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

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.