November  2016, 21(9): 3053-3073. doi: 10.3934/dcdsb.2016087

Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions

1. 

Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, School of Mathematical Science, Nanjing Normal University, Nanjing 210023

2. 

Department of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China

Received  November 2015 Revised  February 2016 Published  October 2016

Three dimensional primitive equations with a small multiplicative noise are studied in this paper. The existence and uniqueness of solutions with small initial value in a fixed probability space are obtained. The proof is based on Galerkin approximation, Itô's formula and weak convergence methods.
Citation: Hongjun Gao, Chengfeng Sun. Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3053-3073. doi: 10.3934/dcdsb.2016087
References:
[1]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Ann. Math., 166 (2007), 245.  doi: 10.4007/annals.2007.166.245.  Google Scholar

[2]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations,, Appl. Math. Optim., 61 (2010), 379.  doi: 10.1007/s00245-009-9091-z.  Google Scholar

[3]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects,, Second edition. With a foreword by John Marshall. International Geophysics Series, (2011).   Google Scholar

[4]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar

[5]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model,, Physica D, 240 (2011), 1123.  doi: 10.1016/j.physd.2011.03.009.  Google Scholar

[6]

A. Debussche, N. Glatt-Holtz, R. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise,, Nonlinearity, 25 (2012), 2093.  doi: 10.1088/0951-7715/25/7/2093.  Google Scholar

[7]

Z. Dong, J. Zhai and R. Zhang, Exponential convergence for 3D stochastic primitive equations of the large scale ocean,, arXiv:1506.08514v1., ().   Google Scholar

[8]

J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences,, Stochastic Processes and Their Applications, 119 (2009), 2052.  doi: 10.1016/j.spa.2008.10.004.  Google Scholar

[9]

B. Ewald, M. Petcu and R. Temam, Stochastic solutions of the two-dimensional primitive equations of the ocean and atmosphere with an additive noise,, Anal. Appl., 5 (2007), 183.  doi: 10.1142/S0219530507000948.  Google Scholar

[10]

M. I. Freidlin and A. D. Wentzell, Reaction-diffusion equation with randomly perturbed boundary condition,, Annals. of Prob., 20 (1992), 963.  doi: 10.1214/aop/1176989813.  Google Scholar

[11]

H. Gao and C. Sun, Random Attractor for the 3D viscous stochastic primitive equations with additive noise,, Stochastics and Dynamics, 9 (2009), 293.  doi: 10.1142/S0219493709002683.  Google Scholar

[12]

H. Gao and C. Sun, Large Deviations for the Stochastic Primitive Equations in Two Space Dimensions,, Comm. Math. Sci., 10 (2012), 575.  doi: 10.4310/CMS.2012.v10.n2.a8.  Google Scholar

[13]

N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensinonal stochastic primitive equations,, J. Math. Phys., 55 (2014).  doi: 10.1063/1.4875104.  Google Scholar

[14]

N. Glatt-Holtz and R. Temam, Pathwise solutions of the 2-d stochastic primitive equations,, Applied Mathematics and Optimization, 63 (2011), 401.  doi: 10.1007/s00245-010-9126-5.  Google Scholar

[15]

N. Glatt-Holtz and M. Ziane, The stochastic primitive equations in two space dimensions with multiplicative noise,, Discrete and Continuous Dynamical Systems Series B, 10 (2008), 801.  doi: 10.3934/dcdsb.2008.10.801.  Google Scholar

[16]

B. Guo and D. Huang, 3D Stochastic Primitive Equations of the Large-Scale Ocean: Global Well-Posedness and Attractors,, Commun. Math. Phys., 286 (2009), 697.  doi: 10.1007/s00220-008-0654-7.  Google Scholar

[17]

F. Guillén-Gonzáez, N. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the Primitive Equations,, Diff. Integral Eq., 14 (2001), 1381.   Google Scholar

[18]

C. Hu, R. Temam and M. Ziane, The primitive equations on the large scale ocean under the small depth hypothesis,, Discrete Contin. Dyn. Syst., 9 (2003), 97.  doi: 10.3934/dcds.2003.9.97.  Google Scholar

[19]

N. Ju, The global attractor for the solutions to the 3d viscous primitive equations,, Discrete Contin. Dyn. Syst., 17 (2007), 159.  doi: 10.3934/dcds.2007.17.159.  Google Scholar

