# American Institute of Mathematical Sciences

April  2021, 41(4): 1579-1604. doi: 10.3934/dcds.2020332

## Well-posedness for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation

 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, P. R. China

Received  March 2020 Revised  August 2020 Published  April 2021 Early access  September 2020

Fund Project: This work was supported by the National Science Foundation of China Grant (11401459, 11871389), the Natural Science Foundation of Shaanxi Province (2018JM1012) and the Fundamental Research Funds for the Central Universities (xjj2018088)

The objective of this paper is to study the well-posedness of solutions for the three dimensional planetary geostrophic equations of large-scale ocean circulation with additive noise. Since strong coupling terms and the noise term create some difficulties in the process of showing the existence of weak solutions, we will first show the existence of weak solutions by the monotonicity methods when the initial data satisfies some "regular" condition. For the case of general initial data, we will establish the existence of weak solutions by taking a sequence of "regular" initial data and proving the convergence in probability as well as some weak convergence of the corresponding solution sequences. Finally, we establish the existence of weak $\mathcal{D}$-pullback mean random attractors in the framework developed in [11,25].

Citation: Bo You. Well-posedness for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1579-1604. doi: 10.3934/dcds.2020332
##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [2] V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1037-5. [3] Z. Brzeźniak, E. Hausenblas and J. H. Zhu, 2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139.  doi: 10.1016/j.na.2012.10.011. [4] C. S. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure and Appl. Math., 56 (2003), 198-233.  doi: 10.1002/cpa.10056. [5] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225. [6] Z. Dong and R. R. Zhang, Long-time behavior of 3D stochastic planetary geostrophic viscous model,, Stoch. Dyn., 18 (2018), 1850038, 48pp. doi: 10.1142/S0219493718500387. [7] J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617. [8] H. J. Gao and H. Liu, Well-posedness and invariant measures for a class of stochastic 3D Navier-Stokes equations with damping driven by jump noise, J. Differential Equations, 267 (2019), 5938-5975.  doi: 10.1016/j.jde.2019.06.015. [9] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing, Tokyo, 1989. [10] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753. [11] P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016. [12] M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Quaderni, Scuola Normale Superiore di Pisa, 1988. [13] J. Pedlosky, The equations for geostrophic motion in the ocean, Journal of Physical Oceanography, 14 (1984), 448-455.  doi: 10.1175/1520-0485(1984)014<0448:TEFGMI>2.0.CO;2. [14] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [15] R. M. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation, Appl. Anal., 70 (1998), 147-173.  doi: 10.1080/00036819808840682. [16] R. M. Samelson, R. Temam and S. Wang, Remarks on the planetary geostrophic model of gyre scale ocean circulation, Differential Integral Equations, 13 (2000), 1-14. [17] R. M. Samelson and G. K. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models, Journal of Physical Oceanography, 27 (1997), 186-194.  doi: 10.1175/1520-0485(1997)027<0186:ASFADS>2.0.CO;2. [18] B. Schmalfuss, Qualitative properties for the stochastic Navier-Stokes equation, Nonlinear Anal., 28 (1997), 1545-1563.  doi: 10.1016/S0362-546X(96)00015-6. [19] A. V. Skorohod, Studies in the Theory of Random Processes, Addison-Wesley Publishing Co., Inc., Reading, Mass, 1965. [20] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [21] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015. [22] B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099. [23] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269. [24] B. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.  doi: 10.1016/j.jde.2019.08.007. [25] B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5. [26] B. You, Random attractors for the three dimensional stochastical planetary geostrophic equations of large-scale ocean circulation, Stochastics, 89 (2017), 766-785.  doi: 10.1080/17442508.2016.1276913. [27] B. You, Large deviation principle for the three dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise, arXiv, (2020), 3312831. [28] B. You and F. Li, Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise, Stoch. Anal. Appl., 34 (2016), 278-292.  doi: 10.1080/07362994.2015.1126184.

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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [2] V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1037-5. [3] Z. Brzeźniak, E. Hausenblas and J. H. Zhu, 2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139.  doi: 10.1016/j.na.2012.10.011. [4] C. S. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure and Appl. Math., 56 (2003), 198-233.  doi: 10.1002/cpa.10056. [5] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225. [6] Z. Dong and R. R. Zhang, Long-time behavior of 3D stochastic planetary geostrophic viscous model,, Stoch. Dyn., 18 (2018), 1850038, 48pp. doi: 10.1142/S0219493718500387. [7] J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617. [8] H. J. Gao and H. Liu, Well-posedness and invariant measures for a class of stochastic 3D Navier-Stokes equations with damping driven by jump noise, J. Differential Equations, 267 (2019), 5938-5975.  doi: 10.1016/j.jde.2019.06.015. [9] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing, Tokyo, 1989. [10] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753. [11] P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016. [12] M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Quaderni, Scuola Normale Superiore di Pisa, 1988. [13] J. Pedlosky, The equations for geostrophic motion in the ocean, Journal of Physical Oceanography, 14 (1984), 448-455.  doi: 10.1175/1520-0485(1984)014<0448:TEFGMI>2.0.CO;2. [14] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [15] R. M. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation, Appl. Anal., 70 (1998), 147-173.  doi: 10.1080/00036819808840682. [16] R. M. Samelson, R. Temam and S. Wang, Remarks on the planetary geostrophic model of gyre scale ocean circulation, Differential Integral Equations, 13 (2000), 1-14. [17] R. M. Samelson and G. K. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models, Journal of Physical Oceanography, 27 (1997), 186-194.  doi: 10.1175/1520-0485(1997)027<0186:ASFADS>2.0.CO;2. [18] B. Schmalfuss, Qualitative properties for the stochastic Navier-Stokes equation, Nonlinear Anal., 28 (1997), 1545-1563.  doi: 10.1016/S0362-546X(96)00015-6. [19] A. V. Skorohod, Studies in the Theory of Random Processes, Addison-Wesley Publishing Co., Inc., Reading, Mass, 1965. [20] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [21] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015. [22] B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099. [23] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269. [24] B. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.  doi: 10.1016/j.jde.2019.08.007. [25] B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5. [26] B. You, Random attractors for the three dimensional stochastical planetary geostrophic equations of large-scale ocean circulation, Stochastics, 89 (2017), 766-785.  doi: 10.1080/17442508.2016.1276913. [27] B. You, Large deviation principle for the three dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise, arXiv, (2020), 3312831. [28] B. You and F. Li, Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise, Stoch. Anal. Appl., 34 (2016), 278-292.  doi: 10.1080/07362994.2015.1126184.
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