Well-posedness by noise for linear advection of $k$-forms

TL;DR

This paper extends well-posedness by noise results to the stochastic linear advection of $k$-forms, demonstrating existence and uniqueness of solutions under certain conditions, thus advancing the understanding of stochastic geometric fluid dynamics.

Contribution

It generalizes well-posedness by noise to the linear advection of $k$-forms, a key structure in geometric fluid dynamics, with new existence and uniqueness results.

Findings

Proves existence and uniqueness of weak $L^p$-solutions.

Extends well-posedness by noise to $k$-forms.

Addresses equations with non-unique solutions without noise.

Abstract

In this work, we extend existing well-posedness by noise results for the stochastic transport and continuity equations by treating them as special cases of the linear advection equation of $k$-forms, which arises naturally in geometric fluid dynamics. In particular, we prove the existence and uniqueness of weak $L^p$-solutions to the stochastic linear advection equation of $k$-forms that is driven by a H\"older continuous, $W^{1,1}_{loc}$ drift and smooth diffusion vector fields, such that the equation without noise admits infinitely many solutions.