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Asymptotic analysis for 1D compressible Navier-Stokes-Vlasov equations
Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process
1. | School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China |
2. | National Engineering Laboratory of, Integrated Transportation Big Data Application Technology, Chengdu, Sichuan 610031, China |
3. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
In this paper, we study the long term behavior of non-autonomous fractional FitzHugh-Nagumo systems with random forcing given by an approximation of white noise, called Wong-Zakai approximation. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation fractional FitzHugh-Nagumo systems, and then establish the upper semicontinuity of attractors of system driven by a linear multiplicative Wong-Zakai approximations as random forcing approaches white noise in some sense.
References:
[1] |
A. Adili and B. Wang,
Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.
doi: 10.3934/dcdsb.2013.18.643. |
[2] |
P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[3] |
P. W. Bates, K. Lu and B. Wang,
Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[4] |
W. J. Beyn, B. Gess, P. Lescot and M. Röckner,
The global random attractor for a class of stochastic porous media equations, Commun. Partial Differ. Equ., 36 (2011), 446-469.
doi: 10.1080/03605302.2010.523919. |
[5] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[6] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[7] |
T. Caraballo, J. Real and I. D. Chueshov,
Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.
doi: 10.3934/dcdsb.2008.9.525. |
[8] |
H. Cui, M. Freitas and J. Langa,
Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1297-1324.
doi: 10.3934/dcdsb.2018152. |
[9] |
H. Cui and J. Langa,
Uniform attractors for non-autonomous random dynamical systems, J. Differ. Equ., 263 (2017), 1225-1268.
doi: 10.1016/j.jde.2017.03.018. |
[10] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[11] |
J. Duan and B. Schmalfuss,
The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Commun. Math. Sci., 1 (2003), 133-151.
|
[12] |
F. Flandoli, Regularity Theory and Stochastic Flows for Parabolic SPDEs, Stochastics Monographs, Vol. 9 Gordon and Breach, London, 1995. |
[13] |
F. Flandoli and B. Schmalfuss,
Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[14] |
A. Gu, D. Li, B. Wang and H. Yang,
Regularity of random attractors for fractional stochastic reaction-diffusion equations on $ \mathbb{R}^n$, J. Differ. Equ., 264 (2018), 7094-7137.
doi: 10.1016/j.jde.2018.02.011. |
[15] |
A. Gu, K. Lu and B. Wang,
Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185-218.
doi: 10.3934/dcds.2019008. |
[16] |
A. Gu and B. Wang,
Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.
doi: 10.3934/dcdsb.2018072. |
[17] |
Q. Guan,
Integration by parts formula for regional fractional Laplacian, Commun. Math. Phys., 266 (2006), 289-329.
doi: 10.1007/s00220-006-0054-9. |
[18] |
Q. Guan and Z. Ma,
Reflected symmetric α-stable processes and regional fractional Laplacian, Probab. Theory Relat. Field, 134 (2006), 649-694.
doi: 10.1007/s00440-005-0438-3. |
[19] |
Q. Guan and Z. Ma,
Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.
doi: 10.1142/S021949370500150X. |
[20] |
J. Huang and W. Shen,
Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.
doi: 10.3934/dcds.2009.24.855. |
[21] |
M. Jara,
Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.
doi: 10.1002/cpa.20253. |
[22] |
P. E. Kloeden and J. A. Langa,
Flattening, squeezing and the existence of random attractors, Proc. R. Soc. A - Math. Phys. Eng. Sci., 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[23] |
D. Li, B. Wang and X. Wang,
Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differ. Equ., 262 (2017), 1575-1602.
doi: 10.1016/j.jde.2016.10.024. |
[24] |
Y. Li and J. Yin,
A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.
doi: 10.3934/dcdsb.2016.21.1203. |
[25] |
H. Lu, P. W. Bates, S. Lu and M. Zhang,
Dynamics of 3D fractional complex Ginzburg-Landau equation, J. Differ. Equ., 259 (2015), 5276-5301.
doi: 10.1016/j.jde.2015.06.028. |
[26] |
H. Lu, P. W. Bates, J. Xin and M. Zhang,
Asymptotic behavior of stochastic fractional power dissipative equations on $ \mathbb{R}^n$, Nonlinear Anal., 128 (2015), 176-198.
