Degrees of Freedom for Long Time Dynamics of Forced Critical Burgers and SQG Equation

Abstract

In this thesis, I investigate the long time dynamics of three equations arising in hydrodynamics: critical Burgers, critical SQG, and Navier-Stokes. To do so, I analyze the compact global atractors for each of these equations. I show that each atractor has finite fractal (Hausdorff) dimension. This dimension in turn gives a bound on the number of degrees of solutions’ long time behavior. Finally, using the results of [1], we attain a single exponential bound on the Lipschitz norm for solutions of forced critical SQG, improving the result of [8].