2019, 27: 37-67. doi: 10.3934/era.2019009

The global conservative solutions for the generalized camassa-holm equation

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

2. 

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China

3. 

College of Mathematics and Statistics, Southwest University, Chongqing 401331, China

* Corresponding author: Li Yang and Chunlai Mu

Received  August 2019 Revised  September 2019 Published  October 2019

Fund Project: The second author is partly supported by NSFC (11771062), the Fundamental Research Funds for the Central Universities (Grant Nos. 106112016CDJXZ238826 and 2019CDJCYJ001). The third author is partly supported by the Science and Technology Research Program of Chongqing Municipal Education Commission, Natural Science Foundation of China(11971082).

This paper deals with the continuation of solutions to the generalized Camassa-Holm equation with higher-order nonlinearity beyond wave breaking. By introducing new variables, we transform the generalized Camassa-Holm equation to a semi-linear system and establish the global solutions to this semi-linear system, and by returning to the original variables, we obtain the existence of global conservative solutions to the original equation. We introduce a set of auxiliary variables tailored to a given conservative solution, which satisfy a suitable semi-linear system, and show that the solution for the semi-linear system is unique. Furthermore, it is obtained that the original equation has a unique global conservative solution. By Thom's transversality lemma, we prove that piecewise smooth solutions with only generic singularities are dense in the whole solution set, which means the generic regularity.

Citation: Li Yang, Chunlai Mu, Shouming Zhou, Xinyu Tu. The global conservative solutions for the generalized camassa-holm equation. Electronic Research Archive, 2019, 27: 37-67. doi: 10.3934/era.2019009
References:
[1]

J. M. Bloom, The local structure of smooth maps of manifolds, B. A. Thesis, Harvard U., 2004. Google Scholar

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rational Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27.  doi: 10.1142/S0219530507000857.  Google Scholar

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A. BressanG. Chen and Q. Zhang, Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics, Discrete. Contin. Dyn. Syst., 35 (2015), 25-42.  doi: 10.3934/dcds.2015.35.25.  Google Scholar

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[7]

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.  Google Scholar

[8]

A. Constantin, The Hamiltonian struction of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85.   Google Scholar

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A. Costantin and D. Lannes, The hydrodynamical of relevance of Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

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A. Costantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422.  doi: 10.1007/s00332-002-0517-x.  Google Scholar

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A. Costantin and W. A. 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.  Google Scholar

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A. Costantin and J. Escher, Global existence of solutions and breaking waves for a shallow water equation, Ann. Sc. Norm. Super. Pisa CL. Sci., 26 (1998), 303-328.   Google Scholar

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C. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbolic Diff. Equat., 8 (2011), 159-168.  doi: 10.1142/S0219891611002366.  Google Scholar

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A. Fokas and B. Fuchssteiner, Symplectic structures, their BĠcklund transformation and hereditary symmetries, Phy. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

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M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, New York, 1973.  Google Scholar

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G. Jamróz, On uniqueness of dissipative solutions of the Camassa-Holm equation, arXiv: 1611.00333v5. Google Scholar

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M. Li and Q. Zhang, Generic regularity of conservative solutions to Camassa-Holm type equations, SIAM. J. Math. Anal., 49 (2017), 2920-2949.  doi: 10.1137/16M1063009.  Google Scholar

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S. Hakkaev and K. Kirchev, Local well-posedness and orbitalstability of solitary wave solutions for the generalized Camassa-Holm equation, Comm. Part. Diff. Equ., 30 (2005), 761-781.  doi: 10.1081/PDE-200059284.  Google Scholar

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J. Lenells, Conservation laws of the Camassa-Holm equation, J. Phys. A, 38 (2005), 869-880.  doi: 10.1088/0305-4470/38/4/007.  Google Scholar

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Y. Mi and C. Mu, Well-posedness and analyticity for the Cauchy problem for the generalized Camassa-Holm equation, J. Math. Anal. Appl., 405 (2013), 173-182.  doi: 10.1016/j.jmaa.2013.03.020.  Google Scholar

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Y. Mi and C. Mu, On the Cauchy problem for the generalized Camassa-Holm equation, Monatsh. Math., 176 (2015), 423-457.  doi: 10.1007/s00605-014-0625-3.  Google Scholar

[22]

L. YangR. ZengS. Zhou and C. Mu, Uniqueness of Conservative solutions to the generalized Camassa-Holm equation via characteristic, Discrete. Contin. Dyn. Syst., 10 (2018), 5205-5220.  doi: 10.3934/dcds.2018230.  Google Scholar

[23]

S. Zhou, Persistence properties for a generalized Camassa-Holm equation in weighted $L^p$ spaces, J. Math. Anal. Appl., 410 (2013), 932-938.  doi: 10.1016/j.jmaa.2013.09.022.  Google Scholar

[24]

S. ZhouC. Mu and L. Wang, Well-posedness, blow up phenomena and global existence for the generalized b-equation with higher-order nonlinearities and weak solution, Discrete. Contin. Dyn. Syst., 34 (2014), 843-867.  doi: 10.3934/dcds.2014.34.843.  Google Scholar

