doi: 10.3934/dcdsb.2022019
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Global well-posedness of the three-dimensional viscous primitive equations with bounded delays

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

*Corresponding author: Wenjun Liu

Received  July 2021 Revised  November 2021 Early access February 2022

Fund Project: This work was supported by the National Natural Science Foundation of China [grant numbers 11771216 and 11901306], the Key Research and Development Program of Jiangsu Province (Social Development) [grant number BE2019725], and the Natural Science Foundation of Jiangsu Province [grant number SBK2017043142]

The study of delay is one of the important problems in fluid mechanics. When we attempt to control the fluid in some sense, this delay may occur by applying a force that takes into account not only the current state of the system, but also the known history. In this paper, the three-dimensional viscous primitive equations with bounded delays are considered. We prove the existence of weak and strong solutions, and obtain the uniqueness of the strong solution. We also obtain the exponential decay behavior of the weak solutions and get some higher order estimates for strong solution. Under appropriate assumptions, we prove that the time-dependent weak solutions converge exponentially to the unique stationary solution.

Citation: Zhenduo Fan, Wenjun Liu, Shengqian Chen. Global well-posedness of the three-dimensional viscous primitive equations with bounded delays. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022019
References:
[1]

M. M. Alam and S. Dubey, Mild solutions of time fractional Navier-Stokes equations driven by finite delayed external forces, arXiv: 1905.13515v2.

[2]

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.

[3]

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.

[4]

C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity, Phys. D, 412 (2020), 132606, 25 pp. doi: 10.1016/j.physd.2020.132606.

[5]

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.

[6]

T. CaraballoA. M. Márquez-Durán and J. Real, Asymptotic behaviour of the three-dimensional $\alpha$-Navier-Stokes model with delays, J. Math. Anal. Appl., 340 (2008), 410-423.  doi: 10.1016/j.jmaa.2007.08.011.

[7]

T. Caraballo, A. M. Márquez-Durán and J. Real, Asymptotic behaviour of the three-dimensional $\alpha$-Navier-Stokes model with locally Lipschitz delay forcing terms, Nonlinear Anal., 71 (2009), e271–e282. doi: 10.1016/j.na.2008.10.048.

[8]

T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807.

[9]

T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.

[10]

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.

[11]

H. GaoŠ. Nečasová and T. Tang, On weak-strong uniqueness and singular limit for the compressible primitive equations, Discrete Contin. Dyn. Syst., 40 (2020), 4287-4305.  doi: 10.3934/dcds.2020181.

[12]

H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3053-3073.  doi: 10.3934/dcdsb.2016087.

[13]

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. 

[14]

B. Guo and G. Zhou, Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4305-4327.  doi: 10.3934/dcdsb.2018160.

[15]

S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238.  doi: 10.3934/dcdsb.2011.16.225.

[16]

S. M. Guzzo and G. Planas, Existence of solutions for a class of Navier-Stokes equations with infinite delay, Appl. Anal., 94 (2015), 840-855.  doi: 10.1080/00036811.2014.905677.

[17]

C. HuR. 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.

[18]

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.

[19]

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.

[20]

G. M. Kobelkov, Existence of a solution `in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286.  doi: 10.1016/j.crma.2006.04.020.

[21]

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.

[22]

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.

[23]

J.-L. LionsR. Temam and S. H. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.  doi: 10.1088/0951-7715/5/2/001.

[24]

J.-L. LionsR. Temam and S. H. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.  doi: 10.1088/0951-7715/5/5/002.

[25]

W. LiuD. Chen and Z. Chen, Long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, Acta Math. Sci. Ser. B (Engl. Ed.), 41 (2021), 609-632.  doi: 10.1007/s10473-021-0220-3.

[26]

W. Liu and H. Zhuang, Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 907-942.  doi: 10.3934/dcdsb.2020147.

[27]

X. Liu and E. S. Titi, Global existence of weak solutions to the compressible primitive equations of atmospheric dynamics with degenerate viscosities, SIAM J. Math. Anal., 51 (2019), 1913-1964.  doi: 10.1137/18M1211994.

