March 2019, 18(2): 643-661. doi: 10.3934/cpaa.2019032

Continuous data assimilation for the 3D primitive equations of the ocean

Department of Mathematics, Western Washington University, 516 High St, Bellingham, WA 98225-9063, USA

Received  December 2017 Revised  May 2018 Published  October 2018

In this article, we show that the continuous data assimilation algorithm is valid for the 3D primitive equations of the ocean. Namely, the assimilated solution converges to the reference solution in $L^2$ norm at an exponential rate in time. We also prove the global existence of strong solution to the assimilated system.

Citation: Yuan Pei. Continuous data assimilation for the 3D primitive equations of the ocean. Communications on Pure & Applied Analysis, 2019, 18 (2) : 643-661. doi: 10.3934/cpaa.2019032
References:
[1]

D. A. AlbanezH. J. Nussenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensional Navier-Stokes-α model, Asymptotic Anal., 97 (2016), 139-164. doi: 10.3233/ASY-151351.

[2]

M. U. AltafE. S. TitiT. GebraelO. M. KnioL. ZhaoM. F. McCabe and I. Hoteit, Downscaling the 2D Bénard convection equations using continuous data assimilation, Computational Geosciences, (2017), 1-18. doi: 10.1007/s10596-017-9619-2.

[3]

A. AzouaniE. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables, J. Nonlinear Sci., 24 (2014), 277-304. doi: 10.1007/s00332-013-9189-y.

[4]

A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters-a reaction-diffusion paradigm, Evol. Equ. Control Theory, 3 (2014), 579-594. doi: 10.3934/eect.2014.3.579.

[5]

H. BessaihE. Olson and E. S. Titi, Continuous data assimilation with stochastically noisy data, Nonlinearity, 28 (2015), 729-753. doi: 10.1088/0951-7715/28/3/729.

[6]

A. Biswas, J. Hudson, A. Larios and Y. Pei, Continuous data assimilation for the magneto-hydrodynamic equations in 2D using one component of the velocity and magnetic fields, Asymptot. Anal., (2018), to appear.

[7]

A. Biswas and V. R. Martinez, Higher-order synchronization for a data assimilation algorithm for the 2D Navier-Stokes equations, Nonlinearity, 35 (2017), 132-157. doi: 10.1016/j.nonrwa.2016.10.005.

[8]

D. Bresch, F. Guillén-González, N. Masmoudi and M. A. Rodríguez-Bellido, Asymptotic derivation of a Navier condition for the primitive equations, Asymptot. Anal., 33 (2003), 237-259

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D. BreschF. Guillén-GonzálezN. Masmoudi and M. A. Rodríguez-Bellido, On the uniqueness of weak solutions of the two-dimensional primitive equations, Differential Integral Equations, 16 (2003), 77-94.

[10]

D. Bresch, F. Guillén-González, N. Masmoudi and M. A. Rodríguez-Bellido, Uniqueness of solution for the 2D primitive equations with friction condition on the bottom, in Seventh Zaragoza-Pau Conference on Applied Mathematics and Statistics (Spanish) (Jaca, 2001) (eds. E.H. Zarantonello and Author 2), Univ. Zaragoza, 27 (2003), 135-143.

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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.

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C. CaoI. G. Kevrekidis and E. S. Titi, Numerical criterion for the stabilization of steady states of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 37-96. doi: 10.1512/iumj.2001.50.2154.

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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.

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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.

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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, Comm. Pure Appl. Math., 69 (2016), 1492-1531. doi: 10.1002/cpa.21576.

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C. CaoJ. Li and E. S. Titi, Strong solutions to the 3D primitive equations with only horizontal dissipation: near $H^1$initial data, J. Funct. Anal., 272 (2017), 4606-4641. doi: 10.1016/j.jfa.2017.01.018.

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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. (2), 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.

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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), 537-563. doi: 10.1007/s00220-011-1409-4.

