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

  • Ionuţ MunteanuEmail author
  • Michael Röckner
Article
  • 21 Downloads

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.

Keywords

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

Mathematics Subject Classification

60H15 35Q30 76F20 76N10 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsAlexandru Ioan Cuza University of IaşiIasiRomania
  2. 2.Octav Mayer Institute of MathematicsRomanian AcademyIasiRomania
  3. 3.Fakultat fur MathematikUniversitat BielefeldBielefeldGermany

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