# American Institute of Mathematical Sciences

July  2019, 24(7): 3157-3174. doi: 10.3934/dcdsb.2018305

## On the backward uniqueness of the stochastic primitive equations with additive noise

 1 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China 2 School of Mathematics and Statistics, Chongqing University, Chongqing city 401331, China

* Corresponding author: Guoli Zhou

Received  May 2018 Revised  July 2018 Published  October 2018

Fund Project: The second author is supported by NSF NNSF of China(Grant No. 11401057), Natural Science Foundation Project of CQ (Grant No. cstc2016jcyjA0326), Fundamental Research Funds for the Central Universities(Grant No. 2018CDXYST0024, ) and China Scholarship Council (Grant No.201506055003)

The previous works focus on the uniqueness for the initial-value problems of stochastic primitive equations. Uniqueness for the initial-value problems means that if the two initial conditions are the same, then the two solutions coincide with each other. However there is no work to answer what will happen to the solutions if the two initial conditions are different. This problem for the stochastic three dimensional primitive equations is addressed by the backward uniqueness established in this article. The backward uniqueness means that if two solutions intersect at time $t>0,$ then they are equal everywhere on the interval $(0, t).$ In other words, given two different initial-value conditions, the corresponding two solutions will never cross in the future. Hence this article can be viewed as a further study of the dependence of the solutions on the initial data.

Citation: Boling Guo, Guoli Zhou. On the backward uniqueness of the stochastic primitive equations with additive noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3157-3174. doi: 10.3934/dcdsb.2018305
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] H. Crauel, Markov measures for random dynamical systems, Stochastics Stochastics Rep., 3 (1991), 153-173. doi: 10.1080/17442509108833733. [3] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. [4] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields., 100 (1994), 365-393. doi: 10.1007/BF01193705. [5] C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1. [6] C. Cao, J. Li and E. S. Titi, Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, J. Differential Equations, 257 (2014), 4108-4132. doi: 10.1016/j.jde.2014.08.003. [7] C. Cao, J. Li and E. S. Titi, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Ration. Mech. Anal., 214 (2014), 35-76. doi: 10.1007/s00205-014-0752-y. [8] C. Cao, J. Li and E. S. Titi, Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion, Communications on Pure and Applied Mathematics, 69 (2016), 1492-1531. doi: 10.1002/cpa.21576. [9] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245. [10] C. Cao and E. S. Titi, Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Comm. Math. Phys., 310 (2012), 537-568. doi: 10.1007/s00220-011-1409-4. [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [12] 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. [13] Z. Dong and R. Zhang, Markov selection and W-strong Feller for 3D stochastic primitive equations, Science China Mathematics, 60 (2017), 1873-1900. doi: 10.1007/s11425-016-0336-y. [14] Z. Dong, J. Zhai and R. Zhang, Large deviation principles for 3D stochastic primitive equations, J. Differential Equations, 263 (2017), 3110-3146. doi: 10.1016/j.jde.2017.04.025. [15] Z. Dong, J. Zhai and R. Zhang, Exponential mixing for 3D stochastic primitive equations of the large scale ocean, Available at arXiv: 1506.08514. [16] H. Gao and C. Sun, Well-posedness and large deviations for the stochastic primitive equations in two space dimensions, Commun. Math. Sci., 10 (2012), 575-593. doi: 10.4310/CMS.2012.v10.n2.a8. [17] H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Disc. and Cont. Dyn. Sys. B., 21 (2016), 3053-3073. doi: 10.3934/dcdsb.2016087. [18] A. E. Gill, Atmosphere-ocean Dynamics, International Geophysics Series, Academic Press, San Diego, 1982. [19] 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. [20] N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensional stochastic primitive equations, J. Math. Phys., 55 (2014), 051504, 34pp. doi: 10.1063/1.4875104. [21] F. Guillén-González, N. Masmoudi and M. A. Rodr$\acute{\mathrm{i}}$guez-Bellido, Anisotropic estimates and strong solutions for the primitive equations, Diff. Int. Equ., 14 (2001), 1381-1408. [22] H. Gao and C. Sun, Hausdorff dimension of random attractor for stochastic Navier-Stokes-Voight equations and primitive equations, Dyn. Partial Differ. Equ., 7 (2010), 307-326. doi: 10.4310/DPDE.2010.v7.n4.a2. [23] G. J. Haltiner, Numerical Weather Prediction, J. W. Wiley & Sons, New York, 1971. [24] G. J. Haltiner and R. T. Williams, Numerical Prediction and Dynamic Meteorology, John Wiley & Sons, New York, 1980. [25] C. Hu, R. Temam and M. Ziane, The primimitive equations of the large scale ocean under the small depth hypothesis, Disc. and Cont. Dyn. Sys., 9 (2003), 97-131. doi: 10.3934/dcds.2003.9.97. [26] 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. [27] K. Liu, Stability of Stochastic Differential Equations in Infinite Dimensions, Springer Verlag, New York, 2004. [28] J. Lions and B. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972. [29] 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. [30] 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. [31] J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean(CAOI), Computational Mechanics Advance, 1 (1993), 120pp. [32] J. L. Lions, R. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAOIII), J. Math. Pures Appl., 74 (1995), 105-163. [33] M. Petcu, On the backward uniqueness of the primitive equations, J. Math. Pures Appl., 87 (2007), 275-289. doi: 10.1016/j.matpur.2007.01.002.

