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Global phase portraits and bifurcation diagrams for reversible equivariant hamiltonian systems of linear plus quartic homogeneous polynomials
Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China |
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 $.
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. 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. |
[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. 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. |
[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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