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

  • Ionuţ MunteanuEmail author
  • Michael Röckner


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.


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

Mathematics Subject Classification

60H15 35Q30 76F20 76N10 



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.


  1. 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.MathSciNetCrossRefGoogle Scholar
  2. 2.
    V. Barbu, M. Röckner, Global solutions to random 3D vorticity equations for small initial data, J. Differ. Equ., 263 (2017), 5395–5411.MathSciNetCrossRefGoogle Scholar
  3. 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.MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Bensoussan, R. Temam, Equations stochastique du type Navier–Stokes. J. Funct. Anal., 13 (1973), 195–222.CrossRefGoogle Scholar
  5. 5.
    O. Bernt, Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin, 2003.zbMATHGoogle Scholar
  6. 6.
    Z. Brzezniak, M. Capinski, F. Flandoli, A convergence result for stochastic partial differential equations, Stochastics 24(4) (1988), 423–445.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Z. Brzezniak, M. Capinski, F. Flandoli, Stochastic partial differential equations and turbulence, Math. Models Methods Appl. Sci., 1 (1991), 41–59.MathSciNetCrossRefGoogle Scholar
  8. 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.MathSciNetCrossRefGoogle Scholar
  9. 9.
    A.P. Carlderon, A. Zygmund, On the existence of certain singular integrals, Acta Matematica, 88(1) (1952), 85–139.MathSciNetCrossRefGoogle Scholar
  10. 10.
    F. Flandoli, D. Gatarek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Probab. Theory Related Fields, 102 (1995), 367–391.MathSciNetCrossRefGoogle Scholar
  11. 11.
    N. Jacob, Pseudo-Differential Operators and Markov Processes, vol 1, Imerial College Press, London, 2001.CrossRefGoogle Scholar
  12. 12.
    T. Kato, H. Fujita, On the nonstationary Navier–Stokes system, Rend. Sem. Mater. Univ. Padova, 32 (1962), 243–260.MathSciNetzbMATHGoogle Scholar
  13. 13.
    R. Mikulevicius, B.L. Rozovskii, Stochastic Navier–Stokes equations for turbulent flows, SIAM J. Math. Anal., 35(5) (2004), 1250–1310.MathSciNetCrossRefGoogle Scholar
  14. 14.
    R. Mikulevicius, B.L. Rozovskii, Global \(L^2\)-solutions of stochastic Navier–Stokes equations, Ann. Probab., 33(1) (2005), 137–176.MathSciNetCrossRefGoogle Scholar
  15. 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.MathSciNetCrossRefGoogle Scholar
  16. 16.
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.zbMATHGoogle Scholar

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

Personalised recommendations