# American Institute of Mathematical Sciences

January  2017, 37(1): 169-188. doi: 10.3934/dcds.2017007

## Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation

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

* Corresponding author: Peng Gao

Received  March 2016 Revised  September 2016 Published  November 2016

This paper is devoted to studying the 1-D viscous Camassa-Holm equation on a bounded interval. We first deduce the existence and uniqueness of strong solution to the viscous Camassa-Holm equation by using Galerkin method. Then we establish an identity for a second order parabolic operator, by applying this identity we obtain two global Carleman estimates for the linear viscous Camassa-Holm operator. Based on these estimates, we obtain two types of Unique Continuation Property for the viscous Camassa-Holm equation.

Citation: 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
##### References:
 [1] L. Baudouin and J. P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), 1537-1554.  doi: 10.1088/0266-5611/18/6/307. [2] M. Bellassoued, Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation, Journal de mathematiques pures et appliquees, 91 (2009), 233-255.  doi: 10.1016/j.matpur.2008.06.002. [3] 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. [4] T. Carleman, Sur un probléme d'unicité pour les systémes d'équations aux derivées partielles á deux variables independentes, Ark. Mat. Astr.Fys., 2B (1939), 1-9. [5] E. Cerpa, A. Mercado and A. F. Pazoto, On the boundary control of a parabolic system coupling KS-KdV and Heat equations, Sci. Ser. A Math. Sci., 22 (2012), 55-74. [6] M. Chen and P. Gao, A New Unique Continuation Property for the Korteweg de-Vries Equation, Bull. Aust. Math. Soc., 90 (2014), 90-98.  doi: 10.1017/S000497271300110X. [7] P. N. da Silva, Unique Continuation for the Kawahara Equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473.  doi: 10.5540/tema.2007.08.03.0463. [8] M. Eller, I. Lasiecka and R. Triggiani, Unique continuation for over-determined Kirchoff plate equations and related thermoelastic systems, J. Inverse Ill-Posed Probl., 9 (2001), 103-148.  doi: 10.1515/jiip.2001.9.2.103. [9] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. [10] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence in: Advances in Nonlinear Mathematics and Science, Phys. D, 152/153 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9. [11] Y. Fu and B. Guo, Time periodic solution of the viscous Camassa-Holm equation, J. Math. Anal. Appl., 313 (2006), 311-321.  doi: 10.1016/j.jmaa.2005.08.073. [12] P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bull. Austral. Math. Soc., 90 (2014), 283-294.  doi: 10.1017/S0004972714000276. [13] P. Gao, Insensitizing controls for the Cahn-Hilliard type equation, Electron. J. Qual. Theory Differ. Equ., 35 (2014), 1-22. [14] 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. [15] P. Gao, M. Chen and Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equations, SIAM J. Control Optim., 53 (2015), 475-500.  doi: 10.1137/130943820. [16] P. Gao, A new global Carleman estimate for Cahn-Hilliard type equation and its applications, J. Differential Equations, 260 (2016), 427-444.  doi: 10.1016/j.jde.2015.08.053. [17] P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Analysis: Theory, Methods & Applications, 139 (2016), 169-195.  doi: 10.1016/j.na.2016.02.023. [18] P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Mathematics of Control, Signals, and Systems, 28 (2016), 1-22.  