On White Noise Solutions of mSQG Equations on $\mathbb{R}^{2}$

TL;DR

This paper establishes the existence of white noise solutions for the weak formulations of mSQG equations on the entire plane, extending previous torus results through a limiting process and compactness techniques.

Contribution

It extends the existence results of white noise solutions for mSQG equations from the torus to the whole plane using a limiting approach and compactness arguments.

Findings

Existence of white noise solutions on 

Extension from torus to  via volume limit

Application of Skorokhod's theorem for compactness

Abstract

In this paper, we show existence of white noise solutions for weak formulations of modified Surface Quasi-Geostrophic (mSQG) equations. Based on previous results (\cite{FS}) on white noise solutions for mSQG equations on the torus $\mathbb{T}^{2}$, we show a similar result for the whole space $\mathbb{R}^{2}$ by letting the volume of the torus go to infinity and applying compactness methods (Skorokhod's theorem).