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doi: 10.3934/dcdsb.2020309

Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay

Tongtong Liang and  Yejuan Wang ,

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

* Corresponding author

Received  June 2020 Revised  September 2020 Published  October 2020

Fund Project: This work was supported by the National Natural Science Foundation of China under grant 41875084

In this paper, we investigate a stochastic fractionally dissipative quasi-geostrophic equation driven by a multiplicative white noise, whose external forces contain hereditary characteristics. The existence and uniqueness of both local martingale and local pathwise solutions are established in $ H^s $ with $ s\geq2-2\alpha $, where $ \alpha\in(\frac{1}{2}, 1) $. For the critical case $ \alpha = \frac12 $, we obtain the similar results in $ H^s $ with $ s>1 $.

Keywords: Stochastic quasi-geostrophic equation, fractional Laplacian, infinite delay, martingale solution, pathwise solution.
Mathematics Subject Classification: Primary: 35Q86, 60H15, 76B03.
Citation: Tongtong Liang, Yejuan Wang. Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020309
References:
[1]

Z. Brzeźniak and E. Motyl, Fractionally dissipative stochastic quasi-geostrophic type equations on $\mathbb{R}^d$, SIAM J. Math. Anal., 51 (2019), 2306-2358.  doi: 10.1137/17M1111589.  Google Scholar

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A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.  Google Scholar

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F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

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N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar

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N. Ju, Global solutions to the two dimensional quasi-geostrophic equation with critical or super-critical dissipation, Math. Ann., 334 (2006), 627-642.  doi: 10.1007/s00208-005-0715-6.  Google Scholar

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T. T. Medjo, Attractors for the multilayer quasi-geostrophic equations of the ocean with delays, Appl. Anal., 87 (2008), 325-347.  doi: 10.1080/00036810701858177.  Google Scholar

[20]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310.  doi: 10.1137/S0036141002409167.  Google Scholar

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C. J. Niche and G. Planas, Existence and decay of solutions to the dissipative quasi-geostrophic equation with delays, Nonlinear Anal., 75 (2012), 3936-3950.  doi: 10.1016/j.na.2012.02.017.  Google Scholar

[22]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167.  doi: 10.1080/17442507908833142.  Google Scholar

[23]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[24]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, in: Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007. doi: 10.1007/978-3-540-70781-3.  Google Scholar

[25]

M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Condensed Matter Phys., 2 (2008), 247-259.   Google Scholar

[26]

M. Röckner, R. Zhu and X. Zhu, Sub and supercritical stochastic quasi-geostrophic equation, Ann. Probab., 43 (2015), 1202-1273.  doi: 10.1214/13-AOP887.  Google Scholar

[27]

M. Röckner, R. Zhu and X. Zhu, Stochastic quasi-geostrophic equation, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 15 (2012), 1-6.  doi: 10.1142/S0219025712500014.  Google Scholar

[28] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.   Google Scholar
[29]

T. Taniguchi, The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier-Stokes equations driven by Levy processes, J. Math. Anal. Appl., 385 (2012), 634-654.  doi: 10.1016/j.jmaa.2011.06.076.  Google Scholar

[30]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. doi: 10.1115/1.3424338.  Google Scholar

[31]

L. Wan and J. Duan, Exponential stability of the multi-layer quasi-geostrophic ocean model with delays, Nonlinear Anal., 71 (2009), 799-811.  doi: 10.1016/j.na.2008.10.107.  Google Scholar

[32]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 56 (2001), 1-13.   Google Scholar

[33]

R. Zhu and X. Zhu, Random attractor associated with the quasi-geostrophic equation, J. Dynam. Differential Equations, 29 (2017), 289-322.  doi: 10.1007/s10884-016-9537-3.  Google Scholar

show all references

References:
[1]

Z. Brzeźniak and E. Motyl, Fractionally dissipative stochastic quasi-geostrophic type equations on $\mathbb{R}^d$, SIAM J. Math. Anal., 51 (2019), 2306-2358.  doi: 10.1137/17M1111589.  Google Scholar

[2]

Z. Brzézniak and S. Peszat, Strong local and global solutions for stochastic Navier-Stokes equations, infinite dimensional stochastic analysis, Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., 52 (2000), 85-98.   Google Scholar

[3]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar

[4]

