# \(\rho \)-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 (Commun Math Phys 129:431–444, 1990) 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 Fokker–Planck 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 Fokker–Planck equation, we discuss how the gradient type estimate may be related to this open problem.

## Keywords

White noise 2D Euler equations Multiplicative noise Fokker–Planck equation Gradient estimates## Mathematics Subject Classification

60H15 60H40 35Q31 35Q84 76D05## Notes

### Acknowledgements

Both authors would like to thank the referees for reading carefully the manuscript and for many valuable comments which improve the presentation of the paper. The second author is grateful to the financial supports of the National Natural Science Foundation of China (Nos. 11571347, 11688101), and the Special Talent Program of the Academy of Mathematics and Systems Science, Chinese Academy of Sciences.

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