Nonlinear maximum principles for dissipative linear nonlocal operators and applications

Abstract

We obtain a family of nonlinear maximum principles for linear dissipa- tive nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2D incompressible Euler equations and generalized fractional dissi- pative 2D Boussinesq equations.