Non-uniqueness in Law of the Surface Quasi-Geostrophic Equations: the Case of Linear Multiplicative Noise

The momentum formulation of the surface quasi-geostrophic equations consists of two nonlinear terms, besides the pressure term, one of which cannot be written in a divergence form. When the anti-divergence operator is applied to such nonlinear terms, in general, one cannot take advantage of the differentiation operator of order minus one unless the nonlinear terms are compactly supported away from the origin in Fourier frequency. Moreover, the two nonlinear terms of the momentum surface quasi-geostrophic equations are one derivative more singular than that of the Navier-Stokes equations. Upon employing the convex integration technique to the random partial differential equations corresponding to the momentum surface quasi-geostrophic equations forced by linear multiplicative noise, these issues create various difficulties, unseen in the deterministic scenario and even in the case the noise is additive. By making a key observation in case the solution is a shear flow and rewriting the difficult stochastic commutator error in terms of the oscillation error, we prove its non-uniqueness in law.