Abstract
In this paper, we consider the global well-posedness and long-time dynamics for the three-dimensional viscous primitive equations describing the large-scale oceanic motion under a random forcing, which is an additive white in time noise. We firstly prove the existence and uniqueness of global strong solutions to the initial boundary value problem for the stochastic primitive equations. Subsequently, by studying the asymptotic behavior of strong solutions, we obtain the existence of random attractors for the corresponding random dynamical system.
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Adams R.A.: Sobolev Space. Academic Press, New York (1975)
Arnold, L.: Random Dynamical System, Springer Monograghs in Mathematics, Berlin: Springer-Verlag 1998
Bourgeois A.J., Beale J.T.: Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean. SIAM J. Math. Anal. 25, 1023–1068 (1994)
Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations (in Russian), Moscow: Nauka 1989 English translation: Amsterdam: North Holland, 1992
Cordoba D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148, 1135–1152 (1998)
Crauel H., Debussche A., Flandoli F.: Random attractors. J. Dyn. Diff. Eq. 29(2), 307–341 (1997)
Crauel H., Flandoli F.: Attractors of random dynamics systems. Prob. Th. Rel. Fields 100, 365–393 (1994)
Charney J.G., Fjortaft R., Von Neumann J.: Numerical integration of the barotropic vorticity equation. Tellus 2, 237–254 (1950)
Constantin, P., Foias, C., Temam, R.: Attractors representing turbolent flows. Memoirs of AMS, Vol. 53, No. 314, 1985
Constantin P., Majda A., Tabak E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994)
Constantin P., Majda A., Tabak E.: Singular front formation in a model for quasigeostrophic flow. Phys. Fluids 6, 9–11 (1994)
Charney J.G., Philips N.A.: Numerical integration of the quasi-geostrophic equations for barotropic simple baroclinic flows. J. Meteor. 10, 71–99 (1953)
Cao C., Titi E.S.: Global well-posedness of the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics. Ann. of Math. 166, 245–267 (2007)
Constantin P., Wu J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (1999)
Duan J., Gao H., Schmalfuss B.: Stochastic dynamics of a coupled atmosphere-ocean model. Stoch. and Dynam. 2(3), 357–380 (2002)
Duan J., Kloeden P.E., Schmalfuss B.: Exponential stability of the quasi-geostrophic equation under random perturbations. Prog. in Probability 49, 241–256 (2001)
Duan J., Schmalfuss B.: The 3D quasi-geostrophic fluid dynamics under random forcing on boundary. Comm. in Math. Sci. 1, 133–151 (2003)
Embid P.F., Majda A.: Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity. Comm. in PDE 21, 619–658 (1996)
Flandoli F.: Dissipativity and invariant measures for stochastic Navier-Stokes equations. Nonliear Diff. Eq. Appl. 1, 403–423 (1994)
Frankignoul C., Hasselmann K.: Stochastic climate models, Part II: Application to sea-surface temperature anomalies and thermocline variability. Tellus 29, 289–305 (1977)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Berlin-Heidelberg-New York: Springer-Verlag, 1994
Guillén-González F., Masmoudi N., Rodríguez-Bellido M.A.: Anisotropic estimates and strong solutions for the primitive equations. Diff. Int. Equ. 14, 1381–1408 (2001)
Hu C., Temam R., Ziane M.: The primimitive equations of the large scale ocean under the small depth hypothesis. Disc. and Cont. Dyn. Sys. 9(1), 97–131 (2003)
Lions J.L., Temam R., Wang S.: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5, 237–288 (1992)
Lions J.L., Temam R., Wang S.: On the equations of the large scale ocean. Nonlinearity 5, 1007–1053 (1992)
Lions J.L., Temam R., Wang S.: Models of the coupled atmosphere and ocean(CAO I). Computational Mechanics Advance 1, 1–54 (1993)
Lions J.L., Temam R., Wang S.: Mathematical theory for the coupled atmosphere-ocean models (CAO III). J. Math. Pures Appl. 74, 105–163 (1995)
Kiselev A., Nazarov F., Volberg A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167, 445–453 (2007)
Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics 9 (2003)
Majda A., Eijnden E.V.: A mathematical framework for stochastic climate models. Comm. Pure Appl. Math 54, 891–974 (2001)
Majda A., Wang X.: The emergence of large-scale coherent structure under small-scale random bombardments. Comm. Pure Appl. Math 59, 467–500 (2001)
Müller, P.: Stochastic forcing of quasi-geostrophic eddies. Stochastic Modelling in Physical Oceanography, edited by R. J. Adler, P. Müller, B. Rozovskii, Basel: Birkhäuser, 1996
Mikolajewicz U., Maier-Reimer E.: Internal secular variability in an OGCM. Climate Dyn. 4, 145–156 (1990)
Pedlosky J.: Geophysical Fluid Dynamics. 2nd Edition. Springer-Verlag, Berlin/New York (1987)
Phillips O.M.: On the generation of waves by turbulent winds. J. Fluid Mech. 2, 417–445 (1957)
Da Prato G., Zabczyk J.: Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Application. Cambridge Univ. Press, Cambridge (1992)
Da Prato G., Zabczyk J.: Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge (1996)
Rubenstein D.: A spectral model of wind-forced internal waves. J. Phys. Oceanogr. 24, 819–831 (1994)
Schmalfuss, B.: Backward cocycle and attractors of stochastic differential equations, In: International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, edited by V. Reitmann, T. Riedrich, N. Kokch, Dresden: Universität 1992, pp. 185–192
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd Edition, Appl. Math. Ser., Vol. 68, New York: Springer-Verlag, 1997
Temam R.: Navier-Stokes Equations: Theory and Numerical Analysis, Revised Edition. North-Holland, Amsterdam (1984)
Samelson R., Temam R., Wang S.: Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation. Appl. Anal. 70(1–2), 147–173 (1998)
Temam, R., Ziane, M.: Some mathematical problems in geophysical fluid dynamics. Handbook of Mathematical Fluid Dynamics, 2004
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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). https://doi.org/10.1007/s00220-008-0654-7
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DOI: https://doi.org/10.1007/s00220-008-0654-7