Renormalization of active scalar equations

TL;DR

This paper establishes new regularity conditions ensuring the conservation of $L^p$ norms in weak solutions of transport equations with incompressible vector fields, especially relevant for active scalar equations.

Contribution

It provides sufficient conditions based on combined regularities of coefficients and scalars, extending classical theories to active scalar equations with less regularity.

Findings

Derived new regularity criteria for conservation of $L^p$ norms.

Extended commutator estimates to handle $p eq 2$ cases.

Applied techniques similar to Constantin-E-Titi in the context of Onsager's conjecture.

Abstract

We consider transport equations with an incompressible transporting vector field. Whereas smooth solutions of such equations conserve every $L^p$ norm simply by the chain rule, the question arises how regular a weak solution needs to be to guarantee this conservation property. Whereas the classical DiPerna-Lions theory gives sufficient conditions in terms of the regularity of the coefficients, with no regularity requirement on the transported scalar, we give here sufficient conditions in terms of the combined regularities of the coefficients and the scalar. This is motivated by the case of active scalar equations, where the transporting vector field has the same regularity as the transported scalar. We use commutator estimates similar to those of Constantin-E-Titi in the context of Onsager's conjecture, but we require novel arguments to handle the case of $L^p$ norms when $p\neq 2$.