Numerical and theoretical investigation of blow-up in various fluid models

The global regularity problem for smooth solutions to the incompressible Euler and Navier-Stokes equations is a major open problem in the analysis partial differential equations and in mathematical physics. In recent years, there have been several significant advances in our understanding of the global regularity problem for the Euler equations both numerically and analytically. Particularly, the Boussinesq system is based on Euler equation and is used to describe the propagation of surface water waves with small amplitude. Some previous results show that the Boussinesq system has local strong solutions which have finite energy but will become singular (or blow-up) in finite time in some specific domain. We will explore the stability and boundary behavior of scale-invariant solutions to Boussinesq system which is possible to write explicitly. Beside, the question that the blow-up property persists or not if the domain (for blow-up solutions ) changes. At last, we hope to calculate the ‘blow-up rate’ at least numerically. For the beta-plane model, basically it is the Euler equation with rotation. We would like to extent the time of existence of the solution when the initial data is small in a periodic domain.

Faculty Supervisor:

Slim Ibrahim

Student:

Partner:

University of California, San Diego

Discipline:

Mathematics

Sector:

Education

University:

University of Victoria

Program: