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

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September 2021, 26(9): 4697-4726. 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 September 2021 Early access 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 and Continuous Dynamical Systems - B, 2021, 26 (9) : 4697-4726. doi: 10.3934/dcdsb.2020309

References:

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[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167. doi: 10.1080/17442507908833142.
[23]	J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.
[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.
[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.
[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.
[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.
[28]	E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[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.
[30]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. doi: 10.1115/1.3424338.
[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.
[32]	J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 56 (2001), 1-13.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167. doi: 10.1080/17442507908833142.
[23]	J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.
[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.
[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.
[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.
[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.
[28]	E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[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.
[30]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. doi: 10.1115/1.3424338.
[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.
[32]	J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 56 (2001), 1-13.
[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.

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