# American Institute of Mathematical Sciences

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 and 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), 245-267. doi: 10.4007/annals.2007.166.245. [2] I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420. doi: 10.1007/s00245-009-9091-z. [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, 101. Elsevier/Academic Press, Amsterdam, 2011. [4] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. [5] A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Physica D, 240 (2011), 1123-1144. doi: 10.1016/j.physd.2011.03.009. [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), 2093-2118. doi: 10.1088/0951-7715/25/7/2093. [7] Z. Dong, J. Zhai and R. Zhang, Exponential convergence for 3D stochastic primitive equations of the large scale ocean,, arXiv:1506.08514v1., (). [8] J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Processes and Their Applications, 119 (2009), 2052-2081. doi: 10.1016/j.spa.2008.10.004. [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), 183-198. doi: 10.1142/S0219530507000948. [10] M. I. Freidlin and A. D. Wentzell, Reaction-diffusion equation with randomly perturbed boundary condition, Annals. of Prob., 20 (1992), 963-986. doi: 10.1214/aop/1176989813. [11] H. Gao and C. Sun, Random Attractor for the 3D viscous stochastic primitive equations with additive noise, Stochastics and Dynamics, 9 (2009), 293-313. doi: 10.1142/S0219493709002683. [12] H. Gao and C. Sun, Large Deviations for the Stochastic Primitive Equations in Two Space Dimensions, Comm. Math. Sci., 10 (2012), 575-593. doi: 10.4310/CMS.2012.v10.n2.a8. [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), 051504, 34 pp, arXiv:1311.4204v1. doi: 10.1063/1.4875104. [14] N. Glatt-Holtz and R. Temam, Pathwise solutions of the 2-d stochastic primitive equations, Applied Mathematics and Optimization, 63 (2011), 401-433. doi: 10.1007/s00245-010-9126-5. [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), 801-822. doi: 10.3934/dcdsb.2008.10.801. [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), 697-723. doi: 10.1007/s00220-008-0654-7. [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), 1381-1408. [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), 97-131. doi: 10.3934/dcds.2003.9.97. [19] N. Ju, The global attractor for the solutions to the 3d viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179. doi: 10.3934/dcds.2007.17.159. [20] G. Kallianpur and J. Xiong, Large deviations for a class of stochastic partial differential equations, Ann. Prob., 24 (1996), 320-345. doi: 10.1214/aop/1042644719. [21] G. Kobelkov, Existence of a solution in "whole" for the large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020. [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), 39-61. doi: 10.1515/rnam.2008.003. [23] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753. doi: 10.1088/0951-7715/20/12/001. [24] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge ; New York : Cambridge University Press, 1990. [25] J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean, Comput. Mech. Adv., 1 (1993), 1-120. [26] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288. doi: 10.1088/0951-7715/5/2/001. [27] J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002. [28] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [29] M. Petcu, Gevrey class regularity for the primitive equations in space dimension 2, Asymptot. Anal., 39 (2004), 1-13. [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), 115-131. doi: 10.3934/cpaa.2004.3.115. [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, volume 14 of Handbook of Numerical Analysis, 577-750, Elsevier, 2009. doi: 10.1016/S1570-8659(08)00212-3. [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), 1636-1659. doi: 10.1016/j.spa.2006.04.001. [33] C. Sun, Random Dynamics and Large Deviation of Some Hydrodynamics Equations, Ph.D thesis, Nanjing Normal University, Nanjing, China, 2010. [34] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of mathematical fluid dynamics, 3 (2004), 535-657.

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##### 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), 245-267. doi: 10.4007/annals.2007.166.245. [2] I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420. doi: 10.1007/s00245-009-9091-z. [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, 101. Elsevier/Academic Press, Amsterdam, 2011. [4] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. [5] A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Physica D, 240 (2011), 1123-1144. doi: 10.1016/j.physd.2011.03.009. [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), 2093-2118. doi: 10.1088/0951-7715/25/7/2093. [7] Z. Dong, J. Zhai and R. Zhang, Exponential convergence for 3D stochastic primitive equations of the large scale ocean,, arXiv:1506.08514v1., (). [8] J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Processes and Their Applications, 119 (2009), 2052-2081. doi: 10.1016/j.spa.2008.10.004. [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), 183-198. doi: 10.1142/S0219530507000948. [10] M. I. Freidlin and A. D. Wentzell, Reaction-diffusion equation with randomly perturbed boundary condition, Annals. of Prob., 20 (1992), 963-986. doi: 10.1214/aop/1176989813. [11] H. Gao and C. Sun, Random Attractor for the 3D viscous stochastic primitive equations with additive noise, Stochastics and Dynamics, 9 (2009), 293-313. doi: 10.1142/S0219493709002683. [12] H. Gao and C. Sun, Large Deviations for the Stochastic Primitive Equations in Two Space Dimensions, Comm. Math. Sci., 10 (2012), 575-593. doi: 10.4310/CMS.2012.v10.n2.a8. [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), 051504, 34 pp, arXiv:1311.4204v1. doi: 10.1063/1.4875104. [14] N. Glatt-Holtz and R. Temam, Pathwise solutions of the 2-d stochastic primitive equations, Applied Mathematics and Optimization, 63 (2011), 401-433. doi: 10.1007/s00245-010-9126-5. [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), 801-822. doi: 10.3934/dcdsb.2008.10.801. [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), 697-723. doi: 10.1007/s00220-008-0654-7. [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), 1381-1408. [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), 97-131. doi: 10.3934/dcds.2003.9.97. [19] N. Ju, The global attractor for the solutions to the 3d viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179. doi: 10.3934/dcds.2007.17.159. [20] G. Kallianpur and J. Xiong, Large deviations for a class of stochastic partial differential equations, Ann. Prob., 24 (1996), 320-345. doi: 10.1214/aop/1042644719. [21] G. Kobelkov, Existence of a solution in "whole" for the large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020. [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), 39-61. doi: 10.1515/rnam.2008.003. [23] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753. doi: 10.1088/0951-7715/20/12/001. [24] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge ; New York : Cambridge University Press, 1990. [25] J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean, Comput. Mech. Adv., 1 (1993), 1-120. [26] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288. doi: 10.1088/0951-7715/5/2/001. [27] J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002. [28] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [29] M. Petcu, Gevrey class regularity for the primitive equations in space dimension 2, Asymptot. Anal., 39 (2004), 1-13. [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), 115-131. doi: 10.3934/cpaa.2004.3.115. [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, volume 14 of Handbook of Numerical Analysis, 577-750, Elsevier, 2009. doi: 10.1016/S1570-8659(08)00212-3. [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), 1636-1659. doi: 10.1016/j.spa.2006.04.001. [33] C. Sun, Random Dynamics and Large Deviation of Some Hydrodynamics Equations, Ph.D thesis, Nanjing Normal University, Nanjing, China, 2010. [34] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of mathematical fluid dynamics, 3 (2004), 535-657.
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