$\rho$-white noise solution to 2D stochastic Euler equations

Abstract

A stochastic version of 2D Euler equations with transport type noise in the vorticity is considered, in the framework of Albeverio--Cruzeiro theory [1] where the equation is considered with random initial conditions related to the so called enstrophy measure. The equation is studied by an approximation scheme based on random point vortices. Stochastic processes solving the Euler equations are constructed and their density with respect to the enstrophy measure is proved to satisfy a continuity equation in weak form. Relevant in comparison with the case without noise is the fact that here we prove a gradient type estimate for the density. Although we cannot prove uniqueness for the continuity equation, we discuss how the gradient type estimate may be related to this open problem.

We do not need hypothesis (H3) in the last version, hence we rewrite the paper under the simple conditions (H1) and (H2) stated below Definition 1.1. The main change is that we only use finite many noises in equation (2.1), so the subsequent proofs are modified accordingly. We add Remarks 1.2 and 4.2 to show that the terms in equations (1.5) and (1.6) are well defined

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