Nonlocal maximum principles for active scalars

TL;DR

This paper introduces a generalized nonlocal maximum principle applicable to active scalar equations in fluid dynamics, demonstrating its effectiveness in proving finite time regularization of weak solutions in supercritical regimes.

Contribution

It develops a broader nonlocal maximum principle framework and applies it to establish regularization results for supercritical active scalar equations.

Findings

Established a new nonlocal maximum principle

Proved finite time regularization for supercritical regimes

Extended applicability to various active scalar equations

Abstract

Active scalars appear in many problems of fluid dynamics. The most common examples of active scalar equations are 2D Euler, Burgers, and 2D surface quasi-geostrophic equations. Many questions about regularity and properties of solutions of these equations remain open. We develop the idea of nonlocal maximum principle, formulating a more general criterion and providing new applications. The most interesting application is finite time regularization of weak solutions in the supercritical regime.