# American Institute of Mathematical Sciences

September  2016, 36(9): 5201-5221. doi: 10.3934/dcds.2016026

## Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing

 1 Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan 610066, China

Received  August 2015 Revised  January 2016 Published  May 2016

In this paper, we study the global well-posedness for the Camassa-Holm(C-H) equation with a forcing in $H^1(\mathbb{R})$ by the characteristic method. Due to the forcing, many important properties to study the well-posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability. By exploiting the balance law and some new estimates, we prove the existence and uniqueness of global weak solutions for the C-H equation with a forcing in $H^1(\mathbb{R})$.
Citation: Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
##### References:
 [1] M. S. Alber, R. Camassa, D. D. Holm and J. E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDE's, Lett. Math. Phys., 32 (1994), 137-151. doi: 10.1007/BF00739423. [2] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math., 57 (1976), 147-190. [3] R. Beals, D. H. Sattinger and J. Szmigielski, Multi-peakons and a theorem of Stieltjes, Inverse Problems, 15 (1999), L1-L4. doi: 10.1088/0266-5611/15/1/001. [4] A. Bressan and A. Constantin, Global conservative solutions to the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. [5] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. [6] A. Bressan, G. Chen and Q. Zhang, Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics, Discr. Cont. Dyn. Syst., 35 (2015), 25-42. [7] A. Bressan, G. Chen and Q. Zhang, Unique conservative solutions to a variational wave equation, Arch. Rat. Mech. Anal., 217 (2015), 1069-1101. doi: 10.1007/s00205-015-0849-y. [8] G. Chen and Y. Shen, Existence and regularity of solutions in nonlinear wave equations, Discr. Cont. Dyn. Syst., 35 (2015), 3327-3342. doi: 10.3934/dcds.2015.35.3327. [9] G. Chen, Y. Shen and S. Zhu, Global well-posedness of weak solutions for a generalized water wave equation, preprint. [10] 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. [11] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [12] A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017. [13] A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res., 40 (2008), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004. [14] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [15] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [16] A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Physica D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6. [17] A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [18] K. E. Dika and L. Molinet, Stability of multipeakons, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 26 (2009), 1517-1532. doi: 10.1016/j.anihpc.2009.02.002. [19] L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [20] B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66. doi: 10.1016/0167-2789(81)90004-X. [21] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation- a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. [22] R. I. Ivanov, Water waves and integrability, Philos. Trans. Roy. Soc. Lond. Ser. A, 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007. [23] M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in: Tsunami and Nonlinear Waves, Springer, Berlin, (2007), 31-49. doi: 10.1007/978-3-540-71256-5_2. [24] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [25] Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2002), 1815-1844. doi: 10.1081/PDE-120016129.

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##### References:
 [1] M. S. Alber, R. Camassa, D. D. Holm and J. E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDE's, Lett. Math. Phys., 32 (1994), 137-151. doi: 10.1007/BF00739423. [2] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math., 57 (1976), 147-190. [3] R. Beals, D. H. Sattinger and J. Szmigielski, Multi-peakons and a theorem of Stieltjes, Inverse Problems, 15 (1999), L1-L4. doi: 10.1088/0266-5611/15/1/001. [4] A. Bressan and A. Constantin, Global conservative solutions to the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. [5] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. [6] A. Bressan, G. Chen and Q. Zhang, Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics, Discr. Cont. Dyn. Syst., 35 (2015), 25-42. [7] A. Bressan, G. Chen and Q. Zhang, Unique conservative solutions to a variational wave equation, Arch. Rat. Mech. Anal., 217 (2015), 1069-1101. doi: 10.1007/s00205-015-0849-y. [8] G. Chen and Y. Shen, Existence and regularity of solutions in nonlinear wave equations, Discr. Cont. Dyn. Syst., 35 (2015), 3327-3342. doi: 10.3934/dcds.2015.35.3327. [9] G. Chen, Y. Shen and S. Zhu, Global well-posedness of weak solutions for a generalized water wave equation, preprint. [10] 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. [11] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [12] A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017. [13] A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res., 40 (2008), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004. [14] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [15] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [16] A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Physica D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6. [17] A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [18] K. E. Dika and L. Molinet, Stability of multipeakons, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 26 (2009), 1517-1532. doi: 10.1016/j.anihpc.2009.02.002. [19] L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [20] B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66. doi: 10.1016/0167-2789(81)90004-X. [21] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation- a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. [22] R. I. Ivanov, Water waves and integrability, Philos. Trans. Roy. Soc. Lond. Ser. A, 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007. [23] M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in: Tsunami and Nonlinear Waves, Springer, Berlin, (2007), 31-49. doi: 10.1007/978-3-540-71256-5_2. [24] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [25] Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2002), 1815-1844. doi: 10.1081/PDE-120016129.
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