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  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 & Continuous Dynamical Systems - A, 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. BrzeźniakE. 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. CrauelA. 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. SamelsonR. 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. SamelsonR. 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

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. BrzeźniakE. 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. CrauelA. 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. SamelsonR. 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. SamelsonR. 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

[1]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[2]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[3]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[4]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[5]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[6]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[7]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[8]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[9]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[10]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[11]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[12]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[13]

Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167

[14]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[15]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105

[16]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329

[17]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111

[18]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[19]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[20]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

2019 Impact Factor: 1.338

Article outline

[Back to Top]