Pathwise solutions of the 2-D stochastic primitive equations

Applied Mathematics and Optimization

Volumn 63, Issue 3, 2011, Pages 401-433

  Glatt Holtz, Nathan ^a   Temam, Roger ^a  

^a INDIANA UNIVERSITY   (United States)

Author keywords

Anisotropic estimates; Cauchy estimates; Mathematical geophysics; Multiplicative noise; Nonlinear spdes; Pathwise solutions; Primitive equations; Stopping time arguments; Strong solutions

Indexed keywords

ANISOTROPIC ESTIMATES; CAUCHY ESTIMATES; MATHEMATICAL GEOPHYSICS; MULTIPLICATIVE NOISE; NONLINEAR SPDES; PATHWISE SOLUTIONS; PRIMITIVE EQUATIONS; STOPPING TIME; STRONG SOLUTION;

ANISOTROPY; ESTIMATION;

STOCHASTIC SYSTEMS;

EID: 79957975942     PISSN: 00954616     EISSN: 14320606     Source Type: Journal    
DOI: 10.1007/s00245-010-9126-5     Document Type: Article

References (50)

1. Bensoussan, A.: Stochastic Navier-Stokes equations. Acta Appl. Math. 38(3), 267-304 (1995).
2. Bensoussan, A., Frehse, J.: Local solutions for stochastic Navier Stokes equations. Math. Model. Numer. Anal. 34(2), 241-273 (2000). Special issue for R. Temam's 60th birthday.
3. Bensoussan, A., Temam, R.: Équations aux dérivées partielles stochastiques non linéaires. I. Israel J. Math. 11, 95-129 (1972).
4. Bensoussan, A., Temam, R.: Équations stochastiques du type Navier-Stokes. J. Functional Analysis 13, 195-222 (1973).
5. Berner, J., Shutts, G.J., Leutbecher, M., Palmer, T.N.: A spectral stochastic kinetic energy backscatter scheme and its impact on flow-dependent predictability in the ecmwf ensemble prediction system. Journal of the Atmospheric Sciences 66(3), 603-626 (2009).
6. Breckner, H.: Galerkin approximation and the strong solution of the Navier-Stokes equation. J. Appl. Math. Stochastic Anal. 13(3), 239-259 (2000).
7. Bresch, D., Kazhikhov, A., Lemoine, J.: On the two-dimensional hydrostatic Navier-Stokes equations. SIAM J. Math. Anal. 36(3), 796-814 (2004/05). (Pubitemid 40672793)
8. Brzézniak, Z., Peszat, S.: Strong local and global solutions for stochastic Navier-Stokes equations. In: Infinite Dimensional Stochastic Analysis, Amsterdam, 1999. Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet, vol. 52, pp. 85-98. R. Neth. Acad. Arts Sci., Amsterdam (2000).
9. Cao, C., Titi, E.: Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. Math. (2) 166(1), 245-267 (2007).
10. Capiński, M., Gatarek, D.: Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension. J. Funct. Anal. 126(1), 26-35 (1994).
11. Cruzeiro, A.B.: Solutions et mesures invariantes pour des équations d'évolution stochastiques du type Navier-Stokes. Exposition. Math. 7(1), 73-82 (1989).
12. Cushman-Roisin, B., Beckers, J.-M.: Introduction to geophysical fluid dynamics: Physical and numerical aspects. To be published by Academic Press (2010).
13. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1992).
14. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series, vol. 229. Cambridge University Press, Cambridge (1996).
15. Ewald, B., Penland, C.: Numerical generation of stochastic differential equations in climate models. In: Special Volume on Computational Methods for the Atmosphere and the Oceans. Handbook of Numerical Analysis, vol. 14, pp. 279-306. Elsevier/North-Holland, Amsterdam (2009).
16. Ewald, B., Petcu, M., Temam, R.: Stochastic solutions of the two-dimensional primitive equations of the ocean and atmosphere with an additive noise. Anal. Appl. (Singap.) 5(2), 183-198 (2007).
17. Flandoli, F.: An introduction to 3d stochastic fluid dynamics. In: SPDE in Hydrodynamic: Recent Progress and Prospects. Lecture Notes in Mathematics, vol. 1942, pp. 51-150. Springer, Berlin (2008).
18. Glatt-Holtz, N., Temam, R.: Cauchy convergence schemes for some nonlinear partial differential equations. Appl. Anal. Special Issue in honor of Prof. V. Solonnikov (to appear).
19. Glatt-Holtz, N., Ziane, M.: The stochastic primitive equations in two space dimensions with multiplicative noise. Discrete Contin. Dyn. Syst. Ser. B 10(4), 801-822 (2008).
20. Glatt-Holtz, N., Ziane, M.: Strong pathwise solutions of the stochastic Navier-Stokes system. Advances in Differential Equations 14(5-6), 567-600 (2009).
21. Glatt-Holtz, N., Temam, R., Tribbia, J.: Some remarks on the role of stochastic parameterization in the equations of the ocean and atmosphere (manuscript in preparation).
