Fractional stochastic active scalar equations generalizing the multi-D-Quasi-Geostrophic & 2D-Navier-Stokes equations. -Short note-

TL;DR

This paper establishes the well-posedness, including existence, uniqueness, and regularity, of solutions for a broad class of fractional stochastic active scalar equations, encompassing models like the stochastic quasi-geostrophic and 2D Navier-Stokes equations.

Contribution

It introduces a unified framework for analyzing the well-posedness of various fractional stochastic active scalar equations with multiplicative noise, extending previous results to more general settings.

Findings

Proved global existence and uniqueness of solutions in the subcritical regime.

Established regularity properties of solutions under different noise and divergence conditions.

Extended the analysis to include equations like fractional Burgers and nonlocal transport equations.

Abstract

We prove the well posedness: global existence, uniqueness and regularity of the solutions, of a class of d-dimensional fractional stochastic active scalar equations. This class includes the stochastic, dD-quasi-geostrophic equation, $ d\geq 1$, fractional Burgers equation on the circle, fractional nonlocal transport equation and the 2D-fractional vorticity Navier-Stokes equation. We consider the multiplicative noise with locally Lipschitz diffusion term in both, the free and no free divergence modes. The random noise is given by an $Q-$Wiener process with the covariance $Q$ being either of finite or infinite trace. In particular, we prove the existence and uniqueness of a global mild solution for the free divergence mode in the subcritical regime ($\alpha>\alpha_0(d)\geq 1$), martingale solutions in the general regime ($\alpha\in (0, 2)$) and free divergence mode, and a local mild solution for the general mode and subcritical regime. Different kinds of regularity are also established for these solutions. The method used here is also valid for other equations like fractional stochastic velocity Navier-Stokes equations (work is in progress). The full paper will be published in Arxiv after a sufficient progress for these equations.