doi: 10.3934/dcdsb.2021288
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Limiting dynamics for stochastic nonclassical diffusion equations

School of Mathematics and Statistics, and Center for Mathematics, and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

Received  June 2021 Revised  October 2021 Early access December 2021

Fund Project: Peng Gao is supported by the Fundamental Research Funds for the Central Universities (2412020FZ022)

In this paper, we are concerned with the dynamical behavior of the stochastic nonclassical parabolic equation, more precisely, it is shown that the inviscid limits of the stochastic nonclassical diffusion equations reduces to the stochastic heat equations. The key points in the proof of our convergence results are establishing some uniform estimates and the regularity theory for the solutions of the stochastic nonclassical diffusion equations which are independent of the parameter. Based on the uniform estimates, the tightness of distributions of the solutions can be obtained.

Citation: Peng Gao. Limiting dynamics for stochastic nonclassical diffusion equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021288
References:
[1]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mechanica, 37 (1980), 265-296.  doi: 10.1007/BF01202949.

[2]

C. T. Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, Nonlinear Anal., 73 (2010), 399-412.  doi: 10.1016/j.na.2010.03.031.

[3]

L. Bai and F. Zhang, Existence of random attractors for 2D-stochastic nonclassical diffusion equations on unbounded domains, Results Math., 69 (2016), 129-160.  doi: 10.1007/s00025-015-0505-8.

[4]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[6]

S. Cerrai and M. 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.

[7]

S. Cerrai and M. Freidlin, Smoluchowski-Kramers approximation for a general class of SPDEs, J. Evol. Equ., 6 (2006), 657-689.  doi: 10.1007/s00028-006-0281-8.

[8]

S. Cerrai and M. Salins, On the Smoluchowski-Kramers approximation for a system with infinite degrees of freedom exposed to a magnetic field, Stochastic Process. Appl., 127 (2017), 273-303.  doi: 10.1016/j.spa.2016.06.008.

[9]

S. Cerrai and M. Salins, Smoluchowski-Kramers approximation and large deviations for infinite-dimensional nongradient systems with applications to the exit problem, Ann. Probab., 44 (2016), 2591-2642.  doi: 10.1214/15-AOP1029.

[10]

S. Cerrai and M. Salins, Smoluchowski-Kramers approximation and large deviations for infinite dimensional gradient systems, Asymptot. Anal., 88 (2014), 201-215.  doi: 10.3233/ASY-141220.

[11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, 2$^{nd}$ edition, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.
[12]

A. DebusscheN. 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.

[13]

G. DeugouéP. A. Razafimandimby and M. Sango, On the 3-D stochastic magnetohydrodynamic-$\alpha$ model, Stochastic Process. Appl., 122 (2012), 2211-2248.  doi: 10.1016/j.spa.2012.03.002.

[14]

G. Deugoue and M. Sango, Weak solutions to stochastic 3D Navier-Stokes-$\alpha$ model of turbulence: $\alpha$-asymptotic behavior, J. Math. Anal. Appl., 384 (2011), 49-62.  doi: 10.1016/j.jmaa.2010.10.048.

[15]

P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bull. Aust. Math. Soc., 90 (2014), 283-294.  doi: 10.1017/S0004972714000276.

[16]

P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Math. Control Signals Systems, 28 (2016), 1-22.  doi: 10.1007/s00498-016-0173-6.

[17]

I. Gyöngy and N. Krylov, Existence of strong solutions for Itö's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833.

[18]

J. U. Kim, Approximate controllability of a stochastic wave equation, Appl. Math. Optim., 49 (2004), 81-98.  doi: 10.1007/s00245-003-0781-7.

[19]

J. U. Kim, Periodic and invariant measures for stochastic wave equations, Electron. J. Differential Equations, (2004), 1–30.

[20]

Y. Lv and A. J. Roberts, Averaging approximation to singularly perturbed nonlinear stochastic wave equations, J. Math. Phys., 53 (2012), 062702.  doi: 10.1063/1.4726175.

[21]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.

[22]

J. C. Peter and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627. 

[23]

D. Pham and P. Nguyen, Stochastic systems of diffusion equations with polynomial reaction terms, Asymptotic Anal., 99 (2016), 125-161.  doi: 10.3233/ASY-161378.

[24]

P. A. Razafimandimby and M. Sango, Weak solutions of a stochastic model for two-dimensional second grade fluids, Bound. Value Probl., 2010 (2010), 1-47.  doi: 10.1155/2010/636140.

[25]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer Science & Business Media, 2006.

[26]

M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Physica D: Nonlinear Phenomena, 239 (2010), 912-923.  doi: 10.1016/j.physd.2010.01.009.

[27]

M. Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations, Forum Math., 25 (2013), 931-965.  doi: 10.1515/form.2011.138.

[28]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, , Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[30]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690.

[31]

C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics, Encyclopedia of Physics, , Springer, Berlin, 1955.

[32]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[33]

W. Wang and Y. Lv, Limit behavior of nonlinear stochastic wave equations with singular perturbation, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 175-193.  doi: 10.3934/dcdsb.2010.13.175.

[34]

E. Waymire and J. Duan, Probability and Partial Differential Equations in Modern Applied Mathematics, , Springer-Verlag, New York, 2005. doi: 10.1007/978-0-387-29371-4.