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I. Chueshov, A squeezing property and its applications to a description of long-time behaviour in the three-dimensional viscous primitive equations, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 711-729. doi: 10.1017/S0308210512001953.

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R. Errico and D. Baumhefner, Predictability experiments using a high-resolution limited-area model, Monthly Weather Review, 115 (1986), 488-505.

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A. FarhatM. Jolly and E. S. Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Phys. D, 303 (2015), 59-66. doi: 10.1016/j.physd.2015.03.011.

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A. FarhatE. Lunasin and E. S. Titi, Abridged continuous data assimilation for the 2D Navier-Stokes Equations utilizing measurements of only one component of the velocity field, J. Math. Fluid Mech., 18 (2016), 1-23. doi: 10.1007/s00021-015-0225-6.

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A. FarhatE. Lunasin and E. S. Titi, Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements, J. of Math. Anal. and Appl., 438 (2016), 492-506. doi: 10.1016/j.jmaa.2016.01.072.

[26]

A. FarhatE. Lunasin and E. S. Titi, Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone, J. of Nonlinear Science, (2017), 1-23. doi: 10.1007/s00332-017-9360-y.

[27]

A. Farhat, E. Lunasin and E. S. Titi, On the Charney Conjecture of Data Assimilation Employing Temperature Measurements Alone: The Paradigm of 3D Planetary Geostrophic Model, preprint arXiv: 1608.04770.

[28]

C. FoiasC. Mondaini and E. S. Titi, A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics, SIAM J. Appl. Dyn. Syst., 15 (2016), 2109-2142. doi: 10.1137/16M1076526.

[29]

M. GeshoE. Olson and E. S. Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier-Stokes equations, Commun. Comput. Phys., 19 (2016), 1094-1110.

[30]

N. Glatt-Holtz and M. Ziane, The stochastic primitive equations in two space dimensions with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 801-822. doi: 10.3934/dcdsb.2008.10.801.

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F. Guillén-GonzálezN. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Differential Integral Equations, 14 (2001), 1381-1408.

[32]

B. Guo and D. Huang, On the 3D viscous primitive equations of the large-scale atmosphere, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 846-866. doi: 10.1016/S0252-9602(09)60074-6.

[33]

K. HaydenE. Olson and E. S. Titi, Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations, Phys. D, 240 (2011), 1416-1425. doi: 10.1016/j.physd.2011.04.021.

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J. Hoke and R. Anthes, The initialization of numerical models by a dynamic relaxation technique, Monthly Weather Review, 104 (1976), 1551-1556.

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C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal., 61 (2005), 425-460. doi: 10.1016/j.na.2004.12.005.

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C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, in Frontiers in mathematical analysis and numerical methods, World Sci. Publ., (2004), 149-170.

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M. JollyV. R. Martinez and E. S. Titi, A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Adv. Nonlinear Stud., 17 (2017), 167-192. doi: 10.1515/ans-2016-6019.

[38]

Don A. Jones and E. S. Titi, Determining finite volume elements for the 2D Navier-Stokes equations, Phys. D, 60 (1992), 165-174, Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991). doi: 10.1016/0167-2789(92)90233-D.

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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.

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N. Ju, Global Uniform Boundedness of Solutions to viscous 3D Primitive Equations with Physical Boundary Conditions, preprint, arXiv: 1710.04622v2.

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N. Ju and R. Temam, Finite dimensions of the global attractor for 3D primitive equations with viscosity, J. Nonlinear Sci., 25 (2015), 131-155. doi: 10.1007/s00332-014-9223-8.

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I. Kukavica and M. Ziane, The regularity of solutions of the primitive equations of the ocean in space dimension three, C. R. Math. Acad. Sci. Paris, 345 (2007), 257-260. doi: 10.1016/j.crma.2007.07.025.

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I. Kukavica and M. Ziane, Uniform gradient bounds for the primitive equations of the ocean, Differential Integral Equations, 21 (2008), 837-849.