doi: 10.1007/s00498-016-0173-6. [19] O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, Journal of Differential Equations, 245 (2008), 1584-1615.  doi: 10.1016/j.jde.2008.06.016. [20] O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal., 60 (2008), 61-100. [21] O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 26 (2009), 2181-2209.  doi: 10.1016/j.anihpc.2009.01.010. [22] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721. [23] D. D. Holm and E. S. Titi, Computational models of turbulence: The lans-$α$ model and the role of global analysis, SIAM News, 38 (2005), 1-5. [24] J. U. Kim, On the Stochastic Wave Equation with Nonlinear Damping, Appl. Math. Optim., 58 (2008), 29-67.  doi: 10.1007/s00245-007-9029-2. [25] K. H. Kwek, H. Gao, W. Zhang and C. Qu, An initial boundary value problem of Camassa-Holm equation, J. Math. Phys., 41 (2000), 8279-8285.  doi: 10.1063/1.1288498. [26] N. A. Larkin, Modified KdV equation with a source term in a bounded domain, Math. Meth. Appl. Sci., 29 (2006), 751-765.  doi: 10.1002/mma.704. [27] I. Lasiecka, R. Triggiani and P. F. Yao, Carleman estimates for a plate equation on a Riemann manifold with energy level terms, Analysis and Applications, 10 (2003), 199-236.  doi: 10.1007/978-1-4757-3741-7_15. [28] W. K. Lim, Global well-posedness for the viscous Camassa-Holm equation, J. Math. Anal. Appl., 326 (2007), 432-442.  doi: 10.1016/j.jmaa.2006.01.095. [29] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, NewYork-Heidelberg, 1972. [30] S. Liu and R. Triggiani, Global uniqueness in determining electric potentials for a system of strongly coupled Schrödinger equations with magnetic potential terms, J. Inverse Ill-Posed Probl., 19 (2011), 223-254.  doi: 10.1515/JIIP.2011.030. [31] J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$α$) equations on bounded domains, Topological Methods in the Physical Sciences, London, 2000, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449-1468.  doi: 10.1098/rsta.2001.0852. [32] A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Problems, 24 (2008), 015017, 18pp.  doi: 10.1088/0266-5611/24/1/015017. [33] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edn, Texts in Applied Mathematics, Vol. 13, Springer-Verlag, New York, 2004. [34] L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956.  doi: 10.1137/050631409. [35] L. Rosier and B.-Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, Journal of Differential Equations, 254 (2013), 141-178.  doi: 10.1016/j.jde.2012.08.014. [36] J. Simon, Compact sets in the space $L^{p}(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [37] L. Tian, C. Shen and D. Ding, Optimal control of the viscous Camassa-Holm equation, Nonlinear Analysis: Real World Applications, 10 (2009), 519-530.  doi: 10.1016/j.nonrwa.2007.10.016. [38] R. Triggiani and P. F. Yao, Inverse/observability estimates for Schrödinger equations with variable coefficients, Control and Cybernetics, 28 (1999), 627-664. [39] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp.  doi: 10.1088/0266-5611/25/12/123013. [40] G. Yuan and M. Yamamoto, Lipschitz stability in inverse problems for a Kirchhoff plate equation, Asymptotic Analysis, 53 (2007), 29-60. [41] X. Zhang, Exact controllability of semilinear plate equations, Asymptotic Analysis, 27 (2001), 95-125. [42] X. Zhang and E. Zuazua, A sharp observability inequality for Kirchhoff plate systems with potentials, Comput. Appl. Math., 25 (2006), 353-373.  doi: 10.1590/S0101-82052006000200013. [43] Z. C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math., 16 (2012), 1991-2017.

