The Cauchy problem for coupled system of the generalized Camassa-Holm equations

Sen Ming; Jiayi Du; Yaxian Ma; Sen Ming; Jiayi Du; Yaxian Ma

doi:10.3934/math.2022810

AIMS Mathematics

2022, Volume 7, Issue 8: 14738-14755. doi: 10.3934/math.2022810

Previous Article Next Article

Research article

The Cauchy problem for coupled system of the generalized Camassa-Holm equations

Department of Mathematics, North University of China, Taiyuan 030051, China

Received: 30 March 2022 Revised: 20 May 2022 Accepted: 30 May 2022 Published: 08 June 2022
MSC : 35G25, 35L15

Local well-posedness for the Cauchy problem of coupled system of generalized Camassa-Holm equations in the Besov spaces is established by employing the Littlewood-Paley theory and a priori estimate of solution to transport equation. Furthermore, the blow-up criterion of solutions to the problem is illustrated. Our main new contribution is that the effects of dissipative coefficient $ \lambda $ and exponent $ b $ in the nonlinear terms to the solutions are analyzed. To the best of our knowledge, the results in Theorems 1.1 and 1.2 are new.
- local well-posedness,
- coupled system,
- generalized Camassa-Holm equations,
- blow-up criterion
Citation: Sen Ming, Jiayi Du, Yaxian Ma. The Cauchy problem for coupled system of the generalized Camassa-Holm equations[J]. AIMS Mathematics, 2022, 7(8): 14738-14755. doi: 10.3934/math.2022810

Related Papers:

Abstract

Local well-posedness for the Cauchy problem of coupled system of generalized Camassa-Holm equations in the Besov spaces is established by employing the Littlewood-Paley theory and a priori estimate of solution to transport equation. Furthermore, the blow-up criterion of solutions to the problem is illustrated. Our main new contribution is that the effects of dissipative coefficient $ \lambda $ and exponent $ b $ in the nonlinear terms to the solutions are analyzed. To the best of our knowledge, the results in Theorems 1.1 and 1.2 are new.

