Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier–Stokes equations

Abstract

We consider a family of stochastic 2D Euler equations in vorticity form on the torus, with transport-type noises and \(L^2\)-initial data. Under a suitable scaling of the noises, we show that the solutions converge weakly to that of the deterministic 2D Navier–Stokes equations. Consequently, we deduce that the weak solutions of the stochastic 2D Euler equations are approximately unique and “weakly quenched exponential mixing.”

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Acknowledgements

The authors would like to thank the referee for many valuable comments on the first version of the paper. The third named author is grateful to the financial supports of the grant “Stochastic models with spatial structure” from the Scuola Normale Superiore di Pisa, the National Natural Science Foundation of China (Nos. 11571347, 11688101), and the Youth Innovation Promotion Association, CAS (2017003).

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Correspondence to Dejun Luo.

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Flandoli, F., Galeati, L. & Luo, D. Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier–Stokes equations. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00592-z

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Keywords

  • 2D Euler equations
  • Vorticity
  • Transport noise
  • Scaling limit
  • 2D Navier–Stokes equations

Mathematics Subject Classification

  • 35R60
  • 35Q31