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Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic
Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains
1. | School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China |
2. | Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA |
3. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
4. | Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA |
In this paper, we study the limiting behavior of dynamics for stochastic reaction-diffusion equations driven by an additive noise and a deterministic non-autonomous forcing on an (n+1)-dimensional thin region when it collapses into an n-dimensional region. We first established the existence of attractors and their properties for these equations on (n+1)-dimensional thin domains. We then show that these attractors converge to the random attractor of the limit equation under the usual semi-distance as the thinness goes to zero.
References:
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F. Antoci and M. Prizzi,
Reaction-diffusion equations on unbounded thin domains, Topol. Methods Nonlinear Anal., 18 (2001), 283-302.
doi: 10.12775/TMNA.2001.035. |
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L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998
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J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz,
Dynamics in dumbbell domains Ⅰ. Continuity of the set of equilibria, J. Differential Equations, 231 (2006), 551-597.
doi: 10.1016/j.jde.2006.06.002. |
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J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz,
Dynamics in dumbbell domains Ⅱ. The limiting problem, J. Differential Equations, 247 (2009), 174-202.
doi: 10.1016/j.jde.2009.03.014. |
[5] |
J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz,
Dynamics in dumbbell domains Ⅲ. Continuity of attractors, J. Differential Equations, 247 (2009), 225-259.
doi: 10.1016/j.jde.2008.12.014. |
[6] |
J. M. Arrieta, A. N. Carvalho, M. C. Pereira and R. P. Da Silva,
Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal., 74 (2011), 5111-5132.
doi: 10.1016/j.na.2011.05.006. |
[7] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[8] |
P. W. Bates, K. Lu and B. Wang,
Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[9] |
T. Caraballo, I. D. Chueshov and P. E. Kloeden,
Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2007), 1489-1507.
doi: 10.1137/050647281. |
[10] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Continuous Dynam. Systems-A, 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[11] |
T. Caraballo, J. Real and I. D. Chueshov,
Pullback attractors for stochastic heat equations in materials with memory, Discrete Continuous Dynam. Systems-B, 9 (2008), 525-539.
doi: 10.3934/dcdsb.2008.9.525. |
[12] |
I. D. Chueshov and S. Kuksin,
Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008), 117-153.
doi: 10.1007/s00205-007-0068-2. |
[13] |
I. D. Chueshov and S. Kuksin,
Stochastic 3D Navier-Stokes equations in a thin domain and its $α$-approximation, Phys. D, 237 (2008), 1352-1367.
doi: 10.1016/j.physd.2008.03.012. |
[14] |
I. S. Ciuperca,
Reaction-diffusion equations on thin domains with varying order of thinness, J. Differential Equations, 126 (1996), 244-291.
doi: 10.1006/jdeq.1996.0051. |
[15] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[16] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[17] |
T. Elsken,
Attractors for reaction-diffusion equations on thin domains whose linear part is non-self-adjoint, J. Differential Equations, 206 (2004), 94-126.
doi: 10.1016/j.jde.2004.07.025. |
[18] |
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. |
[19] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. |
[20] |
J. Hale and G. Raugel,
Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95.
|
[21] |
J. Hale and G. Raugel,
A reaction-diffusion equation on a thin $L$-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283-327.
doi: 10.1017/S0308210500028043. |
[22] |
R. Johnson, M. Kamenskii and P. Nistri,
Existence of periodic solutions of an autonomous damped wave equation in thin domains, J. Dynam. Differential Equations, 10 (1998), 409-424.
doi: 10.1023/A:1022601213052. |
[23] |
P. E. Kloeden and J. A. Langa,
Flattening, squeezing and the existence of random attractors, Proc. R. Soc. London, Ser. A, 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[24] |
D. Li, B. Wang and X. Wang,
Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.
doi: 10.1016/j.jde.2016.10.024. |
[25] |
W. Liu and B. Wang,
Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dynam. Differential Equations, 22 (2010), 413-437.
doi: 10.1007/s10884-010-9186-x. |
[26] |
Y. Morita,
Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain, Japan J. Indust. Appl. Math., 21 (2004), 129-147.
doi: 10.1007/BF03167468. |
[27] |
M. Prizzi and K. P. Rybakowski,
Recent results on thin domain problems, Ⅱ, Topol. Methods Nonlinear Anal., 19 (2002), 199-219.
doi: 10.12775/TMNA.2002.010. |
[28] |
M. Prizzi and K. P. Rybakowski,
The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 237 (2001), 271-320.
doi: 10.1006/jdeq.2000.3917. |
[29] |
G. Raugel and G. Sell,
Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.
doi: 10.2307/2152776. |
[30] |
D. Ruelle,
Characteristic exponents for a viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300.
doi: 10.1007/BF01258529. |
[31] |
B. Schmalfuss,
Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185-192.
|
[32] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997
doi: 10.1007/978-1-4612-0645-3. |
[33] |
S. M. Ulam and J. von Neumann, Random ergodic theorems, Bull. Amer. Math. Soc. , 51 (1945), p660. |
[34] |
B. Wang,
Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[35] |
B. Wang,
Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Continuous Dynam. Systems-A, 34 (2014), 269-300.
