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  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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[23]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[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. Google Scholar

[29]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[23]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[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. Google Scholar

[29]

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

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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 & Continuous Dynamical Systems - B, 2016, 21 (3) : 763-779. doi: 10.3934/dcdsb.2016.21.763

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051

[17]

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

[18]

Peng Gao. Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evolution Equations & Control Theory, 2018, 7 (3) : 465-499. doi: 10.3934/eect.2018023

[19]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43

[20]

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

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (75)
  • HTML views (426)
  • Cited by (0)

Other articles
by authors

[Back to Top]