References

[1]	S. Anco, P. L. Silva, I. L. Freire, A family of wave breaking equations generalizing the Camassa-Holm and Novikov equations, J. Math. Phys., 56 (2015), 091506. https://doi.org/10.1063/1.4929661 doi: 10.1063/1.4929661
[2]	H. Bahouri, J. Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations, Grun. Math. Wiss. Springer, Heidelberg, 343 (2011).
[3]	R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Revi. Lett., 71 (1993), 1661–1664. https://doi.org/10.1103/PhysRevLett.71.1661 doi: 10.1103/PhysRevLett.71.1661
[4]	L. Chen, C. Guan, Global solutions for the generalized Camassa-Holm equation, Nonlinear Anal.-Real, 58 (2021), 103227. https://doi.org/10.1016/j.nonrwa.2020.103227 doi: 10.1016/j.nonrwa.2020.103227
[5]	A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Ann. I. Fourier., 50 (2000), 321–362. https://doi.org/10.5802/aif.1757 doi: 10.5802/aif.1757
[6]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523–535. https://doi.org/10.1007/s00222-006-0002-5 doi: 10.1007/s00222-006-0002-5
[7]	A. Constantin, J. Escher, Particle trajectories in solitary water waves, B. Am. Math. Soc., 44 (2007), 423–431. https://doi.org/10.1090/S0273-0979-07-01159-7 doi: 10.1090/S0273-0979-07-01159-7
[8]	A. Constantin, H. P. Mckean, A shallow water equation on the circle, Commun. Pur. Appl. Math., 52 (1999), 949–982.
[9]	D. C. Ferraioli, I. L. Freire, A generalised multicomponent system of Camassa-Holm-Novikov equations, arXiv: 1608.04604v2, 2017.
[10]	I. L. Freire, A look on some results about Camassa-Holm type equations, Commun. Math., 29 (2021), 115–130. https://doi.org/10.2478/cm-2021-0006 doi: 10.2478/cm-2021-0006
[11]	I. L. Freire, N. S. Filho, L. C. Souza, C. E. Toffoli, Invariants and wave breaking analysis of a Camassa-Holm type equation with quadratic and cubic nonlinearities, J. Differ. Equations, 269 (2020), 56–77. https://doi.org/10.1016/j.jde.2020.04.041 doi: 10.1016/j.jde.2020.04.041
[12]	C. X. Guan, J. M. Wang, Y. P. Meng, Weak well-posedness for a modified two-component Camassa-Holm system, Nonlinear Anal., 178 (2019), 247–265. https://doi.org/10.1016/j.na.2018.07.019 doi: 10.1016/j.na.2018.07.019
[13]	Z. G. Guo, On an integrable Camassa-Holm type equation with cubic nonlinearity, Nonlinear Anal.-Real, 34 (2017), 225–232. https://doi.org/10.1016/j.nonrwa.2016.09.002 doi: 10.1016/j.nonrwa.2016.09.002
[14]	Z. G. Guo, X. G. Li, C. Yu, Some properties of solutions to the Camassa-Holm-type equation with higher order nonlinearities, J. Nonlinear Sci., 28 (2018), 1901–1914. https://doi.org/10.1007/s00332-018-9469-7 doi: 10.1007/s00332-018-9469-7
[15]	A. Himonas, C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity., 25 (2012), 449–479. https://doi.org/10.1088/0951-7715/25/2/449 doi: 10.1088/0951-7715/25/2/449
[16]	A. Himonas, C. Holliman, The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differ. Equ., 144 (2016), 3797–3811. https://doi.org/10.1016/j.jde.2018.11.019 doi: 10.1016/j.jde.2018.11.019
[17]	A. Himonas, G. Misiolek, G. Ponce, Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Commun. Math. Phys., 271 (2007), 511–522. https://doi.org/10.1007/s00220-006-0172-4 doi: 10.1007/s00220-006-0172-4
[18]	A. Himonas, R. Thompson, Persistence properties and unique continuation for a generalized Camassa-Holm equation, J. Math. Phys., 55 (2014), 091503. https://doi.org/10.1063/1.4895572 doi: 10.1063/1.4895572
[19]	A. Hone, H. Lundmark, J. Szmigielski, Explicit multipeakon solutions of Novikov$'$s cubically nonlinear integrable Camassa-Holm equation, Dynam. Part. Differ. Eq., 6 (2009), 253–289. https://doi.org/10.4310/DPDE.2009.v6.n3.a3 doi: 10.4310/DPDE.2009.v6.n3.a3
[20]	A. Hone, J. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. Appl. Math. Theor., 41 (2008), 372002. https://doi.org/10.1088/1751-8113/41/37/372002 doi: 10.1088/1751-8113/41/37/372002
[21]	S. G. Ji, Y. H. Zhou, Wave breaking and global solutions of the weakly dissipative periodic Camassa-Holm type equation, J. Differ. Equ., 306 (2022), 439–455. https://doi.org/10.1016/j.jde.2021.10.035 doi: 10.1016/j.jde.2021.10.035
[22]	J. L. Li, W. Deng, M. Li, Non-uniform dependence for higher dimensional Camassa-Holm equations in Besov spaces, Nonlinear Anal.-Real, 63 (2022), 103420. https://doi.org/10.1016/j.nonrwa.2021.103420 doi: 10.1016/j.nonrwa.2021.103420
[23]	Y. Li, P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ., 162 (2000), 27–63. https://doi.org/10.1006/jdeq.1999.3683 doi: 10.1006/jdeq.1999.3683
[24]	M. Li, Z. Y. Yin, Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation with cubic nonlinearity, Nonlinear Anal., 151 (2017), 208–226. https://doi.org/10.1016/j.na.2016.12.003 doi: 10.1016/j.na.2016.12.003
[25]	J. L. Li, Y. H. Yu, W. P. Zhu, Ill-posedness for the Camassa-Holm and related equations in Besov spaces, J. Differ. Equ., 306 (2022), 403–417. https://doi.org/10.1016/j.jde.2021.10.052 doi: 10.1016/j.jde.2021.10.052
[26]	F. Linares, G. Ponce, Unique continuation properties for solutions to the Camassa-Holm equation and related models, P. Am. Math. Soc., 148 (2020), 3871–3879. https://doi.org/10.1090/proc/15059 doi: 10.1090/proc/15059
