doi: 10.3934/dcdsb.2022086
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Smoluchowski–Kramers approximation with state dependent damping and highly random oscillation

1. 

School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, China

2. 

Department of Mathematics, Nanjing University, Nanjing, China

*Corresponding author: Wei Wang

Received  December 2021 Revised  March 2022 Early access May 2022

Fund Project: The first author is supported by NSFC grant No. 11671204; The second author is supported by NSFC grant No. 11771207

The small mass limit (Smoluchowski–Kramers approximation) of class systems of ordinary differential equations describing motions of small mass particle with state dependent friction and high oscillation is derived by a diffusion approximation approach. In the small mass limit, due to the state dependent damping, one additional term appears in the limit equation, which leads to a stochastic differential equation (SDE) as the highly random oscillation appears as a multiplicative white noise.

Citation: Yan Lv, Wei Wang. Smoluchowski–Kramers approximation with state dependent damping and highly random oscillation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022086
References:
[1]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[2]

S. Cerrai and M. I. Freidlin, On the Smoluchowski–Kramers approximation for a system with an infinite number of degrees of freedom, Probab. Theory Related Fields, 135 (2006), 363-394.  doi: 10.1007/s00440-005-0465-0.

[3]

J. Q. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Insights. Elsevier, Amsterdam, 2014.

[4]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.

[5]

M. I. Freidlin, Some remarks on Smoluchowski–Kramers approximation, J. Stat. Phys., 117 (2004), 617-634.  doi: 10.1007/s10955-004-2273-9.

[6]

M. I. Freidlin and W. Q. Hu, Smoluchowski–Kramers approximation in the case of variable friction, J. Math. Sci., 179 (2011), 184-207.  doi: 10.1007/s10958-011-0589-y.

[7]

S. HottovyA. McDanielJ. Wehr and G. Volpe, The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction, Comm. Math. Phys., 336 (2015), 1259-1283.  doi: 10.1007/s00220-014-2233-4.

[8]

Y. Lv and W. Wang, Diffusion approximation for nonlinear evolutionary equations with large interaction and fast boundary fluctuation, J. Diff. Equa., 266 (2019), 3310-3327.  doi: 10.1016/j.jde.2018.09.001.

[9]

Y. Lv, W. Wang and A. J. Roberts, Approximation of the random inertial manifold of singularly perturbed stochastic wave equations, Stoch. Dyna., 14 (2014), 1350018, 21 pp. doi: 10.1142/S0219493713500184.

[10]

G. A. Pavliotis and A. Stuart, White noise limits for inertial particles in a random field, Multiscale Model. Simul., 1 (2003), 527-533.  doi: 10.1137/S1540345903421076.

[11]

G. A. Pavliotis and A. Stuart, Analysis of white noise limits for stochastic systems with two fast relaxation times, Multiscale Model. Simul., 4 (2005), 1-35.  doi: 10.1137/040610507.

[12]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[13]

W. Wang and A. J. Roberts, Diffusion approximation for self-similarity of stochastic advection in Burgers' equation, Comm. Math. Phys., 333 (2015), 1287-1316.  doi: 10.1007/s00220-014-2117-7.

show all references

References:
[1]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[2]

S. Cerrai and M. I. Freidlin, On the Smoluchowski–Kramers approximation for a system with an infinite number of degrees of freedom, Probab. Theory Related Fields, 135 (2006), 363-394.  doi: 10.1007/s00440-005-0465-0.

[3]

J. Q. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Insights. Elsevier, Amsterdam, 2014.

[4]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.

[5]

M. I. Freidlin, Some remarks on Smoluchowski–Kramers approximation, J. Stat. Phys., 117 (2004), 617-634.  doi: 10.1007/s10955-004-2273-9.

[6]

M. I. Freidlin and W. Q. Hu, Smoluchowski–Kramers approximation in the case of variable friction, J. Math. Sci., 179 (2011), 184-207.  doi: 10.1007/s10958-011-0589-y.

[7]

S. HottovyA. McDanielJ. Wehr and G. Volpe, The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction, Comm. Math. Phys., 336 (2015), 1259-1283.  doi: 10.1007/s00220-014-2233-4.

[8]

Y. Lv and W. Wang, Diffusion approximation for nonlinear evolutionary equations with large interaction and fast boundary fluctuation, J. Diff. Equa., 266 (2019), 3310-3327.  doi: 10.1016/j.jde.2018.09.001.

[9]

Y. Lv, W. Wang and A. J. Roberts, Approximation of the random inertial manifold of singularly perturbed stochastic wave equations, Stoch. Dyna., 14 (2014), 1350018, 21 pp. doi: 10.1142/S0219493713500184.

[10]

G. A. Pavliotis and A. Stuart, White noise limits for inertial particles in a random field, Multiscale Model. Simul., 1 (2003), 527-533.  doi: 10.1137/S1540345903421076.

[11]

G. A. Pavliotis and A. Stuart, Analysis of white noise limits for stochastic systems with two fast relaxation times, Multiscale Model. Simul., 4 (2005), 1-35.  doi: 10.1137/040610507.

[12]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[13]

W. Wang and A. J. Roberts, Diffusion approximation for self-similarity of stochastic advection in Burgers' equation, Comm. Math. Phys., 333 (2015), 1287-1316.  doi: 10.1007/s00220-014-2117-7.

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