[20]

G. Kallianpur and J. Xiong, Large deviations for a class of stochastic partial differential equations,, Ann. Prob., 24 (1996), 320.  doi: 10.1214/aop/1042644719.  Google Scholar

[21]

G. Kobelkov, Existence of a solution in "whole" for the large-scale ocean dynamics equations,, C. R. Math. Acad. Sci. Paris, 343 (2006), 283.  doi: 10.1016/j.crma.2006.04.020.  Google Scholar

[22]

G. M. Kobelkov and V. B. Zalesny, Existence and uniqueness of a solution to primitive equations with stratification 'in the large',, Russian J. Numer. Anal. Math. Modelling, 23 (2008), 39.  doi: 10.1515/rnam.2008.003.  Google Scholar

[23]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean,, Nonlinearity, 20 (2007), 2739.  doi: 10.1088/0951-7715/20/12/001.  Google Scholar

[24]

H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge ; New York : Cambridge University Press, (1990).   Google Scholar

[25]

J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean,, Comput. Mech. Adv., 1 (1993), 1.   Google Scholar

[26]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications,, Nonlinearity, 5 (1992), 237.  doi: 10.1088/0951-7715/5/2/001.  Google Scholar

[27]

J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean,, Nonlinearity, 5 (1992), 1007.  doi: 10.1088/0951-7715/5/5/002.  Google Scholar

[28]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987).   Google Scholar

[29]

M. Petcu, Gevrey class regularity for the primitive equations in space dimension 2,, Asymptot. Anal., 39 (2004), 1.   Google Scholar

[30]

M. Petcu, R. Temam and D. Wirosoetisno, Existence and regularity results for the primitive equations in two space dimensions,, Commun. Pure Appl. Anal., 3 (2004), 115.  doi: 10.3934/cpaa.2004.3.115.  Google Scholar

[31]

M. Petcu, R. Temam and M. Ziane, Some Mathematical Problems in Geophysical Fluid Dynamics,, In Special Volume on Computational Methods for the Atmosphere and the Oceans, (2009), 577.  doi: 10.1016/S1570-8659(08)00212-3.  Google Scholar

[32]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise,, Stoch. Proc. and Appl., 116 (2006), 1636.  doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[33]

C. Sun, Random Dynamics and Large Deviation of Some Hydrodynamics Equations,, Ph.D thesis, (2010).   Google Scholar

[34]

R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics,, Handbook of mathematical fluid dynamics, 3 (2004), 535.   Google Scholar

show all references

References:
[1]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Ann. Math., 166 (2007), 245.  doi: 10.4007/annals.2007.166.245.  Google Scholar

[2]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations,, Appl. Math. Optim., 61 (2010), 379.  doi: 10.1007/s00245-009-9091-z.  Google Scholar

[3]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects,, Second edition. With a foreword by John Marshall. International Geophysics Series, (2011).   Google Scholar

[4]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar

[5]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model,, Physica D, 240 (2011), 1123.  doi: 10.1016/j.physd.2011.03.009.  Google Scholar

[6]

A. Debussche, N. Glatt-Holtz, R. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise,, Nonlinearity, 25 (2012), 2093.  doi: 10.1088/0951-7715/25/7/2093.  Google Scholar

[7]

Z. Dong, J. Zhai and R. Zhang, Exponential convergence for 3D stochastic primitive equations of the large scale ocean,, arXiv:1506.08514v1., ().   Google Scholar

[8]

J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences,, Stochastic Processes and Their Applications, 119 (2009), 2052.  doi: 10.1016/j.spa.2008.10.004.  Google Scholar

[9]

B. Ewald, M. Petcu and R. Temam, Stochastic solutions of the two-dimensional primitive equations of the ocean and atmosphere with an additive noise,, Anal. Appl., 5 (2007), 183.  doi: 10.1142/S0219530507000948.  Google Scholar

[10]

M. I. Freidlin and A. D. Wentzell, Reaction-diffusion equation with randomly perturbed boundary condition,, Annals. of Prob., 20 (1992), 963.  doi: 10.1214/aop/1176989813.  Google Scholar

[11]

H. Gao and C. Sun, Random Attractor for the 3D viscous stochastic primitive equations with additive noise,, Stochastics and Dynamics, 9 (2009), 293.  doi: 10.1142/S0219493709002683.  Google Scholar

