Surface Quasi-Geostrophic Equation driven by Space-Time White Noise

TL;DR

This paper establishes local existence of solutions for the Surface Quasi-Geostrophic equation driven by space-time white noise using regularity structures, addressing challenges posed by Riesz-transforms in the non-linearity.

Contribution

It introduces a novel method to lift Riesz-transforms within regularity structures, enabling the analysis of SQG with space-time white noise.

Findings

Proved local existence of solutions for SQG with white noise.

Developed a technique to handle Riesz-transforms in regularity structures.

Demonstrated the use of inhomogeneous models for spatial operators.

Abstract

We consider the Surface Quasi-Geostrophic equation (SQG) driven by space-time white noise and show the existence of a local in time solution by applying the theory of regularity structures. A main difficulty is the presence of Riesz-transforms in the non-linearity. We show how to lift singular integral operators with a particular structure to the level of regularity structures and using this result we deduce the existence of a solution to SQG by a renormalisation procedure. The fact that the Riesz-transforms act only in the spatial variable makes it necessary to use inhomogeneous models instead of the standard ones.