Global solutions for random vorticity equations perturbed by gradient dependent noise, in two and three dimensions

Abstract

The aim of this work is to prove an existence and uniqueness result of Kato–Fujita type for the Navier–Stokes equations, in vorticity form, in 2D and 3D, perturbed by a gradient-type multiplicative Gaussian noise (for sufficiently small initial vorticity). These equations are considered in order to model hydrodynamic turbulence. The approach was motivated by a recent result by Barbu and Röckner (J Differ Equ 263:5395–5411, 2017) that treats the stochastic 3D Navier–Stokes equations, in vorticity form, perturbed by linear multiplicative Gaussian noise. More precisely, the equation is transformed to a random nonlinear parabolic equation, as in Barbu and Röckner (2017), but the transformation is different and adapted to our gradient-type noise. Then, global unique existence results are proved for the transformed equation, while for the original stochastic Navier–Stokes equations, existence of a solution adapted to the Brownian filtration is obtained up to some stopping time.

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Acknowledgements

I.M. was supported by a grant of the “Alexandru Ioan Cuza” University of Iasi, within the Research Grants program, Grant UAIC, code GI-UAIC-2018-03. Financial support by the DFG through CRC 1283 is gratefully acknowledged by M.R.

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Correspondence to Ionuţ Munteanu.

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Munteanu, I., Röckner, M. Global solutions for random vorticity equations perturbed by gradient dependent noise, in two and three dimensions. J. Evol. Equ. 20, 1173–1194 (2020). https://doi.org/10.1007/s00028-019-00551-3

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Keywords

  • Stochastic Navier–Stokes equation
  • Turbulence
  • Vorticity
  • Biot–Savart operator
  • Gradient-type noise

Mathematics Subject Classification

  • 60H15
  • 35Q30
  • 76F20
  • 76N10