doi: 10.1016/j.na.2015.06.033. |
[27] |
H. Lu, P. W. Bates, S. Lu and M. Zhang,
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.
doi: 10.4310/CMS.2016.v14.n1.a11. |
[28] |
H. Lu, J. Qi, B. Wang and M. Zhang,
Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683-706.
doi: 10.3934/dcds.2019028. |
[29] |
K. Lu and B. Wang,
Wong-Zakai approximations and long term behavior of stochastic partial differential equation, J. Dyn. Differ. Equ., 31 (2019), 1341-1371.
doi: 10.1007/s10884-017-9626-y. |
[30] |
K. Lu and Q. Wang,
Chaotic behavior in differential equations driven by a Brownian motion, J. Differ. Equ., 251 (2011), 2853-2895.
doi: 10.1016/j.jde.2011.05.032. |
[31] |
J. Shen, K. Lu and W. Zhang,
Heteroclinic chaotic behavior driven by a Brownian motion, J. Differ. Equ., 255 (2013), 4185-4225.
doi: 10.1016/j.jde.2013.08.003. |
[32] |
J. Shen, J. Zhao, K. Lu and B. Wang,
The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differ. Equ., 266 (2019), 4568-4623.
doi: 10.1016/j.jde.2018.10.008. |
[33] |
Z. Shen, S. Zhou and W. Shen,
One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differ. Equ., 248 (2010), 1432-1457.
doi: 10.1016/j.jde.2009.10.007. |
[34] |
B. Wang,
Asymptotic behavior of stochastic wave equations with critical exponents on $ \mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[35] |
B. Wang,
Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[36] |
B. Wang,
Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.
doi: 10.3934/dcds.2014.34.269. |
[37] |
B. Wang,
Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.
doi: 10.1016/j.na.2017.04.006. |
[38] |
B. Wang,
Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2011-2051.
doi: 10.3934/dcdsb.2017119. |
[39] |
R. Wang, Y. Li and B. Wang,
Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.
doi: 10.3934/dcds.2019165. |
[40] |
X. Wang, K. Lu and B. Wang,
Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 264 (2018), 378-424.
doi: 10.1016/j.jde.2017.09.006. |
[41] |
Y. Wang and J. Wang,
Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differ. Equ., 259 (2015), 728-776.
doi: 10.1016/j.jde.2015.02.026. |
show all references
References:
[1] |
A. Adili and B. Wang,
Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.
doi: 10.3934/dcdsb.2013.18.643. |
[2] |
P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[3] |
P. W. Bates, K. Lu and B. Wang,
Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[4] |
W. J. Beyn, B. Gess, P. Lescot and M. Röckner,
The global random attractor for a class of stochastic porous media equations, Commun. Partial Differ. Equ., 36 (2011), 446-469.
doi: 10.1080/03605302.2010.523919. |
[5] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[6] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[7] |
T. Caraballo, J. Real and I. D. Chueshov,
Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.
doi: 10.3934/dcdsb.2008.9.525. |
[8] |
H. Cui, M. Freitas and J. Langa,
Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1297-1324.
doi: 10.3934/dcdsb.2018152. |
[9] |
H. Cui and J. Langa,
Uniform attractors for non-autonomous random dynamical systems, J. Differ. Equ., 263 (2017), 1225-1268.
doi: 10.1016/j.jde.2017.03.018. |
[10] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[11] |
J. Duan and B. Schmalfuss,
The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Commun. Math. Sci., 1 (2003), 133-151.
|
[12] |
F. Flandoli, Regularity Theory and Stochastic Flows for Parabolic SPDEs, Stochastics Monographs, Vol. 9 Gordon and Breach, London, 1995. |
[13] |
F. Flandoli and B. Schmalfuss,
Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[14] |
A. Gu, D. Li, B. Wang and H. Yang,
Regularity of random attractors for fractional stochastic reaction-diffusion equations on $ \mathbb{R}^n$, J. Differ. Equ., 264 (2018), 7094-7137.
doi: 10.1016/j.jde.2018.02.011. |
[15] |
A. Gu, K. Lu and B. Wang,
Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185-218.