[28]

X. Liu and E. S. Titi, Local well-posedness of strong solutions to the three-dimensional compressible primitive equations, Arch. Ration. Mech. Anal., 241 (2021), 729-764.  doi: 10.1007/s00205-021-01662-3.

[29]

X. Liu and E. S. Titi, Zero Mach number limit of the compressible primitive equations: Well-prepared initial data, Arch. Ration. Mech. Anal., 238 (2020), 705-747.  doi: 10.1007/s00205-020-01553-z.

[30]

G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258.  doi: 10.3934/dcds.2008.21.1245.

[31]

Y. Qin and K. Su, Upper estimates on Hausdorff and fractal dimensions of global attractors for the 2D Navier-Stokes-Voight equations with a distributed delay, Asymptot. Anal., 111 (2019), 179-199.  doi: 10.3233/ASY-181492.

[32]

K. Su and Y. Qin, Long-time behavior for the Navier-Stokes-Voight equations with delay on a non-smooth domain, J. Partial Differ. Equ., 31 (2018), 281-290.  doi: 10.4208/jpde.v31.n3.7.

[33]

T. Tachim Medjo, The primitive equations of the ocean with delays, Nonlinear Anal. Real World Appl., 10 (2009), 779-797.  doi: 10.1016/j.nonrwa.2007.11.003.

[34]

T. Tachim Medjo, The exponential behavior of the stochastic three-dimensional primitive equations with multiplicative noise, Nonlinear Anal. Real World Appl., 12 (2011), 799-810.  doi: 10.1016/j.nonrwa.2010.08.007.

[35]

B. You, Pullback attractor for the three dimensional nonautonomous primitive equations of large-scale ocean and atmosphere dynamics, Comput. Math. Methods, 2 (2020), e1066, 26 pp. doi: 10.1002/cmm4.1066.

[36]

B. You and F. Li, Global attractor of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics, Z. Angew. Math. Phys., 69 (2018), Paper No. 114, 13 pp. doi: 10.1007/s00033-018-1007-9.

[37]

G. Zhou, Random attractor for the 3D viscous primitive equations driven by fractional noises, J. Differential Equations, 266 (2019), 7569-7637.  doi: 10.1016/j.jde.2018.12.009.

[38]

G. Zhou and B. Guo, The global attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, Commun. Math. Sci., 16 (2018), 2003-2032.  doi: 10.4310/CMS.2018.v16.n7.a11.

[39]

G. Zhou and B. Guo, Global well-posedness of stochastic 2D primitive equations with random initial conditions, Phys. D, 414 (2020), 132713, 24 pp. doi: 10.1016/j.physd.2020.132713.

show all references

References:
[1]

M. M. Alam and S. Dubey, Mild solutions of time fractional Navier-Stokes equations driven by finite delayed external forces, arXiv: 1905.13515v2.

[2]

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.

[3]

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.

[4]

C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity, Phys. D, 412 (2020), 132606, 25 pp. doi: 10.1016/j.physd.2020.132606.

[5]

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.

[6]

T. CaraballoA. M. Márquez-Durán and J. Real, Asymptotic behaviour of the three-dimensional $\alpha$-Navier-Stokes model with delays, J. Math. Anal. Appl., 340 (2008), 410-423.  doi: 10.1016/j.jmaa.2007.08.011.

[7]

T. Caraballo, A. M. Márquez-Durán and J. Real, Asymptotic behaviour of the three-dimensional $\alpha$-Navier-Stokes model with locally Lipschitz delay forcing terms, Nonlinear Anal., 71 (2009), e271–e282. doi: 10.1016/j.na.2008.10.048.

[8]

T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807.

[9]

T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.

[10]

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.

[11]

H. GaoŠ. Nečasová and T. Tang, On weak-strong uniqueness and singular limit for the compressible primitive equations, Discrete Contin. Dyn. Syst., 40 (2020), 4287-4305.  doi: 10.3934/dcds.2020181.

[12]

H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3053-3073.  doi: 10.3934/dcdsb.2016087.

[13]

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. 

