# American Institute of Mathematical Sciences

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
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##### References:
 [1] Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304 [2] Wenxia Chen, Jingyi Liu, Danping Ding, Lixin Tian. Blow-up for two-component Camassa-Holm equation with generalized weak dissipation. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3769-3784. doi: 10.3934/cpaa.2020166 [3] Min Zhao, Changzheng Qu. The two-component Novikov-type systems with peaked solutions and $H^1$-conservation law. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020245 [4] Giuseppe Maria Coclite, Lorenzo Di Ruvo. A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1247-1282. doi: 10.3934/dcds.2017052 [5] Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solution of the Novikov equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2865-2899. doi: 10.3934/dcdsb.2018290 [6] David Henry. Infinite propagation speed for a two component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 597-606. doi: 10.3934/dcdsb.2009.12.597 [7] Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065 [8] Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883 [9] Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 2981-2990. doi: 10.3934/dcds.2016.36.2981 [10] Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613 [11] Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305 [12] Yingying Li, Ying Fu, Changzheng Qu. The two-component $\mu$-Camassa–Holm system with peaked solutions. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5929-5954. doi: 10.3934/dcds.2020253 [13] Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026 [14] Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483 [15] Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194 [16] 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 [17] H. A. Erbay, S. Erbay, A. Erkip. On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3111-3122. doi: 10.3934/dcds.2017133 [18] Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149 [19] Joachim Escher, Tony Lyons. Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach. Journal of Geometric Mechanics, 2015, 7 (3) : 281-293. doi: 10.3934/jgm.2015.7.281 [20] Zeng Zhang, Zhaoyang Yin. Global existence for a two-component Camassa-Holm system with an arbitrary smooth function. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5523-5536. doi: 10.3934/dcds.2018243

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