[25]

S. Zhou and C. Mu, Global conservative solutions and dissipative solutions of the generalized Camassa-Holm equation, Discrete. Contin. Dyn. Syst., 33 (2013), 1713-1739.  doi: 10.3934/dcds.2013.33.1713.  Google Scholar

[26]

S. Zhu, Existence and uniqueness of global weak solutions of the camassa-Holm equation with a forcing, Discrete. Contin. Dyn. Syst., 36 (2016), 5201-5221.  doi: 10.3934/dcds.2016026.  Google Scholar

show all references

References:
[1]

J. M. Bloom, The local structure of smooth maps of manifolds, B. A. Thesis, Harvard U., 2004. Google Scholar

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rational Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27.  doi: 10.1142/S0219530507000857.  Google Scholar

[4]

A. BressanG. Chen and Q. Zhang, Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics, Discrete. Contin. Dyn. Syst., 35 (2015), 25-42.  doi: 10.3934/dcds.2015.35.25.  Google Scholar

[5]

A. Bressan and G. Chen, Generic regularity of conservative solutions to a nonlinear wave equation, Ann. Inst. H. Poincaré Anal. Non linéaire, 34 (2017), 335-354.  doi: 10.1016/j.anihpc.2015.12.004.  Google Scholar

[6]

R. CamassaD. D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.  doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[7]

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.  Google Scholar

[8]

A. Constantin, The Hamiltonian struction of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85.   Google Scholar

[9]

A. Costantin and D. Lannes, The hydrodynamical of relevance of Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[10]

A. Costantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422.  doi: 10.1007/s00332-002-0517-x.  Google Scholar

[11]

A. Costantin and W. A. 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.  Google Scholar

[12]

A. Costantin and J. Escher, Global existence of solutions and breaking waves for a shallow water equation, Ann. Sc. Norm. Super. Pisa CL. Sci., 26 (1998), 303-328.   Google Scholar

[13]

C. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbolic Diff. Equat., 8 (2011), 159-168.  doi: 10.1142/S0219891611002366.  Google Scholar

[14]

A. Fokas and B. Fuchssteiner, Symplectic structures, their BĠcklund transformation and hereditary symmetries, Phy. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[15]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, New York, 1973.  Google Scholar

[16]

G. Jamróz, On uniqueness of dissipative solutions of the Camassa-Holm equation, arXiv: 1611.00333v5. Google Scholar

[17]

M. Li and Q. Zhang, Generic regularity of conservative solutions to Camassa-Holm type equations, SIAM. J. Math. Anal., 49 (2017), 2920-2949.  doi: 10.1137/16M1063009.  Google Scholar

[18]

S. Hakkaev and K. Kirchev, Local well-posedness and orbitalstability of solitary wave solutions for the generalized Camassa-Holm equation, Comm. Part. Diff. Equ., 30 (2005), 761-781.  doi: 10.1081/PDE-200059284.  Google Scholar

[19]

J. Lenells, Conservation laws of the Camassa-Holm equation, J. Phys. A, 38 (2005), 869-880.  doi: 10.1088/0305-4470/38/4/007.  Google Scholar

[20]

Y. Mi and C. Mu, Well-posedness and analyticity for the Cauchy problem for the generalized Camassa-Holm equation, J. Math. Anal. Appl., 405 (2013), 173-182.  doi: 10.1016/j.jmaa.2013.03.020.  Google Scholar

[21]

Y. Mi and C. Mu, On the Cauchy problem for the generalized Camassa-Holm equation, Monatsh. Math., 176 (2015), 423-457.  doi: 10.1007/s00605-014-0625-3.  Google Scholar

[22]

L. YangR. ZengS. Zhou and C. Mu, Uniqueness of Conservative solutions to the generalized Camassa-Holm equation via characteristic, Discrete. Contin. Dyn. Syst., 10 (2018), 5205-5220.  doi: 10.3934/dcds.2018230.  Google Scholar

[23]

S. Zhou, Persistence properties for a generalized Camassa-Holm equation in weighted $L^p$ spaces, J. Math. Anal. Appl., 410 (2013), 932-938.  doi: 10.1016/j.jmaa.2013.09.022.  Google Scholar

[24]

S. ZhouC. Mu and L. Wang, Well-posedness, blow up phenomena and global existence for the generalized b-equation with higher-order nonlinearities and weak solution, Discrete. Contin. Dyn. Syst., 34 (2014), 843-867.  doi: 10.3934/dcds.2014.34.843.  Google Scholar

[25]

S. Zhou and C. Mu, Global conservative solutions and dissipative solutions of the generalized Camassa-Holm equation, Discrete. Contin. Dyn. Syst., 33 (2013), 1713-1739.  doi: 10.3934/dcds.2013.33.1713.  Google Scholar

[26]

S. Zhu, Existence and uniqueness of global weak solutions of the camassa-Holm equation with a forcing, Discrete. Contin. Dyn. Syst., 36 (2016), 5201-5221.  doi: 10.3934/dcds.2016026.  Google Scholar

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