June  2019, 24(6): 2493-2510. doi: 10.3934/dcdsb.2018262

Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations

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

* Corresponding author: Peng Gao

Received  November 2017 Revised  May 2018 Published  June 2019 Early access  October 2018

Fund Project: The author is supported by NSFC Grant (11601073) and NSFC Grant (11701078).

In this paper, we first establish two global Carleman estimates for linear stochastic nonclassical diffusion equations. Based on these estimates, we obtain two types of Unique Continuation Property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations.

Citation: Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262
References:
[1]

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

[2]

V. BarbuA. Răscanu and G. Tessitore, Carleman estimate and controllability of linear stochastic heat equations, Appl. Math. Optim., 47 (2003), 97-120.  doi: 10.1007/s00245-002-0757-z.

[3]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[4]

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

[5]

T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independentes, Ark. Mat. Astr.Fys., 26 (1939), 1-9. 

[6]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[7]

X. Fu and X. Liu, A weighted identity for stochastic partial differential operators and its applications, Journal of Differential Equations, 262 (2017), 3551-3582.  doi: 10.1016/j.jde.2016.11.035.

[8]

X. Fu and X. Liu, Controllability and observability of some stochastic complex ginzburg-landau equations, SIAM Journal on Control and Optimization, 55 (2017), 1102-1127.  doi: 10.1137/15M1039961.

[9]

P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bulletin of the Australian Mathematical Society, 90 (2014), 283-294.  doi: 10.1017/S0004972714000276.

[10]

P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Analysis: Theory, Methods & Applications, 117 (2015), 133-147.  doi: 10.1016/j.na.2015.01.015.

[11]

P. GaoM. Chen and Y. Li, Observability Estimates and Null Controllability for Forward and Backward Linear Stochastic Kuramoto-Sivashinsky Equations, SIAM Journal on Control and Optimization, 53 (2015), 475-500.  doi: 10.1137/130943820.

[12]

P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Mathematics of Control, Signals, and Systems, 28 (2016), Art. 21, 22 pp. doi: 10.1007/s00498-016-0173-6.

[13]

P. Gao, Carleman estimates and unique continuation property for 1-D viscous Camassa-Holm equation, Discrete Contin. Dyn. Syst., 37 (2017), 169-188.  doi: 10.3934/dcds.2017007.

[14]

P. Gao, The stochastic Korteweg-de Vries equation on a bounded domain, Applied Mathematics and Computation, 310 (2017), 97-111.  doi: 10.1016/j.amc.2017.04.031.

[15]

P. Gao, Limiting dynamics for stochastic nonclassical diffusion equations, arXiv: 1703.02790.

[16]

P. Gao, Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications, Evol. Equ. Control Theory, 7 (2018), 465-499.  doi: 10.3934/eect.2018023.

[17]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol.I, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, 1972, translated from the French by P.Kenneth.

[18]

X. Liu, Global Carleman estimate for stochastic parabolic equations, and its application, ESAIM: Control, Optimisation and Calculus of Variations, 20 (2014), 823-839.  doi: 10.1051/cocv/2013085.

[19]

Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems, 28 (2012), 045008, 18 pp. doi: 10.1088/0266-5611/28/4/045008.

[20]

Q. Lü, Observability estimate for stochastic Schröinger equations and its applications, SIAM J. Control Optim., 51 (2013), 121-144.  doi: 10.1137/110830964.

[21]

Q. Lü, Exact controllability for stochastic transport equations, SIAM Journal on Control and Optimization, 52 (2014), 397-419.  doi: 10.1137/130910373.

[22]

S. Micu, On the Controllability of the Linearized Benjamin-Bona-Mahony Equation, SIAM Journal on Control and Optimization, 39 (2001), 1677-1696.  doi: 10.1137/S0363012999362499.

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

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

[25]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM Journal on Control and Optimization, 48 (2009), 2191-2216.  doi: 10.1137/050641508.

[26]

T. W. Ting, Certain non-steady flows of second-order fluids, Archive for Rational Mechanics and Analysis, 14 (1963), 1-26.  doi: 10.1007/BF00250690.

[27]

C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics, Encyclopedia of Physics, Springer, Berlin, 1995.

[28]

X. Zhang and E. Zuazua, A sharp observability inequality for Kirchhoffplate systems with potentials, Computational & Applied Mathematics, 25 (2006), 353-373.  doi: 10.1590/S0101-82052006000200013.

[29]

X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM Journal on Mathematical Analysis, 40 (2008), 851-868.  doi: 10.1137/070685786.

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]

V. BarbuA. Răscanu and G. Tessitore, Carleman estimate and controllability of linear stochastic heat equations, Appl. Math. Optim., 47 (2003), 97-120.  doi: 10.1007/s00245-002-0757-z.

[3]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[4]

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

[5]

T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independentes, Ark. Mat. Astr.Fys., 26 (1939), 1-9. 

[6]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[7]

X. Fu and X. Liu, A weighted identity for stochastic partial differential operators and its applications, Journal of Differential Equations, 262 (2017), 3551-3582.  doi: 10.1016/j.jde.2016.11.035.

[8]

X. Fu and X. Liu, Controllability and observability of some stochastic complex ginzburg-landau equations, SIAM Journal on Control and Optimization, 55 (2017), 1102-1127.  doi: 10.1137/15M1039961.

