Well-posedness of stochastic mSQG equations with Kraichnan noise and $L^p$ data

TL;DR

This paper proves the existence and uniqueness of solutions for stochastic mSQG equations with Kraichnan noise and $L^p$ initial data, advancing understanding of their well-posedness under certain conditions.

Contribution

It establishes weak existence and pathwise uniqueness for stochastic mSQG equations with Kraichnan noise and $L^p$ data, extending prior results to new parameter regimes.

Findings

Proved weak existence of solutions.

Established pathwise uniqueness.

Identified conditions for well-posedness.

Abstract

We consider stochastic mSQG (modified Surface Quasi-Geostrophic) equations with multiplicative transport noise of Kraichnan type, and $L^p$-initial conditions. Inspired by the recent work of Coghi and Maurelli [arXiv:2308.03216], we show weak existence and pathwise uniqueness of solutions to the equations for suitable choices of parameters in the nonlinearity, the noise and the integrability of initial data.