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I. Kukavica and M. Ziane, Primitive equations with continuous initial data, Nonlinearity, 27 (2014), 1135-1155. doi: 10.1088/0951-7715/27/6/1135.

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show all references

References:
[1]

D. A. AlbanezH. J. Nussenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensional Navier-Stokes-α model, Asymptotic Anal., 97 (2016), 139-164. doi: 10.3233/ASY-151351.

[2]

M. U. AltafE. S. TitiT. GebraelO. M. KnioL. ZhaoM. F. McCabe and I. Hoteit, Downscaling the 2D Bénard convection equations using continuous data assimilation, Computational Geosciences, (2017), 1-18. doi: 10.1007/s10596-017-9619-2.

[3]

A. AzouaniE. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables, J. Nonlinear Sci., 24 (2014), 277-304. doi: 10.1007/s00332-013-9189-y.

[4]

A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters-a reaction-diffusion paradigm, Evol. Equ. Control Theory, 3 (2014), 579-594. doi: 10.3934/eect.2014.3.579.

[5]

H. BessaihE. Olson and E. S. Titi, Continuous data assimilation with stochastically noisy data, Nonlinearity, 28 (2015), 729-753. doi: 10.1088/0951-7715/28/3/729.

[6]

A. Biswas, J. Hudson, A. Larios and Y. Pei, Continuous data assimilation for the magneto-hydrodynamic equations in 2D using one component of the velocity and magnetic fields, Asymptot. Anal., (2018), to appear.

[7]

A. Biswas and V. R. Martinez, Higher-order synchronization for a data assimilation algorithm for the 2D Navier-Stokes equations, Nonlinearity, 35 (2017), 132-157. doi: 10.1016/j.nonrwa.2016.10.005.

[8]

D. Bresch, F. Guillén-González, N. Masmoudi and M. A. Rodríguez-Bellido, Asymptotic derivation of a Navier condition for the primitive equations, Asymptot. Anal., 33 (2003), 237-259

[9]

D. BreschF. Guillén-GonzálezN. Masmoudi and M. A. Rodríguez-Bellido, On the uniqueness of weak solutions of the two-dimensional primitive equations, Differential Integral Equations, 16 (2003), 77-94.

[10]

D. Bresch, F. Guillén-González, N. Masmoudi and M. A. Rodríguez-Bellido, Uniqueness of solution for the 2D primitive equations with friction condition on the bottom, in Seventh Zaragoza-Pau Conference on Applied Mathematics and Statistics (Spanish) (Jaca, 2001) (eds. E.H. Zarantonello and Author 2), Univ. Zaragoza, 27 (2003), 135-143.

[11]

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.

[12]

C. CaoI. G. Kevrekidis and E. S. Titi, Numerical criterion for the stabilization of steady states of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 37-96. doi: 10.1512/iumj.2001.50.2154.

[13]

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.

[14]

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.

[15]

C. CaoJ. Li and E. S. Titi, Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion, Comm. Pure Appl. Math., 69 (2016), 1492-1531. doi: 10.1002/cpa.21576.

[16]

C. CaoJ. Li and E. S. Titi, Strong solutions to the 3D primitive equations with only horizontal dissipation: near $H^1$initial data, J. Funct. Anal., 272 (2017), 4606-4641. doi: 10.1016/j.jfa.2017.01.018.

[17]

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. (2), 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.

[18]

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), 537-563. doi: 10.1007/s00220-011-1409-4.

[19]

I. Chueshov, A squeezing property and its applications to a description of long-time behaviour in the three-dimensional viscous primitive equations, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 711-729. doi: 10.1017/S0308210512001953.

[20]

P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, IL, 1988.

[21]

R. Daley, Atmospheric Data Analysis, Cambridge Atmospheric and Space Science Series, 1993.

[22]

R. Errico and D. Baumhefner, Predictability experiments using a high-resolution limited-area model, Monthly Weather Review, 115 (1986), 488-505.

