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  December 2020 Early access  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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