[12]

H. Gao and C. Sun, Large Deviations for the Stochastic Primitive Equations in Two Space Dimensions,, Comm. Math. Sci., 10 (2012), 575.  doi: 10.4310/CMS.2012.v10.n2.a8.  Google Scholar

[13]

N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensinonal stochastic primitive equations,, J. Math. Phys., 55 (2014).  doi: 10.1063/1.4875104.  Google Scholar

[14]

N. Glatt-Holtz and R. Temam, Pathwise solutions of the 2-d stochastic primitive equations,, Applied Mathematics and Optimization, 63 (2011), 401.  doi: 10.1007/s00245-010-9126-5.  Google Scholar

[15]

N. Glatt-Holtz and M. Ziane, The stochastic primitive equations in two space dimensions with multiplicative noise,, Discrete and Continuous Dynamical Systems Series B, 10 (2008), 801.  doi: 10.3934/dcdsb.2008.10.801.  Google Scholar

[16]

B. Guo and D. Huang, 3D Stochastic Primitive Equations of the Large-Scale Ocean: Global Well-Posedness and Attractors,, Commun. Math. Phys., 286 (2009), 697.  doi: 10.1007/s00220-008-0654-7.  Google Scholar

[17]

F. Guillén-Gonzáez, N. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the Primitive Equations,, Diff. Integral Eq., 14 (2001), 1381.   Google Scholar

[18]

C. Hu, R. Temam and M. Ziane, The primitive equations on the large scale ocean under the small depth hypothesis,, Discrete Contin. Dyn. Syst., 9 (2003), 97.  doi: 10.3934/dcds.2003.9.97.  Google Scholar

[19]

N. Ju, The global attractor for the solutions to the 3d viscous primitive equations,, Discrete Contin. Dyn. Syst., 17 (2007), 159.  doi: 10.3934/dcds.2007.17.159.  Google Scholar

[20]

G. Kallianpur and J. Xiong, Large deviations for a class of stochastic partial differential equations,, Ann. Prob., 24 (1996), 320.  doi: 10.1214/aop/1042644719.  Google Scholar

[21]

G. Kobelkov, Existence of a solution in "whole" for the large-scale ocean dynamics equations,, C. R. Math. Acad. Sci. Paris, 343 (2006), 283.  doi: 10.1016/j.crma.2006.04.020.  Google Scholar

[22]

G. M. Kobelkov and V. B. Zalesny, Existence and uniqueness of a solution to primitive equations with stratification 'in the large',, Russian J. Numer. Anal. Math. Modelling, 23 (2008), 39.  doi: 10.1515/rnam.2008.003.  Google Scholar

[23]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean,, Nonlinearity, 20 (2007), 2739.  doi: 10.1088/0951-7715/20/12/001.  Google Scholar

[24]

H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge ; New York : Cambridge University Press, (1990).   Google Scholar

[25]

J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean,, Comput. Mech. Adv., 1 (1993), 1.   Google Scholar

[26]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications,, Nonlinearity, 5 (1992), 237.  doi: 10.1088/0951-7715/5/2/001.  Google Scholar

[27]

J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean,, Nonlinearity, 5 (1992), 1007.  doi: 10.1088/0951-7715/5/5/002.  Google Scholar

[28]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987).   Google Scholar

[29]

M. Petcu, Gevrey class regularity for the primitive equations in space dimension 2,, Asymptot. Anal., 39 (2004), 1.   Google Scholar

[30]

M. Petcu, R. Temam and D. Wirosoetisno, Existence and regularity results for the primitive equations in two space dimensions,, Commun. Pure Appl. Anal., 3 (2004), 115.  doi: 10.3934/cpaa.2004.3.115.  Google Scholar

[31]

M. Petcu, R. Temam and M. Ziane, Some Mathematical Problems in Geophysical Fluid Dynamics,, In Special Volume on Computational Methods for the Atmosphere and the Oceans, (2009), 577.  doi: 10.1016/S1570-8659(08)00212-3.  Google Scholar

[32]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise,, Stoch. Proc. and Appl., 116 (2006), 1636.  doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[33]

C. Sun, Random Dynamics and Large Deviation of Some Hydrodynamics Equations,, Ph.D thesis, (2010).   Google Scholar

[34]

R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics,, Handbook of mathematical fluid dynamics, 3 (2004), 535.   Google Scholar

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