Dissipation vs. quadratic nonlinearity: from a priori energy bound to higher-order regularizing effect

We consider a rather general class of evolutionary PDEs involving dissipation (of possibly fractional order), which competes with quadratic nonlinearities on the regularity of the overall equation. This includes as prototype models, Burgers’ equation, the Navier-Stokes equations, the surface quasi-geostrophic equations and the Keller-Segel model for chemotaxis. Here we establish a Petrowsky type parabolic estimate of such equations which entail a precise time decay of higher-order Sobolev norms for this class of equations. To this end, we introduce as a main new tool, an “infinite order energy functional”, E(t): = \Sigma_n \alpha_n t^n |(-\Delta)^{n\theta/2} u(,t)|_{H^{\beta_c}} for appropriate critical regularity index \beta_c. It captures the regularizing effect of all higher order derivatives of u(,t), by proving — for a carefully, problem-dependent choice of weights {\alpha_n}, that E(t) is non-increasing in time.