doi: 10.3934/dcds.2019008. |
[16] |
A. Gu and B. Wang,
Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.
doi: 10.3934/dcdsb.2018072. |
[17] |
Q. Guan,
Integration by parts formula for regional fractional Laplacian, Commun. Math. Phys., 266 (2006), 289-329.
doi: 10.1007/s00220-006-0054-9. |
[18] |
Q. Guan and Z. Ma,
Reflected symmetric α-stable processes and regional fractional Laplacian, Probab. Theory Relat. Field, 134 (2006), 649-694.
doi: 10.1007/s00440-005-0438-3. |
[19] |
Q. Guan and Z. Ma,
Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.
doi: 10.1142/S021949370500150X. |
[20] |
J. Huang and W. Shen,
Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.
doi: 10.3934/dcds.2009.24.855. |
[21] |
M. Jara,
Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.
doi: 10.1002/cpa.20253. |
[22] |
P. E. Kloeden and J. A. Langa,
Flattening, squeezing and the existence of random attractors, Proc. R. Soc. A - Math. Phys. Eng. Sci., 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[23] |
D. Li, B. Wang and X. Wang,
Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differ. Equ., 262 (2017), 1575-1602.
doi: 10.1016/j.jde.2016.10.024. |
[24] |
Y. Li and J. Yin,
A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.
doi: 10.3934/dcdsb.2016.21.1203. |
[25] |
H. Lu, P. W. Bates, S. Lu and M. Zhang,
Dynamics of 3D fractional complex Ginzburg-Landau equation, J. Differ. Equ., 259 (2015), 5276-5301.
doi: 10.1016/j.jde.2015.06.028. |
[26] |
H. Lu, P. W. Bates, J. Xin and M. Zhang,
Asymptotic behavior of stochastic fractional power dissipative equations on $ \mathbb{R}^n$, Nonlinear Anal., 128 (2015), 176-198.
doi: 10.1016/j.na.2015.06.033. |
[27] |
H. Lu, P. W. Bates, S. Lu and M. Zhang,
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.
doi: 10.4310/CMS.2016.v14.n1.a11. |
[28] |
H. Lu, J. Qi, B. Wang and M. Zhang,
Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683-706.
doi: 10.3934/dcds.2019028. |
[29] |
K. Lu and B. Wang,
Wong-Zakai approximations and long term behavior of stochastic partial differential equation, J. Dyn. Differ. Equ., 31 (2019), 1341-1371.
doi: 10.1007/s10884-017-9626-y. |
[30] |
K. Lu and Q. Wang,
Chaotic behavior in differential equations driven by a Brownian motion, J. Differ. Equ., 251 (2011), 2853-2895.
doi: 10.1016/j.jde.2011.05.032. |
[31] |
J. Shen, K. Lu and W. Zhang,
Heteroclinic chaotic behavior driven by a Brownian motion, J. Differ. Equ., 255 (2013), 4185-4225.
doi: 10.1016/j.jde.2013.08.003. |
[32] |
J. Shen, J. Zhao, K. Lu and B. Wang,
The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differ. Equ., 266 (2019), 4568-4623.
doi: 10.1016/j.jde.2018.10.008. |
[33] |
Z. Shen, S. Zhou and W. Shen,
One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differ. Equ., 248 (2010), 1432-1457.
doi: 10.1016/j.jde.2009.10.007. |
[34] |
B. Wang,
Asymptotic behavior of stochastic wave equations with critical exponents on $ \mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[35] |
B. Wang,
Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[36] |
B. Wang,
Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.
doi: 10.3934/dcds.2014.34.269. |
[37] |
B. Wang,
Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.
doi: 10.1016/j.na.2017.04.006. |
[38] |
B. Wang,
Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2011-2051.
doi: 10.3934/dcdsb.2017119. |
[39] |
R. Wang, Y. Li and B. Wang,
Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.
doi: 10.3934/dcds.2019165. |
[40] |
X. Wang, K. Lu and B. Wang,
Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 264 (2018), 378-424.
doi: 10.1016/j.jde.2017.09.006. |
[41] |
Y. Wang and J. Wang,
Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differ. Equ., 259 (2015), 728-776.
doi: 10.1016/j.jde.2015.02.026. |
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