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] H. Crauel, Markov measures for random dynamical systems, Stochastics Stochastics Rep., 3 (1991), 153-173. doi: 10.1080/17442509108833733. [3] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. [4] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields., 100 (1994), 365-393. doi: 10.1007/BF01193705. [5] C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1. [6] C. Cao, J. Li and E. S. Titi, Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, J. Differential Equations, 257 (2014), 4108-4132. doi: 10.1016/j.jde.2014.08.003. [7] C. Cao, J. Li and E. S. Titi, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Ration. Mech. Anal., 214 (2014), 35-76. doi: 10.1007/s00205-014-0752-y. [8] C. Cao, J. Li and E. S. Titi, Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion, Communications on Pure and Applied Mathematics, 69 (2016), 1492-1531. doi: 10.1002/cpa.21576. [9] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245. [10] C. Cao and E. S. Titi, Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Comm. Math. Phys., 310 (2012), 537-568. doi: 10.1007/s00220-011-1409-4. [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [12] 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. [13] Z. Dong and R. Zhang, Markov selection and W-strong Feller for 3D stochastic primitive equations, Science China Mathematics, 60 (2017), 1873-1900. doi: 10.1007/s11425-016-0336-y. [14] Z. Dong, J. Zhai and R. Zhang, Large deviation principles for 3D stochastic primitive equations, J. Differential Equations, 263 (2017), 3110-3146. doi: 10.1016/j.jde.2017.04.025. [15] Z. Dong, J. Zhai and R. Zhang, Exponential mixing for 3D stochastic primitive equations of the large scale ocean, Available at arXiv: 1506.08514. [16] H. Gao and C. Sun, Well-posedness and large deviations for the stochastic primitive equations in two space dimensions, Commun. Math. Sci., 10 (2012), 575-593. doi: 10.4310/CMS.2012.v10.n2.a8. [17] H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Disc. and Cont. Dyn. Sys. B., 21 (2016), 3053-3073. doi: 10.3934/dcdsb.2016087. [18] A. E. Gill, Atmosphere-ocean Dynamics, International Geophysics Series, Academic Press, San Diego, 1982. [19] 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. [20] N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensional stochastic primitive equations, J. Math. Phys., 55 (2014), 051504, 34pp. doi: 10.1063/1.4875104. [21] F. Guillén-González, N. Masmoudi and M. A. Rodr$\acute{\mathrm{i}}$guez-Bellido, Anisotropic estimates and strong solutions for the primitive equations, Diff. Int. Equ., 14 (2001), 1381-1408. [22] H. Gao and C. Sun, Hausdorff dimension of random attractor for stochastic Navier-Stokes-Voight equations and primitive equations, Dyn. Partial Differ. Equ., 7 (2010), 307-326. doi: 10.4310/DPDE.2010.v7.n4.a2. [23] G. J. Haltiner, Numerical Weather Prediction, J. W. Wiley & Sons, New York, 1971. [24] G. J. Haltiner and R. T. Williams, Numerical Prediction and Dynamic Meteorology, John Wiley & Sons, New York, 1980. [25] C. Hu, R. Temam and M. Ziane, The primimitive equations of the large scale ocean under the small depth hypothesis, Disc. and Cont. Dyn. Sys., 9 (2003), 97-131. doi: 10.3934/dcds.2003.9.97. [26] 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. [27] K. Liu, Stability of Stochastic Differential Equations in Infinite Dimensions, Springer Verlag, New York, 2004. [28] J. Lions and B. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972. [29] 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. [30] 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. [31] J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean(CAOI), Computational Mechanics Advance, 1 (1993), 120pp. [32] J. L. Lions, R. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAOIII), J. Math. Pures Appl., 74 (1995), 105-163. [33] M. Petcu, On the backward uniqueness of the primitive equations, J. Math. Pures Appl., 87 (2007), 275-289. doi: 10.1016/j.matpur.2007.01.002.
 [1] T. Tachim Medjo. Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491 [2] Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002 [3] 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 [4] T. Tachim Medjo. The exponential behavior of the stochastic primitive equations in two dimensional space with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 177-197. doi: 10.3934/dcdsb.2010.14.177 [5] Nathan Glatt-Holtz, Mohammed Ziane. The stochastic primitive equations in two space dimensions with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 801-822. doi: 10.3934/dcdsb.2008.10.801 [6] Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075 [7] Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803 [8] Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447 [9] Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115 [10] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929 [11] Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285 [12] Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565 [13] Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013 [14] Qi Lü, Xu Zhang. Transposition method for backward stochastic evolution equations revisited, and its application. Mathematical Control & Related Fields, 2015, 5 (3) : 529-555. doi: 10.3934/mcrf.2015.5.529 [15] Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905 [16] Tianxiao Wang, Yufeng Shi. Symmetrical solutions of backward stochastic Volterra integral equations and their applications. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 251-274. doi: 10.3934/dcdsb.2010.14.251 [17] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613 [18] Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585 [19] Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379 [20] Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-23. doi: 10.3934/dcdsb.2019054

2018 Impact Factor: 1.008

Article outline