equations, namely quasi--geostrophic equation (SQG), Boussinesq system (BSQ) and plane wave of general reaction-diffusion models. In addition, in each case, we provide the dominant part of the solution which leads to the long term asymptotic profiles of each model. The first two equations, arising in fluid dynamics, model some aspect of the shallow waters with horizontal and vertical structures. Indeed, quasi--geostrophis equation models the horizontal inertia forces of a flow. As a result of that, atmospheric and oceanographic flows which take place over horizontal length scales, which are very large compare to their vertical length scales, are studied by SQG equation. On the other hand BSQ system models some vertical aspect of the flow, namely the speed, pressure and the temperature of the flow. In coastal engineering, BSQ type equations have a vast application in computer modeling. Lastly, a plane wave is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector. In order to study these equations, we made some developments in the "scaling variable" methods, so that it fits over models. In particular, we now have a good understanding of this method when it is applied to the equations with fractional dissipations.