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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

Bo You

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

Received  March 2020 Revised  August 2020 Published  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].

Keywords: Well-posedness, Planetary geostrophic equations, Additive noise, Weak $ \mathcal{D} $-pullback mean random attractors.
Mathematics Subject Classification: Primary: 35R60, 37L55, 60H30; Secondary: 35Q86.
Citation: Bo You. Well-posedness for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing, Tokyo, 1989.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[12]

M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Quaderni, Scuola Normale Superiore di Pisa, 1988.  Google Scholar

[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.  Google Scholar

[14]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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B. Schmalfuss, Qualitative properties for the stochastic Navier-Stokes equation, Nonlinear Anal., 28 (1997), 1545-1563.  doi: 10.1016/S0362-546X(96)00015-6.  Google Scholar

[19]

A. V. Skorohod, Studies in the Theory of Random Processes, Addison-Wesley Publishing Co., Inc., Reading, Mass, 1965.  Google Scholar

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing, Tokyo, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Quaderni, Scuola Normale Superiore di Pisa, 1988.  Google Scholar

[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.  Google Scholar

[14]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. V. Skorohod, Studies in the Theory of Random Processes, Addison-Wesley Publishing Co., Inc., Reading, Mass, 1965.  Google Scholar

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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