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On the Navier–Stokes equation perturbed by rough transport noise

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Abstract

We consider the Navier–Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space–time-dependent noise that is smooth in space and rough in time. We study the system within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of a weak solution of the Navier–Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we prove that the solution is unique and stable with respect to the driving noise.

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References

  1. T. J. Lyons. Differential equations driven by rough signals. Revista Matemática Iberoamericana, 14(2):215–310, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Caruana and P. Friz. Partial differential equations driven by rough paths. J. Differential Equations, 247(1):140–173, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Caruana, P. K. Friz, and H. Oberhauser. A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. In Annales de l’Institut Henri Poincare (C) Non Linear Analysis, volume 28, pages 27–46. Elsevier, 2011.

  4. M. Gubinelli and S. Tindel. Rough evolution equations. The Annals of Probability, 38(1):1–75, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Deya, M. Gubinelli, and S. Tindel. Non-linear rough heat equations. Probability Theory and Related Fields, 153(1):97–147, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Gubinelli, A. Lejay, and S. Tindel. Young integrals and SPDEs. Potential Analysis, 25(4):307–326, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Gubinelli, P. Imkeller, and N. Perkowski. Paracontrolled distributions and singular pdes. In Forum of Mathematics, Pi, volume 3, page e6. Cambridge Univ Press, 2015.

  8. M. Hairer. A theory of regularity structures. Invent. Math., 198(2):269–504, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Bailleul and M. Gubinelli. Unbounded rough drivers. Annales de la Faculté des Sciences de Toulouse, 26(4):795–830, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Deya, M. Gubinelli, M. Hofmanová, and S. Tindel. A priori estimates for rough PDEs with application to rough conservation laws. arXiv preprint arXiv:1604.00437, 2016.

  11. A. Hocquet and M. Hofmanová. An energy method for rough partial differential equations. Journal of Differential Equations, 265(4):1407–1466, 2018.

    Article  MathSciNet  Google Scholar 

  12. A. Deya, M. Gubinelli, M. Hofmanová, and S. Tindel. One-dimensional reflected rough differential equations. arXiv preprint arXiv:1610.07481, 2016 (To appear in Stochastic processes and their applications).

  13. L. C. Young. An inequality of the Hölder type, connected with Stieltjes integration. Acta Mathematica, 67(1):251–282, 1936.

    Article  MathSciNet  MATH  Google Scholar 

  14. Colin J Cotter, Georg A Gottwald, and Darryl D Holm. Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics. Proc. R. Soc. A, 473(2205):20170388, 2017.

  15. R. H. Kraichnan. Small-scale structure of a scalar field convected by turbulence. The Physics of Fluids, 11(5):945–953, 1968.

    Article  MATH  Google Scholar 

  16. K. Gawedzki and A. Kupiainen. University in turbulence: An exactly solvable model. Low-dimensional models in statistical physics and quantum field theory, pages 71–105, 1996.

  17. K. Gawȩdzki and M. Vergassola. Phase transition in the passive scalar advection. Physica D: Nonlinear Phenomena, 138(1):63–90, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  18. Z. Brzeźniak, M. Capiński, and F. Flandoli. Stochastic partial differential equations and turbulence. Mathematical Models and Methods in Applied Sciences, 1(01):41–59, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  19. Z. Brzeźniak, M. Capiński, and F. Flandoli. Stochastic Navier-Stokes equations with multiplicative noise. Stochastic Analysis and Applications, 10(5):523–532, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Mikulevicius and B. L. Rozovskii. Stochastic Navier–Stokes equations for turbulent flows. SIAM Journal on Mathematical Analysis, 35(5):1250–1310, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Mikulevicius and B. L. Rozovskii. Global L2-solutions of stochastic Navier–Stokes equations. The Annals of Probability, 33(1):137–176, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Flandoli and D. Gatarek. Martingale and stationary solutions for stochastic Navier-Stokes equations. Probability Theory and Related Fields, 102(3):367–391, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  23. I. Chueshov and A. Millet. Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem. Stoch. Anal. Appl., 29(4):570–611, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  24. I. Chueshov and A. Millet. Stochastic 2D hydrodynamical type systems: well posedness and large deviations. Appl. Math. Optim., 61(3):379–420, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  25. R Mikulevicius. On the Cauchy problem for the stochastic Stokes equations. SIAM Journal on Mathematical Analysis, 34(1):121–141, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  26. R Mikulevicius, B Rozovskii, et al. A note on Krylov’s \(l\_p\)-theory for systems of SPDEs. Electronic Journal of Probability, 6, 2001.

  27. R. Temam. Navier-Stokes equations and nonlinear functional analysis, volume 41 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983.

  28. Emmanuel Hebey. Nonlinear analysis on manifolds: Sobolev spaces and inequalities, volume 5. American Mathematical Soc., 2000.

  29. P. K. Friz and N. B. Victoir. Multidimensional stochastic processes as rough paths: theory and applications, volume 120. Cambridge University Press, 2010.

    Book  MATH  Google Scholar 

  30. T. J. Lyons, M. Caruana, and T. Lévy. Differential equations driven by rough paths, volume 1908 of Lecture Notes in Mathematics. Springer, Berlin, 2007. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With an introduction concerning the Summer School by Jean Picard.

  31. P. K. Friz and M. Hairer. A course on rough paths: with an introduction to regularity structures. Universitext. Springer, Cham, 2014.

    Book  MATH  Google Scholar 

  32. P. K. Friz and H. Oberhauser. Rough path stability of (semi-) linear spdes. Probability Theory and Related Fields, 158(1-2):401–434, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Diehl, P. K. Friz, and H. Oberhauser. Regularity theory for rough partial differential equations and parabolic comparison revisited. In Stochastic analysis and applications 2014, volume 100 of Springer Proc. Math. Stat., pages 203–238. Springer, Cham, 2014.

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Authors and Affiliations

  1. Institute of Mathematics, Technical University of Berlin, Berlin, Germany

    Martina Hofmanová & Torstein Nilssen

  2. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Martina Hofmanová

  3. Department of Mathematics, University of Southern California, Los Angeles, USA

    James-Michael Leahy

  4. Department of Mathematics, University of Oslo, Oslo, Norway

    Torstein Nilssen

Authors
  1. Martina Hofmanová
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  2. James-Michael Leahy
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  3. Torstein Nilssen
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Correspondence to Torstein Nilssen.

Additional information

M. Hofmanová and T. Nilssen: Financial support by the DFG via Research Unit FOR 2402 is gratefully acknowledged. T. Nilssen: Funded by the Norwegian Research Council (Project 230448/F20).

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Hofmanová, M., Leahy, JM. & Nilssen, T. On the Navier–Stokes equation perturbed by rough transport noise. J. Evol. Equ. 19, 203–247 (2019). https://doi.org/10.1007/s00028-018-0473-z

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  • DOI: https://doi.org/10.1007/s00028-018-0473-z

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