[14]

B. Guo and G. Zhou, Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4305-4327.  doi: 10.3934/dcdsb.2018160.

[15]

S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238.  doi: 10.3934/dcdsb.2011.16.225.

[16]

S. M. Guzzo and G. Planas, Existence of solutions for a class of Navier-Stokes equations with infinite delay, Appl. Anal., 94 (2015), 840-855.  doi: 10.1080/00036811.2014.905677.

[17]

C. HuR. 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.

[18]

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.

[19]

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.

[20]

G. M. Kobelkov, Existence of a solution `in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286.  doi: 10.1016/j.crma.2006.04.020.

[21]

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.

[22]

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.

[23]

J.-L. LionsR. Temam and S. H. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.  doi: 10.1088/0951-7715/5/2/001.

[24]

J.-L. LionsR. Temam and S. H. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.  doi: 10.1088/0951-7715/5/5/002.

[25]

W. LiuD. Chen and Z. Chen, Long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, Acta Math. Sci. Ser. B (Engl. Ed.), 41 (2021), 609-632.  doi: 10.1007/s10473-021-0220-3.

[26]

W. Liu and H. Zhuang, Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 907-942.  doi: 10.3934/dcdsb.2020147.

[27]

X. Liu and E. S. Titi, Global existence of weak solutions to the compressible primitive equations of atmospheric dynamics with degenerate viscosities, SIAM J. Math. Anal., 51 (2019), 1913-1964.  doi: 10.1137/18M1211994.

[28]

X. Liu and E. S. Titi, Local well-posedness of strong solutions to the three-dimensional compressible primitive equations, Arch. Ration. Mech. Anal., 241 (2021), 729-764.  doi: 10.1007/s00205-021-01662-3.

[29]

X. Liu and E. S. Titi, Zero Mach number limit of the compressible primitive equations: Well-prepared initial data, Arch. Ration. Mech. Anal., 238 (2020), 705-747.  doi: 10.1007/s00205-020-01553-z.

[30]

G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258.  doi: 10.3934/dcds.2008.21.1245.

[31]

Y. Qin and K. Su, Upper estimates on Hausdorff and fractal dimensions of global attractors for the 2D Navier-Stokes-Voight equations with a distributed delay, Asymptot. Anal., 111 (2019), 179-199.  doi: 10.3233/ASY-181492.

[32]

K. Su and Y. Qin, Long-time behavior for the Navier-Stokes-Voight equations with delay on a non-smooth domain, J. Partial Differ. Equ., 31 (2018), 281-290.  doi: 10.4208/jpde.v31.n3.7.

[33]

T. Tachim Medjo, The primitive equations of the ocean with delays, Nonlinear Anal. Real World Appl., 10 (2009), 779-797.  doi: 10.1016/j.nonrwa.2007.11.003.

[34]

T. Tachim Medjo, The exponential behavior of the stochastic three-dimensional primitive equations with multiplicative noise, Nonlinear Anal. Real World Appl., 12 (2011), 799-810.  doi: 10.1016/j.nonrwa.2010.08.007.

[35]

B. You, Pullback attractor for the three dimensional nonautonomous primitive equations of large-scale ocean and atmosphere dynamics, Comput. Math. Methods, 2 (2020), e1066, 26 pp. doi: 10.1002/cmm4.1066.

[36]

B. You and F. Li, Global attractor of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics, Z. Angew. Math. Phys., 69 (2018), Paper No. 114, 13 pp. doi: 10.1007/s00033-018-1007-9.

[37]

G. Zhou, Random attractor for the 3D viscous primitive equations driven by fractional noises, J. Differential Equations, 266 (2019), 7569-7637.  doi: 10.1016/j.jde.2018.12.009.

[38]

G. Zhou and B. Guo, The global attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, Commun. Math. Sci., 16 (2018), 2003-2032.  doi: 10.4310/CMS.2018.v16.n7.a11.

[39]

G. Zhou and B. Guo, Global well-posedness of stochastic 2D primitive equations with random initial conditions, Phys. D, 414 (2020), 132713, 24 pp. doi: 10.1016/j.physd.2020.132713.

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