Probability Theory and Related Fields

, Volume 175, Issue 3–4, pp 783–832 | Cite as

\(\rho \)-White noise solution to 2D stochastic Euler equations

  • Franco FlandoliEmail author
  • Dejun Luo


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.


White noise 2D Euler equations Multiplicative noise Fokker–Planck equation Gradient estimates 

Mathematics Subject Classification

60H15 60H40 35Q31 35Q84 76D05 



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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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Scuola Normale Superiore of PisaPisaItaly
  2. 2.RCSDS, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesUniversity of the Chinese Academy of SciencesBeijingChina

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