December  2020, 28(4): 1545-1562. doi: 10.3934/era.2020081

Global weak solutions for the two-component Novikov equation

School of Mathematics and Statistics and Center for Nonlinear Studies, Ningbo University, Ningbo 315211, China

Received  May 2020 Revised  July 2020 Published  July 2020

Fund Project: This work is partially supported by the NSF-China grant-11631007 and grant-11971251

The two-component Novikov equation is an integrable generalization of the Novikov equation, which has the peaked solitons in the sense of distribution as the Novikov and Camassa-Holm equations. In this paper, we prove the existence of the $ H^1 $-weak solution for the two-component Novikov equation by the regular approximation method due to the existence of three conserved densities. The key elements in our approach are some a priori estimates on the approximation solutions.

Citation: Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
References:
[1]

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

[2]

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

[3]

R. M. ChenF. GuoY. Lu and C. Qu, Analysis on the blow-up of solutions to a class of interable peakon equations, J. Funct. Anal., 270 (2016), 2343-2374.  doi: 10.1016/j.jfa.2016.01.017.  Google Scholar

[4]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[5]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.   Google Scholar

[6]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[7]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Commun. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.  Google Scholar

[8]

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.  Google Scholar

[9]

A. S. Fokas, P. J. Olver and P. Rosenau, A plethora of integrable bi-Hamiltonian equations, in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., 26, Birkhäuser, Boston, MA, 1997, 93–101. doi: 10.1007/978-1-4612-2434-1_5.  Google Scholar

[10]

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

[11]

X. Geng and B. Xue, An extension of integrable peakon equations with cubic nonlinearity, Nonlinearity, 22 (2009), 1847-1856.  doi: 10.1088/0951-7715/22/8/004.  Google Scholar

[12]

C. He, X. C. Liu and C. Qu, Orbital stability of peakons and the trains of peakons for an integrable two-component Novikov system, work in progress. Google Scholar

[13]

A. Himonas and D. Mantzavinos, The initial value problem for a Novikov system, J. Math. Phys. 57 (2016), 21pp. doi: 10.1063/1.4959774.  Google Scholar

[14]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 10pp. doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[15]

Z. Jiang and L. Ni, Blow-up phenomenon for the integrable Novikov equation, J. Math. Anal. Appl., 385 (2012), 551-558.  doi: 10.1016/j.jmaa.2011.06.067.  Google Scholar

[16]

J. Kang, X. Liu, P. J. Olver and C. Qu, Liouville correspondences between integrable hierarchies, SIGMA Symmetry Integrability Geom. Methods Appl., 13 (2017), 26pp. doi: 10.3842/SIGMA.2017.035.  Google Scholar

[17]

S. Lai, Global weak solutions to the Novikov equation, J. Funct. Anal., 265 (2013), 520-544.  doi: 10.1016/j.jfa.2013.05.022.  Google Scholar

[18]

H. Li, Two-component generalizations of the Novikov equation, J. Nonlinear Math. Phys., 26 (2019), 390-403.  doi: 10.1080/14029251.2019.1613048.  Google Scholar

[19]

N. Li and Q. P. Liu, On bi-Hamiltonian structure of two-component Novikov equation, Phys. Lett. A, 377 (2013), 257-261.  doi: 10.1016/j.physleta.2012.11.023.  Google Scholar

[20]

X. LiuY. Liu and C. Qu, Stability of peakons for the Novikov equation, J. Math. Pures Appl., 101 (2014), 172-187.  doi: 10.1016/j.matpur.2013.05.007.  Google Scholar

[21]

H. Lundmark and J. Szmigielski, An inverse spectral problem related to the Geng-Xue two-component peakon equation, Mem. Amer. Math. Soc., 244 (2016), 87pp. doi: 10.1090/memo/1155.  Google Scholar

[22]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.  Google Scholar

[23]

A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.  doi: 10.1088/0305-4470/35/22/309.  Google Scholar

[24]

V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 14pp. doi: 10.1088/1751-8113/42/34/342002.  Google Scholar

[25]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/physreve.53.1900.  Google Scholar

