Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. Barbu, M. Röckner, An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise, J. Eur. Math. Soc., 17 (2015), 1789–1815.

    Article  MathSciNet  Google Scholar 

  2. V. Barbu, M. Röckner, Global solutions to random 3D vorticity equations for small initial data, J. Differ. Equ., 263 (2017), 5395–5411.

    Article  MathSciNet  Google Scholar 

  3. V. Barbu, M. Röckner, Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise, Arch. Ration. Mech. Anal., 209(3) (2013), 797–834.

    Article  MathSciNet  Google Scholar 

  4. A. Bensoussan, R. Temam, Equations stochastique du type Navier–Stokes. J. Funct. Anal., 13 (1973), 195–222.

    Article  Google Scholar 

  5. O. Bernt, Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin, 2003.

    MATH  Google Scholar 

  6. Z. Brzezniak, M. Capinski, F. Flandoli, A convergence result for stochastic partial differential equations, Stochastics 24(4) (1988), 423–445.

    Article  MathSciNet  Google Scholar 

  7. Z. Brzezniak, M. Capinski, F. Flandoli, Stochastic partial differential equations and turbulence, Math. Models Methods Appl. Sci., 1 (1991), 41–59.

    Article  MathSciNet  Google Scholar 

  8. Z. Brzezniak, E. Motyl, Existence of a martingale solution of the stochastic Navier–Stokes equations in unbounded 2D and 3D domains, J. Differ. Equ., 254 (2013), 1627-1685.

    Article  MathSciNet  Google Scholar 

  9. A.P. Carlderon, A. Zygmund, On the existence of certain singular integrals, Acta Matematica, 88(1) (1952), 85–139.

    Article  MathSciNet  Google Scholar 

  10. F. Flandoli, D. Gatarek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Probab. Theory Related Fields, 102 (1995), 367–391.

    Article  MathSciNet  Google Scholar 

  11. N. Jacob, Pseudo-Differential Operators and Markov Processes, vol 1, Imerial College Press, London, 2001.

    Book  Google Scholar 

  12. T. Kato, H. Fujita, On the nonstationary Navier–Stokes system, Rend. Sem. Mater. Univ. Padova, 32 (1962), 243–260.

    MathSciNet  MATH  Google Scholar 

  13. R. Mikulevicius, B.L. Rozovskii, Stochastic Navier–Stokes equations for turbulent flows, SIAM J. Math. Anal., 35(5) (2004), 1250–1310.

    Article  MathSciNet  Google Scholar 

  14. R. Mikulevicius, B.L. Rozovskii, Global \(L^2\)-solutions of stochastic Navier–Stokes equations, Ann. Probab., 33(1) (2005), 137–176.

    Article  MathSciNet  Google Scholar 

  15. I. Munteanu, M. Röckner, The total variation flow perturbed by gradient linear multiplicative noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21(1) (2018), 28pp.

    Article  MathSciNet  Google Scholar 

  16. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.

    MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, Alexandru Ioan Cuza University of Iaşi, Blvd. Carol I, No.11, 700506, Iasi, Romania

    Ionuţ Munteanu

  2. Octav Mayer Institute of Mathematics, Romanian Academy, Blvd. Carol I, No.8, 700505, Iasi, Romania

    Ionuţ Munteanu

  3. Fakultat fur Mathematik, Universitat Bielefeld, 33501, Bielefeld, Germany

    Michael Röckner

Authors
  1. Ionuţ Munteanu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Röckner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ionuţ Munteanu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-019-00551-3

Keywords

Mathematics Subject Classification

Search

Navigation