22. Guo, B., Huang, D.: 3d stochastic primitive equations of the large-scale ocean: global well-posedness and attractors. Commun. Math. Phys. 286, 697-723 (2009).
23. Jacod, J.: Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979).
24. Kobelkov, G.M.: Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations. C. R. Math. Acad. Sci. Paris 343(4), 283-286 (2006). (Pubitemid 44103602)
25. Kobelkov, G.M.: Existence of a solution "in the large" for ocean dynamics equations. J. Math. Fluid Mech. 9(4), 588-610 (2007). (Pubitemid 351169494)
26. Kukavica, I., Ziane, M.: On the regularity of the primitive equations of the ocean. Nonlinearity 20(12), 2739-2753 (2007).
27. Leslie, D.C., Quarini, G.L.: The application of turbulence theory to the formulation of subgrid modelling procedures. Journal of Fluid Mechanics 91, 65-91 (1979). (Pubitemid 9447453)
28. Lions, J.-L., Temam, R., Wang, S.H.: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5(2), 237-288 (1992).
29. Lions, J.-L., Temam, R., Wang, S.H.: On the equations of the large-scale ocean. Nonlinearity 5(5), 1007-1053 (1992).
30. Lions, J.-L., Temam, R., Wang, S.:Models for the coupled atmosphere and ocean (CAO I, II). Comput. Mech. Adv. 1(1), 120 (1993).
31. Mason, P.J., Thomson, D.J.: Stochastic backscatter in large-eddy simulations of boundary layers. Journal of Fluid Mechanics 242, 51-78 (1992).
32. Mikulevicius, R., Rozovskii, B.L.: Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal. 35(5), 1250-1310 (2004). (Pubitemid 39288776)
33. Mikulevicius, R., Rozovskii, B.L.: Global L2-solutions of stochastic Navier-Stokes equations. Ann. Probab. 33(1), 137-176 (2005). (Pubitemid 40367060)
34. Pedlosky, J.: Geophysical Fluid Dynamics. Springer, Berlin (1982).
35. Penland, C., Ewald, B.D.: On modelling physical systems with stochastic models: diffusion versus Lévy processes. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366(1875), 2457-2476 (2008).
36. Penland, C., Sardeshmukh, P.D.: The optimal growth of tropical sea surface temperature anomalies. Journal of Climate 8(8), 1999-2024 (1995). (Pubitemid 26472943)
37. Petcu, M., Temam, R., Wirosoetisno, D.: Existence and regularity results for the primitive equations in two space dimensions. Commun. Pure Appl. Anal. 3(1), 115-131 (2004). (Pubitemid 41301984)
38. Petcu, M., Temam, R., Ziane, M.: Some mathematical problems in geophysical fluid dynamics. In: Special Volume on Computational Methods for the Atmosphere and the Oceans. Handbook of Numerical Analysis, vol. 14, pp. 577-750. Elsevier, Amsterdam (2008).
39. Potter, T., Colman, B. (eds.): Handbook of Weather, Climate and Water: Atmospheric Chemistry, Hydrology and Societal Impacts. Wiley, New York (2003).
40. Potter, T., Colman, B. (eds.): Handbook ofWeather, Climate andWater: Dynamics, Climate, Physical Meteorology, Weather Systems and Measurements. Wiley, New York (2003).
41. Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2007).
42. Rose, H.A.: Eddy diffusivity, eddy noise and subgrid-scale modelling. Journal of Fluid Mechanics 81, 719-734 (1977).
43. Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. AMS/Chelsea, Providence (2001). Reprint of the 1984 edition.
44. Temam, R., Tribbia, J. (eds.): Computational Methods for the Oceans and the Atmosphere. Special Volume of the Handbook of Numerical Analysis. Elsevier, Amsterdam (2008).
45. Temam, R., Ziane, M.: Some mathematical problems in geophysical fluid dynamics. In: Handbook of Mathematical Fluid Dynamics, vol. III, pp. 535-657. North-Holland, Amsterdam (2004).
46. Temam, R., Rousseau, A., Tribbia, J.: Boundary value problems for the inviscid primitive equations in limited domain. In: Special Volume on Computational Methods for the Atmosphere and the Oceans. Handbook of Numerical Analysis, vol. 14, pp. 577-750. Elsevier, Amsterdam (2008).
47. Trenberth, K. (ed.): Climate System Modeling, 1st edn. Cambridge University Press, Cambridge (1993).
48. Viot, M.: Solutions faibles d'équations aux dérivées partielles non linéaires. Thèse, Université Pierre et Marie Curie, Paris (1976).
49. Ziane, M.: Regularity results for Stokes type systems. Appl. Anal. 58(3-4), 263-292 (1995).
50. Zidikheri, M.J., Frederiksen, J.S.: Stochastic subgrid-scale modelling for non-equilibrium geophysical flows. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368(1910), 145-160 (2010).