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Seminar on Stochastic Analysis, Random Fields and Applications VI pp 227–244Cite as

The Martingale Problem for Markov Solutions to the Navier-Stokes Equations

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Part of the book series: Progress in Probability ((PRPR,volume 63))

Abstract

Under suitable assumptions of regularity and non-degeneracy on the covariance of the driving additive noise, any Markov solution to the stochastic Navier-Stokes equations has an associated generator of the diffusion and is the unique solution to the corresponding martingale problem. Some elementary examples are discussed to interpret these results.

Mathematics Subject Classification (2000). Primary: 76D05; Secondary: 60H15, 35Q30, 60H30, 76M35.

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References

  1. D. Bl¨omker, F. Flandoli, and M. Romito, Markovianity and ergodicity for a surface growth PDE, Ann. Probab., 37 (1) (2009), 275–313.

    Google Scholar 

  2. S. Cerrai, Second-order PDE’s in finite and infinite dimension, Lecture Notes in Mathematics, 1762 (2001), Springer-Verlag, Berlin.

    Google Scholar 

  3. G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl., 82 (2003), 877–947.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Da Prato and A. Debussche, On the martingale problem associated to the 2D and 3D Stochastic Navier-Stokes equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (3) (2008), 247–264.

    Google Scholar 

  5. A. Debussche and C. Odasso, Markov solutions for the 3D stochastic Navier-Stokes equations with state dependent noise, J. Evol. Equ., 6 (2) (2006), 305–324.

    Article  MathSciNet  MATH  Google Scholar 

  6. H.J. Engelbert and W. Schmidt, On solutions of one-dimensional stochastic differential equations without drift, Z. Wahrsch. Verw. Gebiete, 68 (3) (1985), 287–314.

    Article  MathSciNet  MATH  Google Scholar 

  7. C.L. Feffermann, Existence and smoothness of the Navier-Stokes equation, in: The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, 2006, 57–67.

    Google Scholar 

  8. F. Flandoli, Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations, Metrika, 69 (2) (2009), 101–123.

    Article  MathSciNet  Google Scholar 

  9. F. Flandoli, M. Gubinelli, and E. Priola, Well-posedness of the transport equation by stochastic perturbation, to appear in Inven. Math.

    Google Scholar 

  10. F. Flandoli and M. Romito, Partial regularity for the stochastic Navier-Stokes equations, Trans. Amer. Math. Soc., 354 (6) (2002), 2207–2241.

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Flandoli and M. Romito, Markov selections and their regularity for the threedimensional stochastic Navier-Stokes equations, C. R. Math. Acad. Sci. Paris, Ser. I, 343 (2006), 47–50.

    Google Scholar 

  12. F. Flandoli and M. Romito, Regularity of transition semigroups associated to a 3D stochastic Navier-Stokes equation, in: P.H. Baxendale and S.V. Lototski, Eds., Stochastic Differential Equations: Theory and Applications, Interdisciplinary Mathematical Sciences vol. 2, World Scientific Singapore, 2007.

    Google Scholar 

  13. F. Flandoli and M. Romito, Markov selections for the 3D stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 140 (3–4) (2008), 407–458.

    MathSciNet  MATH  Google Scholar 

  14. I.V. Girsanov, An example of non-uniqueness of the solution of the stochastic equation of K. Ito (English, Russian original), Theor. Probab. Appl., 7 (1962), 325–331.

    Google Scholar 

  15. B. Goldys, M. R¨ockner, and X. Zhang, Martingale solutions and Markov selections for stochastic partial differential equations, preprint, 2008.

    Google Scholar 

  16. M. Hairer and J.C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math. (2), 164 (3) (2006), 993–1032.

    Google Scholar 

  17. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113 (second edition), Springer-Verlag, New York, 1991.

    Google Scholar 

  18. N.V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 691–708.

    MathSciNet  Google Scholar 

  19. H.P. McKean, Stochastic Integrals, Probability and Mathematical Statistics, 5, Academic Press, New York, 1969.

    Google Scholar 

  20. C. Odasso, Exponential mixing for the 3D stochastic Navier-Stokes equations, Comm. Math. Phys., 270 (1) (2007), 109–139.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Romito, Analysis of equilibrium states of Markov solutions to the 3D Navier- Stokes equations driven by additive noise, J. Stat. Phys., 131 (3) (2008), 415–444.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Romito and L. Xu, Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise, to appear in Stoch. Proc. Appl.

    Google Scholar 

  23. D.W. Stroock and S.R.S. Varadhan, Diffusion processes with boundary conditions, Comm. Pure Appl. Math., 24 (1971), 147–225.

    Article  MathSciNet  MATH  Google Scholar 

  24. D.W. Stroock and S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin, 1979.

    Google Scholar 

  25. D.W. Stroock and M. Yor, On extremal solutions of martingale problems, Ann. Sci. ´ Ecole Norm. Sup. (4), 13 (1) (1980), 95–164.

    Google Scholar 

  26. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis (second edition), CBMS-NSF Regional Conference Series in Applied Mathematics, 66, SIAM, Philadelphia, 1995.

    Google Scholar 

  27. K. Yosida, Functional Analysis, Grundlehren der Mathematischen Wissenschaften, 123, Springer-Verlag, Berlin-New York, 1980.

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Firenze, viale Morgagni 67/A, I-50134, Firenze, Italia

    Marco Romito

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  1. Marco Romito
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Correspondence to Marco Romito .

Editor information

Editors and Affiliations

  1. Fac. Sciences de Base (FSB), Institut de Mathématiques (IMA), Ecole Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland

    Robert Dalang

  2. Institut Élie Cartan IECN, Nancy Université, Vandoeuvre-les-Nancy CX, France

    Marco Dozzi

  3. , Unité de Mathématiques appliquées, Ecole Nationale Supérieure des Technique, Boulevard Victor 32, Paris Cedex 15, 75739, France

    Francesco Russo

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Romito, M. (2011). The Martingale Problem for Markov Solutions to the Navier-Stokes Equations. In: Dalang, R., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability, vol 63. Springer, Basel. https://doi.org/10.1007/978-3-0348-0021-1_15

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