Regularization by rough Kraichnan noise for the generalised SQG equations

TL;DR

This paper demonstrates that adding rough Kraichnan-type noise to the generalized SQG equations regularizes the PDE, ensuring existence, uniqueness, and continuous dependence on initial data for a broad class of initial conditions.

Contribution

It introduces a novel stochastic regularization technique for the gSQG equations, establishing strong solutions and uniqueness in cases where deterministic solutions are not unique.

Findings

Regularization of gSQG equations via Kraichnan noise.

Existence and uniqueness of solutions for broad initial data.

Solutions depend continuously on initial conditions.

Abstract

We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in $\mathbb R^2$ with parameter $\beta\in (0,1)$, an active scalar model interpolating between SQG ($\beta=1$) and the 2D Euler equations ($\beta=0$) in vorticity form. Existence of weak $(L^1\cap L^p)$-valued solutions in the deterministic setting is known, but their uniqueness is open. We show that the addition of a rough Stratonovich transport noise of Kraichnan type regularizes the PDE, providing strong existence and pathwise uniqueness of solutions for initial data $\theta_0\in L^1\cap L^p$, for suitable values $p\in[2,\infty]$ related to the regularity degree $\alpha$ of the noise and the singularity degree $\beta$ of the velocity field; in particular, we can cover any $\beta\in (0,1)$ for suitable $\alpha$ and $p$ and we can reach a suitable ("critical") threshold. The result also holds in the presence of external forcing $f\in L^1_t (L^1\cap L^p)$ and solutions are shown to depend continuously on the data of the problem; furthermore, they are well approximated by vanishing viscosity and regular approximations. With similar techniques, we also show well-posedness for two-dimensional linear transport equation with random drift, with the same noise.