[23]

A. FarhatM. Jolly and E. S. Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Phys. D, 303 (2015), 59-66. doi: 10.1016/j.physd.2015.03.011.

[24]

A. FarhatE. Lunasin and E. S. Titi, Abridged continuous data assimilation for the 2D Navier-Stokes Equations utilizing measurements of only one component of the velocity field, J. Math. Fluid Mech., 18 (2016), 1-23. doi: 10.1007/s00021-015-0225-6.

[25]

A. FarhatE. Lunasin and E. S. Titi, Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements, J. of Math. Anal. and Appl., 438 (2016), 492-506. doi: 10.1016/j.jmaa.2016.01.072.

[26]

A. FarhatE. Lunasin and E. S. Titi, Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone, J. of Nonlinear Science, (2017), 1-23. doi: 10.1007/s00332-017-9360-y.

[27]

A. Farhat, E. Lunasin and E. S. Titi, On the Charney Conjecture of Data Assimilation Employing Temperature Measurements Alone: The Paradigm of 3D Planetary Geostrophic Model, preprint arXiv: 1608.04770.

[28]

C. FoiasC. Mondaini and E. S. Titi, A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics, SIAM J. Appl. Dyn. Syst., 15 (2016), 2109-2142. doi: 10.1137/16M1076526.

[29]

M. GeshoE. Olson and E. S. Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier-Stokes equations, Commun. Comput. Phys., 19 (2016), 1094-1110.

[30]

N. Glatt-Holtz and M. Ziane, The stochastic primitive equations in two space dimensions with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 801-822. doi: 10.3934/dcdsb.2008.10.801.

[31]

F. Guillén-GonzálezN. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Differential Integral Equations, 14 (2001), 1381-1408.

[32]

B. Guo and D. Huang, On the 3D viscous primitive equations of the large-scale atmosphere, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 846-866. doi: 10.1016/S0252-9602(09)60074-6.

[33]

K. HaydenE. Olson and E. S. Titi, Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations, Phys. D, 240 (2011), 1416-1425. doi: 10.1016/j.physd.2011.04.021.

[34]

J. Hoke and R. Anthes, The initialization of numerical models by a dynamic relaxation technique, Monthly Weather Review, 104 (1976), 1551-1556.

[35]

C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal., 61 (2005), 425-460. doi: 10.1016/j.na.2004.12.005.

[36]

C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, in Frontiers in mathematical analysis and numerical methods, World Sci. Publ., (2004), 149-170.

[37]

M. JollyV. R. Martinez and E. S. Titi, A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Adv. Nonlinear Stud., 17 (2017), 167-192. doi: 10.1515/ans-2016-6019.

[38]

Don A. Jones and E. S. Titi, Determining finite volume elements for the 2D Navier-Stokes equations, Phys. D, 60 (1992), 165-174, Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991). doi: 10.1016/0167-2789(92)90233-D.

[39]

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.

[40]

N. Ju, Global Uniform Boundedness of Solutions to viscous 3D Primitive Equations with Physical Boundary Conditions, preprint, arXiv: 1710.04622v2.

[41]

N. Ju and R. Temam, Finite dimensions of the global attractor for 3D primitive equations with viscosity, J. Nonlinear Sci., 25 (2015), 131-155. doi: 10.1007/s00332-014-9223-8.

[42]

E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2003.

[43]

G. M. Kobelkov, Existence of a solution "in the large" for ocean dynamics equations, J. Math. Fluid Mech., 9 (2007), 588-610. doi: 10.1007/s00021-006-0228-4.

[44]

I. Kukavica and M. Ziane, The regularity of solutions of the primitive equations of the ocean in space dimension three, C. R. Math. Acad. Sci. Paris, 345 (2007), 257-260. doi: 10.1016/j.crma.2007.07.025.

[45]

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.

[46]

I. Kukavica and M. Ziane, Uniform gradient bounds for the primitive equations of the ocean, Differential Integral Equations, 21 (2008), 837-849.

[47]

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