A Study in the Asymptotic Behavior of Nonlinear Evolution Equations with Nonlocal Operators

Abstract

Nonlinear evolution equations appear in a very wide variety of physical, economical, and numerical models. Many exotic phenomena demand the use of nonlocal operators in these models. This thesis focuses on investigating the asymptotic behavior of two such classes of equations: the surface quasigeostrophic (SQG) equation, of hydrodynamic origin, and the fractional Fisher-KPP equation, a reaction-diffusion equation with a non-standard diffusion process. We first prove the absence of anomalous dissipation of energy for the forced critical SQG equation with vanishing hyperviscosity through the analysis of stationary statistical solutions. Then we use precise nonlinear lower bounds on the fractional Laplacian to prove global regularity for the forced critical SQG equation (bootstrapping directly from L∞ to H¨older continuity) and use this to further prove the existence of a compact global attractor (of finite fractal dimension) for the associated dynamics. Lastly, we prove a comparative exponential decay estimate on the derivatives of the solution to the fractional Fisher- KPP equation (starting from decaying initial data), which then proves a flattening/symmetrization result for the reaction fronts.