doi: 10.3934/dcds.2014.34.269. |
[36] |
X. Wang, K. Lu and B. Wang,
Long term behavior of delay parabolic equations with additive noise and deterministic time dependent forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047.
doi: 10.1137/140991819. |
show all references
References:
[1] |
F. Antoci and M. Prizzi,
Reaction-diffusion equations on unbounded thin domains, Topol. Methods Nonlinear Anal., 18 (2001), 283-302.
doi: 10.12775/TMNA.2001.035. |
[2] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998
doi: 10.1007/978-3-662-12878-7. |
[3] |
J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz,
Dynamics in dumbbell domains Ⅰ. Continuity of the set of equilibria, J. Differential Equations, 231 (2006), 551-597.
doi: 10.1016/j.jde.2006.06.002. |
[4] |
J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz,
Dynamics in dumbbell domains Ⅱ. The limiting problem, J. Differential Equations, 247 (2009), 174-202.
doi: 10.1016/j.jde.2009.03.014. |
[5] |
J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz,
Dynamics in dumbbell domains Ⅲ. Continuity of attractors, J. Differential Equations, 247 (2009), 225-259.
doi: 10.1016/j.jde.2008.12.014. |
[6] |
J. M. Arrieta, A. N. Carvalho, M. C. Pereira and R. P. Da Silva,
Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal., 74 (2011), 5111-5132.
doi: 10.1016/j.na.2011.05.006. |
[7] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[8] |
P. W. Bates, K. Lu and B. Wang,
Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[9] |
T. Caraballo, I. D. Chueshov and P. E. Kloeden,
Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2007), 1489-1507.
doi: 10.1137/050647281. |
[10] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Continuous Dynam. Systems-A, 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[11] |
T. Caraballo, J. Real and I. D. Chueshov,
Pullback attractors for stochastic heat equations in materials with memory, Discrete Continuous Dynam. Systems-B, 9 (2008), 525-539.
doi: 10.3934/dcdsb.2008.9.525. |
[12] |
I. D. Chueshov and S. Kuksin,
Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008), 117-153.
doi: 10.1007/s00205-007-0068-2. |
[13] |
I. D. Chueshov and S. Kuksin,
Stochastic 3D Navier-Stokes equations in a thin domain and its $α$-approximation, Phys. D, 237 (2008), 1352-1367.
doi: 10.1016/j.physd.2008.03.012. |
[14] |
I. S. Ciuperca,
Reaction-diffusion equations on thin domains with varying order of thinness, J. Differential Equations, 126 (1996), 244-291.
doi: 10.1006/jdeq.1996.0051. |
[15] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[16] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[17] |
T. Elsken,
Attractors for reaction-diffusion equations on thin domains whose linear part is non-self-adjoint, J. Differential Equations, 206 (2004), 94-126.
doi: 10.1016/j.jde.2004.07.025. |
[18] |
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. |
[19] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. |
[20] |
J. Hale and G. Raugel,
Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95.
|
[21] |
J. Hale and G. Raugel,
A reaction-diffusion equation on a thin $L$-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283-327.
doi: 10.1017/S0308210500028043. |
[22] |
R. Johnson, M. Kamenskii and P. Nistri,
Existence of periodic solutions of an autonomous damped wave equation in thin domains, J. Dynam. Differential Equations, 10 (1998), 409-424.
doi: 10.1023/A:1022601213052. |
[23] |
P. E. Kloeden and J. A. Langa,
Flattening, squeezing and the existence of random attractors, Proc. R. Soc. London, Ser. A, 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[24] |
D. Li, B. Wang and X. Wang,
Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.
doi: 10.1016/j.jde.2016.10.024. |
[25] |
W. Liu and B. Wang,
Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dynam. Differential Equations, 22 (2010), 413-437.
doi: 10.1007/s10884-010-9186-x. |
[26] |
Y. Morita,
Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain, Japan J. Indust. Appl. Math., 21 (2004), 129-147.
doi: 10.1007/BF03167468. |
[27] |
M. Prizzi and K. P. Rybakowski,
Recent results on thin domain problems, Ⅱ, Topol. Methods Nonlinear Anal., 19 (2002), 199-219.
doi: 10.12775/TMNA.2002.010. |
[28] |
M. Prizzi and K. P. Rybakowski,
The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 237 (2001), 271-320.
doi: 10.1006/jdeq.2000.3917. |
[29] |
G. Raugel and G. Sell,
Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.
doi: 10.2307/2152776. |
[30] |
D. Ruelle,
Characteristic exponents for a viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300.
doi: 10.1007/BF01258529. |
[31] |
B. Schmalfuss,
Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185-192.
|
[32] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997
doi: 10.1007/978-1-4612-0645-3. |
[33] |
S. M. Ulam and J. von Neumann, Random ergodic theorems, Bull. Amer. Math. Soc. , 51 (1945), p660. |
[34] |
B. Wang,
Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[35] |
B. Wang,
Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Continuous Dynam. Systems-A, 34 (2014), 269-300.
doi: 10.3934/dcds.2014.34.269. |
[36] |
X. Wang, K. Lu and B. Wang,
Long term behavior of delay parabolic equations with additive noise and deterministic time dependent forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047.
doi: 10.1137/140991819. |
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