Global regularity or finite time singularity of the surface quasi-geostrophic equations.

摘要

It remains open if all classical solutions of the surface quasi-geostrophic (SQG) equation are global in time. We try to solve this issue through numerical computations and geometric approach. In addition, we present numerical results of a model equation generalizing the 2D Euler equation and the SQG equation.;This thesis mainly contains four chapters. The first chapter introduces the SQG equation, and presents various open problems and existing results. The second chapter presents new numerical computations of the solutions to the SQG equation corresponding to several classes of initial data previously proposed by Constantin, Majda and Tabak [18]. By parallelizing the serial pseudo-spectral codes through slab decompositions and applying suitable filters, we are able to simulate these solutions with great precision and on large time intervals. These computations reveal detailed finite-time behavior, large-time asymptotics and key parameter dependence of the solutions and provide valuable information for further investigations on the global regularity issue concerning the SQG equation. The third chapter presents several geometric criteria under which the solutions of the SQG equation become regular for all time. The relation between the geometry of the level curves and the regularity of the solutions is the central focus of this part. The last chapter presents numerical studies of a general model equation that represents the 2D Euler equation and the SQG equation depending upon a parameter involved in the equation. This model equation reduces to the 2D Euler equation when the parameter is 0 and to the SQG equation when it is 1. We are interested in how the regularity of the solutions is affected by the parameter. In particular, we would like to know if the known global regularity of the 2D Euler equation can be used to understand the regularity of the SQG equation. Using the same parallel pseudo-spectral method and smooth initial data as in the second chapter, we are able to compare the solutions for different values of the parameter at various times.