[27]	W. Luo, Z. Y. Yin, Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space, Nonlinear Anal., 122 (2015), 1–22. https://doi.org/10.1016/j.na.2015.03.022 doi: 10.1016/j.na.2015.03.022
[28]	A. Madiyeva, D. E. Pelinovsky, Growth of perturbations to the peaked periodic waves in the Camassa-Holm equation, SIAM J. Math. Anal., 53 (2021), 3016–3039. https://doi.org/10.1137/20M1347474 doi: 10.1137/20M1347474
[29]	S. Ming, S. Y. Lai, Y. Q. Su, The Cauchy problem of a weakly dissipative shallow water equation, Appl. Anal., 98 (2019), 1387–1402. https://doi.org/10.1080/00036811.2017.1422728 doi: 10.1080/00036811.2017.1422728
[30]	S. Ming, S. Y. Lai, Y. Q. Su, Well-posedness and behaviors of solutions to an integrable evolution equation, Boun. Valu. Prob., 165 (2020), 1–22. https://doi.org/10.1186/s13661-020-01460-y doi: 10.1186/s13661-020-01460-y
[31]	S. Ming, H. Yang, Z. L. Chen, L. Yong, The properties of solutions to the dissipative two-component Camassa-Holm system, Appl. Anal., 95 (2016), 1165–1183. https://doi.org/10.1080/00036811.2015.1055557 doi: 10.1080/00036811.2015.1055557
[32]	N. D. Mutlubas, I. L. Freire, The Cauchy problem and continuation of periodic solutions for a generalized Camassa-Holm equation, arXiv: 2202.07110v1, 2022.
[33]	V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002. https://doi.org/10.1088/1751-8113/42/34/342002 doi: 10.1088/1751-8113/42/34/342002
[34]	E. Novruzova, A. Hagverdiyevb, On the behavior of the solution of the dissipative Camassa-Holm equation with the arbitrary dispersion coefficient, J. Differ. Equ., 257 (2014), 4525–4541. https://doi.org/10.1016/j.jde.2014.08.016 doi: 10.1016/j.jde.2014.08.016
[35]	H. M. Qiu, L. Y. Zhong, J. H. Shen, Traveling waves in a generalized Camassa-Holm equation involving dual power law nonlinearities, Commun. Nonlinear Sci., 106 (2022), 106106. https://doi.org/10.1016/j.cnsns.2021.106106 doi: 10.1016/j.cnsns.2021.106106
[36]	P. L. Silva, I. L. Freire, Strict self-adjointness and shallow water models, arXiv: 1312.3992v1, 2013.
[37]	P. L. Silva, I. L. Freire, An equation unifying both Camassa-Holm and Novikov equations, Proc. 10th AIMS Int. Conf., 2015. https://doi.org/10.3934/proc.2015.0304
[38]	P. L. Silva, I. L. Freire, Existence, continuation and dynamics of solutions for the generalized 0-Holm-Staley equation, arXiv: 2008.11848, 2020.
[39]	H. Tang, Z. R. Liu, The Cauchy problem for a two-component Novikov equation inthe critical Besov space, J. Math. Anal. Appl., 423 (2015), 120–135. https://doi.org/10.1016/j.jmaa.2014.09.032 doi: 10.1016/j.jmaa.2014.09.032
[40]	R. Thompson, Decay properties of solutions to a four-parameter family of wave equations, J. Math. Anal. Appl., 451 (2017), 393–404. https://doi.org/10.1016/j.jmaa.2017.02.002 doi: 10.1016/j.jmaa.2017.02.002
[41]	F. Wang, F. Q. Li, Z. J. Qiao, Well-posedness and peakons for a higher order $\mu$-Camassa-Holm equation, Nonlinear Anal., 175 (2018), 210–236. https://doi.org/10.1016/j.na.2018.06.001 doi: 10.1016/j.na.2018.06.001
[42]	Y. Wang, M. Zhu, Blow-up issues for a two-component system modelling water waves with constant vorticity, Nonlinear Anal., 172 (2018), 163–179. https://doi.org/10.1016/j.na.2018.02.010 doi: 10.1016/j.na.2018.02.010
[43]	S. Y. Wu, Z. Y. Yin, Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation, J. Differ. Equ., 246 (2009), 4309–4321. https://doi.org/10.1016/j.jde.2008.12.008 doi: 10.1016/j.jde.2008.12.008
[44]	K. Yan, Wave breaking and global existence for a family of peakon equations with high order nonlinearity, Nonlinear Anal.-Real, 45 (2019), 721–735. https://doi.org/10.1016/j.nonrwa.2018.07.032 doi: 10.1016/j.nonrwa.2018.07.032
[45]	W. Yan, Y. S. Li, Y. M. Zhang, Global existence and blow-up phenomena for the weakly dissipative Novikov equation, Nonlinear Anal., 75 (2012), 2464–2473. https://doi.org/10.1016/j.na.2011.10.044 doi: 10.1016/j.na.2011.10.044
[46]	W. Yan, Y. S. Li, Y. M. Zhang, The Cauchy problem for the integrable Novikov equation, J. Differ. Equ., 253 (2012), 298–318. https://doi.org/10.1016/j.jde.2012.03.015 doi: 10.1016/j.jde.2012.03.015
[47]	W. Yan, Y. S. Li, Y. Zhang, The Cauchy problem for the generalized Camassa-Holm equation in Besov space, J. Differ. Equ., 256 (2014), 2876–2901. https://doi.org/10.1016/j.jde.2014.01.023 doi: 10.1016/j.jde.2014.01.023
[48]	Z. Y. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649–666. https://doi.org/10.1215/ijm/1258138186 doi: 10.1215/ijm/1258138186
[49]	L. Zhang, B. Liu, On the Cauchy problem for a class of shallow water wave equations with $(k+1)$-order nonlinearities, J. Math. Anal. Appl., 445 (2017), 151–185. https://doi.org/10.1016/j.jmaa.2016.07.056 doi: 10.1016/j.jmaa.2016.07.056
[50]	Y. Zhou, H. P. Chen, Wave breaking and propagation speed for the Camassa-Holm equation with $k\neq 0$, Nonlinear Anal.-Real, 12 (2011), 1875–1882. https://doi.org/10.1016/j.nonrwa.2010.12.005 doi: 10.1016/j.nonrwa.2010.12.005
[51]	S. M. Zhou, Z. J. Qiao, C. L. Mu, Continuity for a generalized cross-coupled Camassa-Holm system with waltzing and higher order nonlinearities, Nonlinear Anal.-Real, 51 (2020), 102970. https://doi.org/10.1016/j.nonrwa.2019.102970 doi: 10.1016/j.nonrwa.2019.102970

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)