Paracontrolled quasi-geostrophic equation with space-time white noise

TL;DR

This paper develops a paracontrolled calculus approach to solve a highly singular stochastic quasi-geostrophic equation driven by space-time white noise on a 2D torus, for fractional Laplacian orders above 7/4, without renormalization.

Contribution

It formulates and solves the stochastic quasi-geostrophic equation in a new regime using paracontrolled calculus, avoiding renormalization for certain fractional Laplacian orders.

Findings

Successfully formulated the equation in the paracontrolled framework.

Proved local well-posedness for fractional Laplacian order > 7/4.

No renormalization needed for the model in this setting.

Abstract

We study the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus when the differential order of the main term, the fractional Laplacian, is larger than $7/4$. No renormalization has to be done for this model.