[26]

F. Tiğlay, The periodic Cauchy problem for Novikov's equation, Int. Math. Res. Not., 2011 (2011), 4633-4648.  doi: 10.1093/imrn/rnq267.  Google Scholar

[27]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation, J. Phys. A., 44 (2011), 17pp. doi: 10.1088/1751-8113/44/5/055202.  Google Scholar

[28]

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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

R. M. ChenF. GuoY. Lu and C. Qu, Analysis on the blow-up of solutions to a class of interable peakon equations, J. Funct. Anal., 270 (2016), 2343-2374.  doi: 10.1016/j.jfa.2016.01.017.  Google Scholar

[4]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[5]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.   Google Scholar

[6]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[7]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Commun. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.  Google Scholar

[8]

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.  Google Scholar

[9]

A. S. Fokas, P. J. Olver and P. Rosenau, A plethora of integrable bi-Hamiltonian equations, in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., 26, Birkhäuser, Boston, MA, 1997, 93–101. doi: 10.1007/978-1-4612-2434-1_5.  Google Scholar

[10]

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

[11]

X. Geng and B. Xue, An extension of integrable peakon equations with cubic nonlinearity, Nonlinearity, 22 (2009), 1847-1856.  doi: 10.1088/0951-7715/22/8/004.  Google Scholar

[12]

C. He, X. C. Liu and C. Qu, Orbital stability of peakons and the trains of peakons for an integrable two-component Novikov system, work in progress. Google Scholar

[13]

A. Himonas and D. Mantzavinos, The initial value problem for a Novikov system, J. Math. Phys. 57 (2016), 21pp. doi: 10.1063/1.4959774.  Google Scholar

[14]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 10pp. doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[15]

Z. Jiang and L. Ni, Blow-up phenomenon for the integrable Novikov equation, J. Math. Anal. Appl., 385 (2012), 551-558.  doi: 10.1016/j.jmaa.2011.06.067.  Google Scholar

[16]

J. Kang, X. Liu, P. J. Olver and C. Qu, Liouville correspondences between integrable hierarchies, SIGMA Symmetry Integrability Geom. Methods Appl., 13 (2017), 26pp. doi: 10.3842/SIGMA.2017.035.  Google Scholar

[17]

S. Lai, Global weak solutions to the Novikov equation, J. Funct. Anal., 265 (2013), 520-544.  doi: 10.1016/j.jfa.2013.05.022.  Google Scholar

[18]

H. Li, Two-component generalizations of the Novikov equation, J. Nonlinear Math. Phys., 26 (2019), 390-403.  doi: 10.1080/14029251.2019.1613048.  Google Scholar

[19]

N. Li and Q. P. Liu, On bi-Hamiltonian structure of two-component Novikov equation, Phys. Lett. A, 377 (2013), 257-261.  doi: 10.1016/j.physleta.2012.11.023.  Google Scholar

[20]

X. LiuY. Liu and C. Qu, Stability of peakons for the Novikov equation, J. Math. Pures Appl., 101 (2014), 172-187.  doi: 10.1016/j.matpur.2013.05.007.  Google Scholar

[21]

H. Lundmark and J. Szmigielski, An inverse spectral problem related to the Geng-Xue two-component peakon equation, Mem. Amer. Math. Soc., 244 (2016), 87pp. doi: 10.1090/memo/1155.  Google Scholar

[22]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.  Google Scholar

[23]

A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.  doi: 10.1088/0305-4470/35/22/309.  Google Scholar

[24]

V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 14pp. doi: 10.1088/1751-8113/42/34/342002.  Google Scholar

[25]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/physreve.53.1900.  Google Scholar

[26]

F. Tiğlay, The periodic Cauchy problem for Novikov's equation, Int. Math. Res. Not., 2011 (2011), 4633-4648.  doi: 10.1093/imrn/rnq267.  Google Scholar

[27]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation, J. Phys. A., 44 (2011), 17pp. doi: 10.1088/1751-8113/44/5/055202.  Google Scholar

[28]

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.  Google Scholar

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