[9]

P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bulletin of the Australian Mathematical Society, 90 (2014), 283-294.  doi: 10.1017/S0004972714000276.

[10]

P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Analysis: Theory, Methods & Applications, 117 (2015), 133-147.  doi: 10.1016/j.na.2015.01.015.

[11]

P. GaoM. Chen and Y. Li, Observability Estimates and Null Controllability for Forward and Backward Linear Stochastic Kuramoto-Sivashinsky Equations, SIAM Journal on Control and Optimization, 53 (2015), 475-500.  doi: 10.1137/130943820.

[12]

P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Mathematics of Control, Signals, and Systems, 28 (2016), Art. 21, 22 pp. doi: 10.1007/s00498-016-0173-6.

[13]

P. Gao, Carleman estimates and unique continuation property for 1-D viscous Camassa-Holm equation, Discrete Contin. Dyn. Syst., 37 (2017), 169-188.  doi: 10.3934/dcds.2017007.

[14]

P. Gao, The stochastic Korteweg-de Vries equation on a bounded domain, Applied Mathematics and Computation, 310 (2017), 97-111.  doi: 10.1016/j.amc.2017.04.031.

[15]

P. Gao, Limiting dynamics for stochastic nonclassical diffusion equations, arXiv: 1703.02790.

[16]

P. Gao, Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications, Evol. Equ. Control Theory, 7 (2018), 465-499.  doi: 10.3934/eect.2018023.

[17]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol.I, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, 1972, translated from the French by P.Kenneth.

[18]

X. Liu, Global Carleman estimate for stochastic parabolic equations, and its application, ESAIM: Control, Optimisation and Calculus of Variations, 20 (2014), 823-839.  doi: 10.1051/cocv/2013085.

[19]

Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems, 28 (2012), 045008, 18 pp. doi: 10.1088/0266-5611/28/4/045008.

[20]

Q. Lü, Observability estimate for stochastic Schröinger equations and its applications, SIAM J. Control Optim., 51 (2013), 121-144.  doi: 10.1137/110830964.

[21]

Q. Lü, Exact controllability for stochastic transport equations, SIAM Journal on Control and Optimization, 52 (2014), 397-419.  doi: 10.1137/130910373.

[22]

S. Micu, On the Controllability of the Linearized Benjamin-Bona-Mahony Equation, SIAM Journal on Control and Optimization, 39 (2001), 1677-1696.  doi: 10.1137/S0363012999362499.

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

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

[25]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM Journal on Control and Optimization, 48 (2009), 2191-2216.  doi: 10.1137/050641508.

[26]

T. W. Ting, Certain non-steady flows of second-order fluids, Archive for Rational Mechanics and Analysis, 14 (1963), 1-26.  doi: 10.1007/BF00250690.

[27]

C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics, Encyclopedia of Physics, Springer, Berlin, 1995.

[28]

X. Zhang and E. Zuazua, A sharp observability inequality for Kirchhoffplate systems with potentials, Computational & Applied Mathematics, 25 (2006), 353-373.  doi: 10.1590/S0101-82052006000200013.

[29]

X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM Journal on Mathematical Analysis, 40 (2008), 851-868.  doi: 10.1137/070685786.

[1]

Qiangheng Zhang. Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021293

[2]

Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851

[3]

Yangrong Li, Renhai Wang, Jinyan Yin. Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2569-2586. doi: 10.3934/dcdsb.2017092

[4]

Zhongqi Yin. A quantitative internal unique continuation for stochastic parabolic equations. Mathematical Control and Related Fields, 2015, 5 (1) : 165-176. doi: 10.3934/mcrf.2015.5.165

[5]

Anne-Sophie de Suzzoni. Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2905-2920. doi: 10.3934/dcds.2015.35.2905

[6]

Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824

[7]

Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583

[8]

Peng Gao. Limiting dynamics for stochastic nonclassical diffusion equations. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021288

[9]

C. H. Arthur Cheng, John M. Hong, Ying-Chieh Lin, Jiahong Wu, Juan-Ming Yuan. Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 763-779. doi: 10.3934/dcdsb.2016.21.763

[10]

Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171

[11]

Wenxia Chen, Ping Yang, Weiwei Gao, Lixin Tian. The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium. Communications on Pure and Applied Analysis, 2018, 17 (3) : 823-848. doi: 10.3934/cpaa.2018042

[12]

Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623

[13]

Peng Gao. Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 169-188. doi: 10.3934/dcds.2017007

[14]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515

[15]

Jianbo Yuan, Shixuan Zhang, Yongqin Xie, Jiangwei Zhang. Attractors for a class of perturbed nonclassical diffusion equations with memory. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021261

[16]

Muriel Boulakia. Quantification of the unique continuation property for the nonstationary Stokes problem. Mathematical Control and Related Fields, 2016, 6 (1) : 27-52. doi: 10.3934/mcrf.2016.6.27

[17]

Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control and Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012

[18]

Gunther Uhlmann, Jenn-Nan Wang. Unique continuation property for the elasticity with general residual stress. Inverse Problems and Imaging, 2009, 3 (2) : 309-317. doi: 10.3934/ipi.2009.3.309

[19]

Hui Yin, Huijiang Zhao. Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space. Kinetic and Related Models, 2009, 2 (3) : 521-550. doi: 10.3934/krm.2009.2.521

[20]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (267)
  • HTML views (503)
  • Cited by (0)

Other articles
by authors

[Back to Top]