American Institute of Mathematical Sciences

September  2018, 17(5): 1853-1874. doi: 10.3934/cpaa.2018088

Global Carleman estimate for the Kawahara equation and its applications

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

* Corresponding author

Received  July 2017 Revised  November 2017 Published  April 2018

Fund Project: This work is supported by NSFC Grant (11601073), NSFC Grant (11701078), China Postdoctoral Science Foundation (2017M611292), the Fundamental Research Funds for the Central Universities (2412017QD002), and NSFC Grant (11626043).

In this paper, we establish a global Carleman estimate for the Kawahara equation. Based on this estimate, we obtain the Unique Continuation Property (UCP) for this equation and the global exponential stability for the Kawahara equation with a very weak localized dissipation.

Citation: Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088
References:
 [1] T. B. Benjamin, Lectures on nonlinear wave motion, Lecture notes in applied mathematics, 15 (1974), 3-47. [2] D. J. Benney, Long waves on liquid films, J. Math. and Phys., 45 (1966), 150-155. [3] H. A. Biagioni and F. Linares, On the Benney-Lin and Kawahara equations, J. Math. Anal. Appl., 211 (1997), 131-152. [4] J. P. Boyd, Weakly non-local solitons for capillary-gravity waves: fifth degree Korteweg-de Vries equation, Phys. D, 48 (1991), 129-146. [5] T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independentes, Ark. Mat. Astr.Fys., 2B (1939), 1-9. [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. [7] S. B. Cui, D. G. Deng and S. P. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with $L^{2}$ initial data, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1457-1466. [8] P. N. da Silva, Unique continuation for the Kawahara equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473. [9] L. Dawson, Uniqueness properties of higher order dispersive equations, J. Differential Equations, 236 (2007), 199-236. [10] G. Doronin and N. Larkin, Kawahara equation in a bounded domain, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 783-799. [11] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), 1578-1614. [12] P. Gao, Insensitizing controls for the Cahn-Hilliard type equation, Electron. J. Qual. Theory Differ. Equ, 35 (2014), 1-22. [13] 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. [14] 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. [15] O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2181-2209. [16] T. Iguchi, A long wave approximation for capillary-gravity waves and the Kawahara equations, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 179-220. [17] O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Lecture Notes in Pure and Appl. Math., 218 (2001), 113-137. [18] O. Y. Imanuvilov, On Carleman estimates for hyperbolic equations, Asymptot. Anal., 32 (2002), 185C-220. [19] T. Kakutani, Axially symmetric stagnation-point flow of an electrically conducting fluid under transverse magnetic field, J. Phys. Soc. Japan, 15 (1960), 688-695. [20] C. P. Massarolo, G. P. Menzala and A. F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping, Math. Methods Appl. Sci., 12 (2007), 1419-1435. [21] P. G. Meléndez, Lipschitz stability in an inverse problem for the mian coefficient of a Kuramoto-Sivashinsky type equation, J. Math. Anal. Appl., 408 (2013), 275-290. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. [23] Y. Pomeau, A. Ramani and B. Grammaticos, Structural stability of the Korteweg-de Vries solitons under a singular perturbation, Phys. D, 31 (1988), 127-134. [24] 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. [25] J.C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. [26] J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous fluid, J. Phys. Soc. Japan, 44 (1978), 663-666. [27] C. F. Vasconcellos and P. N. da Silva, Stabilization of the linear Kawahara equation with localized damping, Asymptot. Anal., 58 (2008), 229-252. [28] C. F. Vasconcellos and P. N. da Silva, Stabilization of the Kawahara equation with localized damping, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 102-116. [29] B. Y. Zhang and X. Q. Zhao, Control and stabilization of the Kawahara equation on a periodic domain, Communications in Information and Systems, 12 (2012), 77-96.

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References:
 [1] T. B. Benjamin, Lectures on nonlinear wave motion, Lecture notes in applied mathematics, 15 (1974), 3-47. [2] D. J. Benney, Long waves on liquid films, J. Math. and Phys., 45 (1966), 150-155. [3] H. A. Biagioni and F. Linares, On the Benney-Lin and Kawahara equations, J. Math. Anal. Appl., 211 (1997), 131-152. [4] J. P. Boyd, Weakly non-local solitons for capillary-gravity waves: fifth degree Korteweg-de Vries equation, Phys. D, 48 (1991), 129-146. [5] T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independentes, Ark. Mat. Astr.Fys., 2B (1939), 1-9. [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. [7] S. B. Cui, D. G. Deng and S. P. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with $L^{2}$ initial data, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1457-1466. [8] P. N. da Silva, Unique continuation for the Kawahara equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473. [9] L. Dawson, Uniqueness properties of higher order dispersive equations, J. Differential Equations, 236 (2007), 199-236. [10] G. Doronin and N. Larkin, Kawahara equation in a bounded domain, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 783-799. [11] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), 1578-1614. [12] P. Gao, Insensitizing controls for the Cahn-Hilliard type equation, Electron. J. Qual. Theory Differ. Equ, 35 (2014), 1-22. [13] 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. [14] 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. [15] O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2181-2209. [16] T. Iguchi, A long wave approximation for capillary-gravity waves and the Kawahara equations, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 179-220. [17] O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Lecture Notes in Pure and Appl. Math., 218 (2001), 113-137. [18] O. Y. Imanuvilov, On Carleman estimates for hyperbolic equations, Asymptot. Anal., 32 (2002), 185C-220. [19] T. Kakutani, Axially symmetric stagnation-point flow of an electrically conducting fluid under transverse magnetic field, J. Phys. Soc. Japan, 15 (1960), 688-695. [20] C. P. Massarolo, G. P. Menzala and A. F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping, Math. Methods Appl. Sci., 12 (2007), 1419-1435. [21] P. G. Meléndez, Lipschitz stability in an inverse problem for the mian coefficient of a Kuramoto-Sivashinsky type equation, J. Math. Anal. Appl., 408 (2013), 275-290. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. [23] Y. Pomeau, A. Ramani and B. Grammaticos, Structural stability of the Korteweg-de Vries solitons under a singular perturbation, Phys. D, 31 (1988), 127-134. [24] 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. [25] J.C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. [26] J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous fluid, J. Phys. Soc. Japan, 44 (1978), 663-666. [27] C. F. Vasconcellos and P. N. da Silva, Stabilization of the linear Kawahara equation with localized damping, Asymptot. Anal., 58 (2008), 229-252. [28] C. F. Vasconcellos and P. N. da Silva, Stabilization of the Kawahara equation with localized damping, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 102-116. [29] B. Y. Zhang and X. Q. Zhao, Control and stabilization of the Kawahara equation on a periodic domain, Communications in Information and Systems, 12 (2012), 77-96.
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