Surface quasi-geostrophic equations forced by random noise: prescribed energy and non-unique Markov selections

TL;DR

This paper studies the 2D surface quasi-geostrophic equations with random forcing, demonstrating the existence of solutions with prescribed energy and establishing non-uniqueness of Markov selections for weak solutions.

Contribution

It constructs solutions with fixed energy for stochastic surface quasi-geostrophic equations and proves the non-uniqueness of Markov selections in this context.

Findings

Solutions with prescribed energy exist under certain conditions.

Non-uniqueness of Markov selections is established for these stochastic equations.

The results apply to both additive and multiplicative noise cases.

Abstract

We consider the momentum formulation of the two-dimensional surface quasi-geostrophic equations forced by random noise, of both additive and linear multiplicative types. For any prescribed deterministic function under some conditions, we construct solutions to each system whose energy is the fixed function. Consequently, we prove non-uniqueness of almost sure Markov selections of suitable class of weak solutions associated to the momentum surface quasi-geostrophic equations in both cases of noise.