Non-uniqueness for the stochastic incompressible Euler equations with a passive tracer

TL;DR

This paper demonstrates pathwise non-uniqueness for stochastic incompressible Euler equations with a passive tracer, extending previous deterministic results to stochastic settings and applying the findings to stochastic ideal MHD equations.

Contribution

It introduces a novel approach to establish non-uniqueness of solutions for stochastic Euler equations using the Baire category method and transforms SPDEs into PDEs with random coefficients.

Findings

Constructs infinitely many global weak solutions in any dimension ≥ 2.

Establishes pathwise non-uniqueness for the original stochastic PDEs.

Extends non-uniqueness results to stochastic ideal MHD equations.

Abstract

In this work we investigate the phenomenon of pathwise non-uniqueness for the stochastic incompressible Euler equations with a passive tracer on the whole Euclidean space. The stochastic perturbations are interpreted as a transport noise and a linear multiplicative noise in the Stratonovich sense. In both cases, via classical transformations, we convert the SPDEs into PDEs with random coefficients. Using the Baire category method developed by De Lellis and Sz\'ekelyhidi Jr., we then construct infinitely many global-in-time weak solutions to the random PDEs in any spatial dimension greater than or equal to two. By applying the inverse transformations, we obtain pathwise non-uniqueness for the original SPDEs. Finally, we present an application of our result to the three-dimensional stochastic ideal MHD equations. This study can be regarded as a stochastic counterpart of Bronzi et al.~[\textit{Commun.~Math.~Sci.},~2015]. In particular, our non-uniqueness result in the random setting extends theirs from a constant energy profile equal to one to arbitrary positive, bounded, and continuous time-dependent profiles, originally established in two dimensions via the convex integration technique.