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

TL;DR

This paper investigates how dissipation and quadratic nonlinearities in PDEs influence regularity, introducing an infinite order energy functional to establish decay estimates for higher-order Sobolev norms across various models.

Contribution

It introduces a novel infinite order energy functional that captures the regularizing effects of higher derivatives in PDEs with dissipation and quadratic nonlinearities.

Findings

Established a Petrowsky type parabolic estimate for the class of PDEs.

Proved the non-increasing property of the infinite order energy functional over time.

Derived precise decay rates for higher-order Sobolev norms.

Abstract

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.