T. Caraballo and X. Han, A survey on Navier-Stokes models with delays: existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 1079-1101.  doi: 10.3934/dcdss.2015.8.1079.  Google Scholar

[5]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., vol. 152, Cambridge University Press, 2014. doi: 10.1017/CBO9780511666223.  Google Scholar

[6]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.  Google Scholar

[7]

T. Dlotko, T. Liang and Y. Wang, Critical and super-critical abstract parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1517-1541.  doi: 10.3934/dcdsb.2019238.  Google Scholar

[8]

R. Farwig and C. Qian, Asymptotic behavior for the quasi-geostrophic equations with fractional dissipation in $\mathbb{R}^2$, J. Differential Equations, 266 (2019), 6525-6579.  doi: 10.1016/j.jde.2018.11.009.  Google Scholar

[9]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[10]

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.  Google Scholar

[11]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar

[12]

N. Ju, Global solutions to the two dimensional quasi-geostrophic equation with critical or super-critical dissipation, Math. Ann., 334 (2006), 627-642.  doi: 10.1007/s00208-005-0715-6.  Google Scholar

[13]

N. Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Comm. Math. Phys., 251 (2004), 365-376.  doi: 10.1007/s00220-004-1062-2.  Google Scholar

[14]

N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161-181.  doi: 10.1007/s00220-004-1256-7.  Google Scholar

[15]

T. G. Kurtz, The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities, Electr. J. Probab., 12 (2007), 951-965.  doi: 10.1214/EJP.v12-431.  Google Scholar

[16]

L. Liu and T. Caraballo, Analysis of a stochastic 2D-Navier-Stokes model with infinite delay, J. Dynam. Differential Equations, 31 (2019), 2249-2274.  doi: 10.1007/s10884-018-9703-x.  Google Scholar

[17]

H. Lu, S. Lü, J. Xin and D. Huang, A random attractor for the stochastic quasi-geostrophic dynamical system on unbounded domains, Nonlinear Anal., 90 (2013), 96-112.  doi: 10.1016/j.na.2013.05.020.  Google Scholar

[18]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655-673.  doi: 10.3934/dcdsb.2010.14.655.  Google Scholar

[19]

T. T. Medjo, Attractors for the multilayer quasi-geostrophic equations of the ocean with delays, Appl. Anal., 87 (2008), 325-347.  doi: 10.1080/00036810701858177.  Google Scholar

[20]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310.  doi: 10.1137/S0036141002409167.  Google Scholar

[21]

C. J. Niche and G. Planas, Existence and decay of solutions to the dissipative quasi-geostrophic equation with delays, Nonlinear Anal., 75 (2012), 3936-3950.  doi: 10.1016/j.na.2012.02.017.  Google Scholar

[22]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167.  doi: 10.1080/17442507908833142.  Google Scholar

[23]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[24]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, in: Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007. doi: 10.1007/978-3-540-70781-3.  Google Scholar

[25]

M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Condensed Matter Phys., 2 (2008), 247-259.   Google Scholar

[26]

M. Röckner, R. Zhu and X. Zhu, Sub and supercritical stochastic quasi-geostrophic equation, Ann. Probab., 43 (2015), 1202-1273.  doi: 10.1214/13-AOP887.  Google Scholar

[27]

M. Röckner, R. Zhu and X. Zhu, Stochastic quasi-geostrophic equation, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 15 (2012), 1-6.  doi: 10.1142/S0219025712500014.  Google Scholar

[28] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.   Google Scholar
[29]

T. Taniguchi, The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier-Stokes equations driven by Levy processes, J. Math. Anal. Appl., 385 (2012), 634-654.  doi: 10.1016/j.jmaa.2011.06.076.  Google Scholar

[30]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. doi: 10.1115/1.3424338.  Google Scholar

[31]

L. Wan and J. Duan, Exponential stability of the multi-layer quasi-geostrophic ocean model with delays, Nonlinear Anal., 71 (2009), 799-811.  doi: 10.1016/j.na.2008.10.107.  Google Scholar

[32]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 56 (2001), 1-13.   Google Scholar

[33]

R. Zhu and X. Zhu, Random attractor associated with the quasi-geostrophic equation, J. Dynam. Differential Equations, 29 (2017), 289-322.  doi: 10.1007/s10884-016-9537-3.  Google Scholar

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