We obtain a family of nonlinear maximum principles for linear dissipative
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 dissipative 2d Boussinesq equations
