Strong pathwise solutions of the stochastic Navier-Stokes system

Advances in Differential Equations

Volumn 14, Issue 5-6, 2009, Pages 567-600

  Glatt Holtz, Nathan ^a   Ziane, Mohammed ^b  

^a INDIANA UNIVERSITY   (United States)

^b UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 73249124953     PISSN: 10799389     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (31)

1. R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Pure and Applied Mathematics, 140, Academic Press, 2003.
2. A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Functional Analysis, 13 (1973), 195-222.
3. A. Bensoussan and J. Frehse, Local solutions for stochastic Navier Stokes equations, Special issue for R. Temam's 60th birthday, M2AN Math. Model. Numer. Anal., 34 (2000), 241-273.
4. H. Breckner, Galerkin approximation and the strong solution of the Navier-Stokes equation, J. Appl. Math. Stochastic Anal., 13 (2000), 239-259.
5. Brzézniak, Z. and Peszat, S., Strong local and global solutions for stochastic Navier-Stokes equations, Infinite dimensional stochastic analysis (Amsterdam, 1999), Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., 52 (2000), 85-98.
6. M. Capínski and D. Gatarek, Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension, J. Funct. Anal., 126 (1994), 26-35.
7. M. Capínski and S. Peszat, Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 185-200.
8. P. Constantin and C. Foias, "Navier-Stokes equations," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
9. B. Cushman-Roisin,and J.-M. Beckers, "Introduction To Geophysical Fluid Dynamics: Physical and Numerical Aspects," Academic Press, 2007.
10. A. B. Cruzeiro, Solutions et mesures invariantes pour des équations d'évolution stochastiques du type Navier-Stokes, Exposition. Math., 7 (1989), 73-82.
11. G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.
12. G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems," London Mathematical Society Lecture Note Series, 229, Cambridge University Press, Cambridge, 1996.
13. J. L. Doob, "Measure Theory," Graduate Texts in Mathematics, 143, Springer-Verlag, New York, 1994.
14. N. Glatt-Holtz, "Well Posedness and Asymptotic Analysis for the Stochastic Equations of Geophysical Fluid Dynamics," PhD Thesis, University of Southern California, May, 2008.
15. F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), no. 3, 367-391.
16. F. Flandoli, An Introduction to 3D Stochastic Fluid Dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics, 1942 (2008), 51-150.
17. N. Glatt-Holtz and M. Ziane, The stochastic primitive equations in two space dimensions with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 801-822.
18. J. Jacod, "Calcul stochastique et problèmes de martingales," Lecture Notes in Mathematics, 714, Springer, Berlin, 1979.
19. I. Karatzas and S.E. Shreve, "Brownian Motion and Stochastic Calculus," Graduate Texts in Mathematics, 113, 2nd Edition, Springer-Verlag, New York, 1991.
20. R. E. Megginson, "An Introduction to Banach Space Theory," Graduate Texts in Mathematics, 183, Springer-Verlag, New York, 1998.
21. J.-L. Menaldi and S.S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim., 46 2002, 31-53.
22. R. Mikulevicius and B. L. Rozovskii, On equations of stochastic fluid mechanics, in "Stochastics in finite and infinite dimensions," Trends Math., 285-302, Birkhäuser Boston, Boston, MA, 2001.
23. R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310.
24. R. Mikulevicius and B. L. Rozovskii, Martingale problems for stochastic PDE's, in "Stochastic partial differential equations: six perspectives," Math. Surveys Monogr., 64, 243-325, Amer. Math. Soc., Providence, RI, 1999.
25. R. Mikulevicius and B. L. Rozovskii, Global L2-solutions of stochastic Navier-Stokes equations, Ann. Probab., 33 (2005), 137-176.
26. B. Oksendal, "Stochastic Differential Equations," Universitext, 6th Edition, Springer-Verlag, Berlin, 2003.
27. C. Prévôt and M. R̈ockner, "A Concise Course on Stochastic Partial Differential Equations," Lecture Notes in Mathematics, 1905, Springer, Berlin, 2007.
28. B. L. Rozovskii, "Stochastic evolution systems," Mathematics and its Applications (Soviet Series), 35, Kluwer Academic Publishers Group, Dordrecht, 1990.
29. R. Temam, "Navier-Stokes equations: Theory and numerical analysis: Reprint of the 1984 edition," AMS Chelsea Publishing, Providence, RI, 2001.
30. M. Viot, "Solutions faibles d'équations aux dérivées partielles non linéaires," Th'ese, Universit́e Pierre et Marie Curie, Paris, 1976.
31. K. Yosida, "Functional Analysis," Classics in Mathematics, Reprint of the sixth (1980) edition, Springer-Verlag, Berlin, 1995.