On the conservation laws and the structure of the nonlinearity for SQG and its generalizations

TL;DR

This paper introduces a new approach to defining the nonlinear term in SQG equations, proving conservation laws for weak solutions, and establishing the optimality and uniqueness of the nonlinearity characterization.

Contribution

It presents a novel definition for the nonlinear term, proves conservation of angular momentum and Hamiltonian for weak solutions, and characterizes the mSQG nonlinearity uniquely among active scalars.

Findings

Weak solutions conserve angular momentum.

Hamiltonian conservation under optimal assumptions.

New estimate for nonlinearity is optimal and characterizes mSQG uniquely.

Abstract

Using a new definition for the nonlinear term, we prove that all weak solutions to the SQG equation (and mSQG) conserve the angular momentum. This result is new for the weak solutions of [Resnick, '95] and rules out the possibility of anomalous dissipation of angular momentum. We also prove conservation of the Hamiltonian under conjecturally optimal assumptions, sharpening a well-known criterion of [Cheskidov-Constantin-Friedlander-Shvydkoy, '08]. Moreover, we show that our new estimate for the nonlinearity is optimal and that it characterizes the mSQG nonlinearity uniquely among active scalar nonlinearities with a scaling symmetry.