Logarithmic spikes of gradients and uniqueness of weak solutions to a class of active scalar equations

TL;DR

This paper investigates the uniqueness of weak solutions to a class of active scalar equations with Fourier multiplier relations, exploring integrability conditions on gradients and the inviscid limit, revealing new insights into solution behavior.

Contribution

It introduces weaker integrability conditions on gradients to ensure uniqueness and analyzes the inviscid limit for these active scalar equations.

Findings

Weak solutions are unique under certain integrability conditions on the gradient.

The paper establishes results on the inviscid limit behavior.

Logarithmic spikes of gradients are significant in solution analysis.

Abstract

We study the question weather weak solutions to a class of active scalar equations, with the drift velocity and the active scalar related via a Fourier multiplier of order zero, are unique. Due to some recent results we cannot expect weak solutions to be unique without additional conditions. We analyze the case of some integrability conditions on the gradient of the solutions. The condition is weaker than simply imposing $\nabla \theta \in L^\infty$. Lastly, we consider the inviscid limit for the studied class of equations.