English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We consider a rather general class of evolutionary PDEs involving dissipation(of possibly fractional order), which competes with quadratic nonlinearities onthe regularity of the overall equation. This includes as prototype models,Burgers' equation, the Navier-Stokes equations, the surface quasi-geostrophicequations and the Keller-Segel model for chemotaxis. Here we establish aPetrowsky type parabolic estimate of such equations which entail a precise timedecay 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}} forappropriate critical regularity index \beta_c. It captures the regularizingeffect of all higher order derivatives of u(*,t), by proving --- for acarefully, problem-dependent choice of weights {\alpha_n}, that E(t) isnon-increasing in time.

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