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:
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R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar

[2]

H. Crauel, Markov measures for random dynamical systems, Stochastics Stochastics Rep., 3 (1991), 153-173.  doi: 10.1080/17442509108833733.  Google Scholar

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

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C. CaoS. IbrahimK. 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.  Google Scholar

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C. CaoJ. 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.  Google Scholar

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C. CaoJ. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. Google Scholar

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A. DebusscheN. Glatt-HoltzR. 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.  Google Scholar

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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.  Google Scholar

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Z. DongJ. 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.  Google Scholar

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Z. Dong, J. Zhai and R. Zhang, Exponential mixing for 3D stochastic primitive equations of the large scale ocean, Available at arXiv: 1506.08514. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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A. E. Gill, Atmosphere-ocean Dynamics, International Geophysics Series, Academic Press, San Diego, 1982. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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F. Guillén-GonzálezN. 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.   Google Scholar

[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.  Google Scholar

[23]

G. J. Haltiner, Numerical Weather Prediction, J. W. Wiley & Sons, New York, 1971. Google Scholar

[24]

G. J. Haltiner and R. T. Williams, Numerical Prediction and Dynamic Meteorology, John Wiley & Sons, New York, 1980. Google Scholar

[25]

C. HuR. 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.  Google Scholar

[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.  Google Scholar

[27]

K. Liu, Stability of Stochastic Differential Equations in Infinite Dimensions, Springer Verlag, New York, 2004. Google Scholar

[28]

J. Lions and B. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972.  Google Scholar

[29]

J. L. LionsR. 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.  Google Scholar

[30]

J. L. LionsR. 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.  Google Scholar

[31]

J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean(CAOI), Computational Mechanics Advance, 1 (1993), 120pp.  Google Scholar

[32]

J. L. LionsR. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAOIII), J. Math. Pures Appl., 74 (1995), 105-163.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar

[2]

H. Crauel, Markov measures for random dynamical systems, Stochastics Stochastics Rep., 3 (1991), 153-173.  doi: 10.1080/17442509108833733.  Google Scholar

[3]

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

[4]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields., 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[5]

C. CaoS. IbrahimK. 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.  Google Scholar

[6]

C. CaoJ. 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.  Google Scholar

[7]

C. CaoJ. 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.  Google Scholar

[8]

C. CaoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. Google Scholar

[12]

A. DebusscheN. Glatt-HoltzR. 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.  Google Scholar

[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.  Google Scholar

[14]

Z. DongJ. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. E. Gill, Atmosphere-ocean Dynamics, International Geophysics Series, Academic Press, San Diego, 1982. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

F. Guillén-GonzálezN. 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.   Google Scholar

[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.  Google Scholar

[23]

G. J. Haltiner, Numerical Weather Prediction, J. W. Wiley & Sons, New York, 1971. Google Scholar

[24]

G. J. Haltiner and R. T. Williams, Numerical Prediction and Dynamic Meteorology, John Wiley & Sons, New York, 1980. Google Scholar

[25]

C. HuR. 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.  Google Scholar

[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.  Google Scholar

[27]

K. Liu, Stability of Stochastic Differential Equations in Infinite Dimensions, Springer Verlag, New York, 2004. Google Scholar

[28]

J. Lions and B. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972.  Google Scholar

[29]

J. L. LionsR. 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.  Google Scholar

[30]

J. L. LionsR. 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.  Google Scholar

[31]

J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean(CAOI), Computational Mechanics Advance, 1 (1993), 120pp.  Google Scholar

[32]

J. L. LionsR. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAOIII), J. Math. Pures Appl., 74 (1995), 105-163.   Google Scholar

[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.  Google Scholar

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