The existence of weak solutions for a generalized Camassa-Holm equation

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January 2011, 10(1): 45-57. doi: 10.3934/cpaa.2011.10.45

The existence of weak solutions for a generalized Camassa-Holm equation

Shaoyong Lai ^1, , Qichang Xie ^2, , Yunxi Guo ^2, and YongHong Wu ^2,

1.	Department of Mathematics,Sichuan Normal University, Chengdu, Department of Mathematics and Statistics,Curtin University of Technology, Perth, China
2.	Department of Applied Mathematics, Southwestern University of Finance and Economics, 610074, Chengdu, China, China, China

Received October 2009 Revised August 2010 Published November 2010

Abstract
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A Camassa-Holm type equation containing nonlinear dissipative effect is investigated. A sufficient condition which guarantees the existence of weak solutions of the equation in lower order Sobolev space $H^s$ with $1 \leq s \leq \frac{3}{2}$ is established by using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself.

Keywords: high order nonlinear terms, pseudoparabolic regularization technique., weak solution, Generalized Camassa-Holm equation.

Mathematics Subject Classification: Primary: 35Q53; Secondary: 35L0.

Citation: Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45

References:

[1]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: doi:10.1103/PhysRevLett.71.1661.
[2]	Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: doi:10.1006/jdeq.1999.3683.
[3]	Z. H. Guo, M. Jiang, Z. Wang and G. F. Zheng, Global weak solutions to the Camassa-Holm equation,, Discrete and Continuous Dynamical Systems, 21 (2008), 883. doi: doi:10.3934/dcds.2008.21.883.
[4]	A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: doi:10.1007/s002200050801.
[5]	R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429. doi: doi:10.1016/S0022-0396(03)00096-2.
[6]	A. M. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions,, Appl. Math. Comput., 165 (2005), 485. doi: doi:10.1016/j.amc.2004.04.029.
[7]	A. M. Wazwaz, The tanh-coth and the sine-cosine methods for kinks, solitons and periodic solutions for the Pochhammer-Chree equations,, Appl. Math. Comput., 195 (2008), 24. doi: doi:10.1016/j.amc.2007.04.066.
[8]	L. X. Tian and X. Y. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation,, Chaos, 19 (2004), 621. doi: doi:10.1016/S0960-0779(03)00192-9.
[9]	Y. Zheng and S. Y. Lai, Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations,, Phys. Lett. A, 372 (2008), 4141. doi: doi:10.1016/j.physleta.2007.03.096.
[10]	S. Hakkaev and K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation,, Communications in Partial Differential Equations, 30 (2005), 761. doi: doi:10.1081/PDE-200059284.
[11]	T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation,, Commun. Pure Appl. Math., 41 (1988), 891. doi: doi:10.1002/cpa.3160410704.
[12]	J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation,, Phil. Trans. Roy. Soc. London Ser. A, 278 (1975), 555. doi: doi:10.1098/rsta.1975.0035.
[13]	A. Moameni, Soliton solutions for quasilinear Schr$\ddoto$dinger equations involving supercritical exponent in $R^N$,, Communications on Pure and Applied Analysis, 7 (2008), 89. doi: doi:10.3934/cpaa.2008.7.89.
[14]	R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source,, Communications on Pure and Applied Analysis, 7 (2008), 1077. doi: doi:10.3934/cpaa.2008.7.1077.
[15]	D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation,, Communications on Pure and Applied Analysis, 7 (2008), 867. doi: doi:10.3934/cpaa.2008.7.867.
[16]	K. Y. Wang, Global well-posedness for a transport equation with non-local velocity and critical diffusion,, Communications on Pure and Applied Analysis, 7 (2008), 1203. doi: doi:10.3934/cpaa.2008.7.1203.
[17]	C. L. He, D. X. Kong and K. F. Liu, Hyperbolic mean curvature flow,, J. Differential Equations, 246 (2009), 373. doi: doi:10.1016/j.jde.2008.06.026.

show all references

References:

[1]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: doi:10.1103/PhysRevLett.71.1661.
[2]	Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: doi:10.1006/jdeq.1999.3683.
[3]	Z. H. Guo, M. Jiang, Z. Wang and G. F. Zheng, Global weak solutions to the Camassa-Holm equation,, Discrete and Continuous Dynamical Systems, 21 (2008), 883. doi: doi:10.3934/dcds.2008.21.883.
[4]	A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: doi:10.1007/s002200050801.
[5]	R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429. doi: doi:10.1016/S0022-0396(03)00096-2.
[6]	A. M. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions,, Appl. Math. Comput., 165 (2005), 485. doi: doi:10.1016/j.amc.2004.04.029.
[7]	A. M. Wazwaz, The tanh-coth and the sine-cosine methods for kinks, solitons and periodic solutions for the Pochhammer-Chree equations,, Appl. Math. Comput., 195 (2008), 24. doi: doi:10.1016/j.amc.2007.04.066.
[8]	L. X. Tian and X. Y. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation,, Chaos, 19 (2004), 621. doi: doi:10.1016/S0960-0779(03)00192-9.
[9]	Y. Zheng and S. Y. Lai, Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations,, Phys. Lett. A, 372 (2008), 4141. doi: doi:10.1016/j.physleta.2007.03.096.
[10]	S. Hakkaev and K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation,, Communications in Partial Differential Equations, 30 (2005), 761. doi: doi:10.1081/PDE-200059284.
[11]	T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation,, Commun. Pure Appl. Math., 41 (1988), 891. doi: doi:10.1002/cpa.3160410704.
[12]	J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation,, Phil. Trans. Roy. Soc. London Ser. A, 278 (1975), 555. doi: doi:10.1098/rsta.1975.0035.
[13]	A. Moameni, Soliton solutions for quasilinear Schr$\ddoto$dinger equations involving supercritical exponent in $R^N$,, Communications on Pure and Applied Analysis, 7 (2008), 89. doi: doi:10.3934/cpaa.2008.7.89.
[14]	R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source,, Communications on Pure and Applied Analysis, 7 (2008), 1077. doi: doi:10.3934/cpaa.2008.7.1077.
[15]	D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation,, Communications on Pure and Applied Analysis, 7 (2008), 867. doi: doi:10.3934/cpaa.2008.7.867.
[16]	K. Y. Wang, Global well-posedness for a transport equation with non-local velocity and critical diffusion,, Communications on Pure and Applied Analysis, 7 (2008), 1203. doi: doi:10.3934/cpaa.2008.7.1203.
[17]	C. L. He, D. X. Kong and K. F. Liu, Hyperbolic mean curvature flow,, J. Differential Equations, 246 (2009), 373. doi: doi:10.1016/j.jde.2008.06.026.

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