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

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.

33 pages. We have updated the relation of ${\beta }_N$ and ${\beta }_L$ in Lemma 2.2, following Proposition 2.7 in arXiv:2308.03216v2. Moreover, we have simplified the statements of Theorem 1.4, covering slightly wider range of parameters