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On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations
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 |
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.
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.
|
show all references
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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