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

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.