[35]

F. H. Zhang and W. Han, Pullback attractors for nonclassical diffusion delay equations on unbounded domains with non-autonomous deterministic and stochastic forcing terms, Electron. J. Differential Equations, 2016 (2016), Paper No. 139, 28 pp.

[36]

W. Zhao and S. Song, Dynamics of stochastic nonclassical diffusion equations on unbounded domains, Electron. J. Differential Equations, 282 (2015), 1-22. 

show all references

References:
[1]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mechanica, 37 (1980), 265-296.  doi: 10.1007/BF01202949.

[2]

C. T. Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, Nonlinear Anal., 73 (2010), 399-412.  doi: 10.1016/j.na.2010.03.031.

[3]

L. Bai and F. Zhang, Existence of random attractors for 2D-stochastic nonclassical diffusion equations on unbounded domains, Results Math., 69 (2016), 129-160.  doi: 10.1007/s00025-015-0505-8.

[4]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[6]

S. Cerrai and M. 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.

[7]

S. Cerrai and M. Freidlin, Smoluchowski-Kramers approximation for a general class of SPDEs, J. Evol. Equ., 6 (2006), 657-689.  doi: 10.1007/s00028-006-0281-8.

[8]

S. Cerrai and M. Salins, On the Smoluchowski-Kramers approximation for a system with infinite degrees of freedom exposed to a magnetic field, Stochastic Process. Appl., 127 (2017), 273-303.  doi: 10.1016/j.spa.2016.06.008.

[9]

S. Cerrai and M. Salins, Smoluchowski-Kramers approximation and large deviations for infinite-dimensional nongradient systems with applications to the exit problem, Ann. Probab., 44 (2016), 2591-2642.  doi: 10.1214/15-AOP1029.

[10]

S. Cerrai and M. Salins, Smoluchowski-Kramers approximation and large deviations for infinite dimensional gradient systems, Asymptot. Anal., 88 (2014), 201-215.  doi: 10.3233/ASY-141220.

[11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, 2$^{nd}$ edition, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.
[12]

A. DebusscheN. 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.

[13]

G. DeugouéP. A. Razafimandimby and M. Sango, On the 3-D stochastic magnetohydrodynamic-$\alpha$ model, Stochastic Process. Appl., 122 (2012), 2211-2248.  doi: 10.1016/j.spa.2012.03.002.

[14]

G. Deugoue and M. Sango, Weak solutions to stochastic 3D Navier-Stokes-$\alpha$ model of turbulence: $\alpha$-asymptotic behavior, J. Math. Anal. Appl., 384 (2011), 49-62.  doi: 10.1016/j.jmaa.2010.10.048.

[15]

P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bull. Aust. Math. Soc., 90 (2014), 283-294.  doi: 10.1017/S0004972714000276.

[16]

P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Math. Control Signals Systems, 28 (2016), 1-22.  doi: 10.1007/s00498-016-0173-6.

[17]

I. Gyöngy and N. Krylov, Existence of strong solutions for Itö's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833.

[18]

J. U. Kim, Approximate controllability of a stochastic wave equation, Appl. Math. Optim., 49 (2004), 81-98.  doi: 10.1007/s00245-003-0781-7.

[19]

J. U. Kim, Periodic and invariant measures for stochastic wave equations, Electron. J. Differential Equations, (2004), 1–30.

[20]

Y. Lv and A. J. Roberts, Averaging approximation to singularly perturbed nonlinear stochastic wave equations, J. Math. Phys., 53 (2012), 062702.  doi: 10.1063/1.4726175.

[21]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.

[22]

J. C. Peter and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627. 

[23]

D. Pham and P. Nguyen, Stochastic systems of diffusion equations with polynomial reaction terms, Asymptotic Anal., 99 (2016), 125-161.  doi: 10.3233/ASY-161378.

[24]

P. A. Razafimandimby and M. Sango, Weak solutions of a stochastic model for two-dimensional second grade fluids, Bound. Value Probl., 2010 (2010), 1-47.  doi: 10.1155/2010/636140.

[25]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer Science & Business Media, 2006.

[26]

M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Physica D: Nonlinear Phenomena, 239 (2010), 912-923.  doi: 10.1016/j.physd.2010.01.009.

[27]

M. Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations, Forum Math., 25 (2013), 931-965.  doi: 10.1515/form.2011.138.

[28]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, , Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[30]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690.

[31]

C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics, Encyclopedia of Physics, , Springer, Berlin, 1955.

[32]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[33]

W. Wang and Y. Lv, Limit behavior of nonlinear stochastic wave equations with singular perturbation, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 175-193.  doi: 10.3934/dcdsb.2010.13.175.

[34]

E. Waymire and J. Duan, Probability and Partial Differential Equations in Modern Applied Mathematics, , Springer-Verlag, New York, 2005. doi: 10.1007/978-0-387-29371-4.

[35]

F. H. Zhang and W. Han, Pullback attractors for nonclassical diffusion delay equations on unbounded domains with non-autonomous deterministic and stochastic forcing terms, Electron. J. Differential Equations, 2016 (2016), Paper No. 139, 28 pp.

[36]

W. Zhao and S. Song, Dynamics of stochastic nonclassical diffusion equations on unbounded domains, Electron. J. Differential Equations, 282 (2015), 1-22. 

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