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##### References:
 [1] L. Baudouin and J. P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), 1537-1554.  doi: 10.1088/0266-5611/18/6/307. [2] M. Bellassoued, Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation, Journal de mathematiques pures et appliquees, 91 (2009), 233-255.  doi: 10.1016/j.matpur.2008.06.002. [3] 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. [4] T. Carleman, Sur un probléme d'unicité pour les systémes d'équations aux derivées partielles á deux variables independentes, Ark. Mat. Astr.Fys., 2B (1939), 1-9. [5] E. Cerpa, A. Mercado and A. F. Pazoto, On the boundary control of a parabolic system coupling KS-KdV and Heat equations, Sci. Ser. A Math. Sci., 22 (2012), 55-74. [6] M. Chen and P. Gao, A New Unique Continuation Property for the Korteweg de-Vries Equation, Bull. Aust. Math. Soc., 90 (2014), 90-98.  doi: 10.1017/S000497271300110X. [7] P. N. da Silva, Unique Continuation for the Kawahara Equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473.  doi: 10.5540/tema.2007.08.03.0463. [8] M. Eller, I. Lasiecka and R. Triggiani, Unique continuation for over-determined Kirchoff plate equations and related thermoelastic systems, J. Inverse Ill-Posed Probl., 9 (2001), 103-148.  doi: 10.1515/jiip.2001.9.2.103. [9] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. [10] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence in: Advances in Nonlinear Mathematics and Science, Phys. D, 152/153 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9. [11] Y. Fu and B. Guo, Time periodic solution of the viscous Camassa-Holm equation, J. Math. Anal. Appl., 313 (2006), 311-321.  doi: 10.1016/j.jmaa.2005.08.073. [12] P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bull. Austral. Math. Soc., 90 (2014), 283-294.  doi: 10.1017/S0004972714000276. [13] P. Gao, Insensitizing controls for the Cahn-Hilliard type equation, Electron. J. Qual. Theory Differ. Equ., 35 (2014), 1-22. [14] 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. [15] P. Gao, M. Chen and Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equations, SIAM J. Control Optim., 53 (2015), 475-500.  doi: 10.1137/130943820. [16] P. Gao, A new global Carleman estimate for Cahn-Hilliard type equation and its applications, J. Differential Equations, 260 (2016), 427-444.  doi: 10.1016/j.jde.2015.08.053. [17] P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Analysis: Theory, Methods & Applications, 139 (2016), 169-195.  doi: 10.1016/j.na.2016.02.023. [18] P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Mathematics of Control, Signals, and Systems, 28 (2016), 1-22.  doi: 10.1007/s00498-016-0173-6. [19] O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, Journal of Differential Equations, 245 (2008), 1584-1615.  doi: 10.1016/j.jde.2008.06.016. [20] O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal., 60 (2008), 61-100. [21] O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 26 (2009), 2181-2209.  doi: 10.1016/j.anihpc.2009.01.010. [22] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721. [23] D. D. Holm and E. S. Titi, Computational models of turbulence: The lans-$α$ model and the role of global analysis, SIAM News, 38 (2005), 1-5. [24] J. U. Kim, On the Stochastic Wave Equation with Nonlinear Damping, Appl. Math. Optim., 58 (2008), 29-67.  doi: 10.1007/s00245-007-9029-2. [25] K. H. Kwek, H. Gao, W. Zhang and C. Qu, An initial boundary value problem of Camassa-Holm equation, J. Math. Phys., 41 (2000), 8279-8285.  doi: 10.1063/1.1288498. [26] N. A. Larkin, Modified KdV equation with a source term in a bounded domain, Math. Meth. Appl. Sci., 29 (2006), 751-765.  doi: 10.1002/mma.704. [27] I. Lasiecka, R. Triggiani and P. F. Yao, Carleman estimates for a plate equation on a Riemann manifold with energy level terms, Analysis and Applications, 10 (2003), 199-236.  doi: 10.1007/978-1-4757-3741-7_15. [28] W. K. Lim, Global well-posedness for the viscous Camassa-Holm equation, J. Math. Anal. Appl., 326 (2007), 432-442.  doi: 10.1016/j.jmaa.2006.01.095. [29] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, NewYork-Heidelberg, 1972. [30] S. Liu and R. Triggiani, Global uniqueness in determining electric potentials for a system of strongly coupled Schrödinger equations with magnetic potential terms, J. Inverse Ill-Posed Probl., 19 (2011), 223-254.  doi: 10.1515/JIIP.2011.030. [31] J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$α$) equations on bounded domains, Topological Methods in the Physical Sciences, London, 2000, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449-1468.  doi: 10.1098/rsta.2001.0852. [32] A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Problems, 24 (2008), 015017, 18pp.  doi: 10.1088/0266-5611/24/1/015017. [33] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edn, Texts in Applied Mathematics, Vol. 13, Springer-Verlag, New York, 2004. [34] L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956.  doi: 10.1137/050631409. [35] L. Rosier and B.-Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, Journal of Differential Equations, 254 (2013), 141-178.  doi: 10.1016/j.jde.2012.08.014. [36] J. Simon, Compact sets in the space $L^{p}(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [37] L. Tian, C. Shen and D. Ding, Optimal control of the viscous Camassa-Holm equation, Nonlinear Analysis: Real World Applications, 10 (2009), 519-530.  doi: 10.1016/j.nonrwa.2007.10.016. [38] R. Triggiani and P. F. Yao, Inverse/observability estimates for Schrödinger equations with variable coefficients, Control and Cybernetics, 28 (1999), 627-664. [39] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp.  doi: 10.1088/0266-5611/25/12/123013. [40] G. Yuan and M. Yamamoto, Lipschitz stability in inverse problems for a Kirchhoff plate equation, Asymptotic Analysis, 53 (2007), 29-60. [41] X. Zhang, Exact controllability of semilinear plate equations, Asymptotic Analysis, 27 (2001), 95-125. [42] X. Zhang and E. Zuazua, A sharp observability inequality for Kirchhoff plate systems with potentials, Comput. Appl. Math., 25 (2006), 353-373.  doi: 10.1590/S0101-82052006000200013. [43] Z. C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math., 16 (2012), 1991-2017.
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