The entropy weak solution to a nonlinear shallow water wave equation including the Degasperis-Procesi model

Mingming Li; Shaoyong Lai; Mingming Li; Shaoyong Lai

doi:10.3934/math.2024086

AIMS Mathematics

2024, Volume 9, Issue 1: 1772-1782. doi: 10.3934/math.2024086

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The entropy weak solution to a nonlinear shallow water wave equation including the Degasperis-Procesi model

Mingming Li ¹,
Shaoyong Lai ^{2
,
,}

1.
School of Mathematics and Statistics, Kashi University, Kashi, 844006, China
2.
School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

Received: 24 October 2023 Revised: 18 November 2023 Accepted: 22 November 2023 Published: 14 December 2023
MSC : 35Q35, 76B25

A nonlinear model, which characterizes motions of shallow water waves and includes the famous Degasperis-Procesi equation, is considered. The essential step is the derivation of the $ L^2(\mathbb{R}) $ uniform bound of solutions for the nonlinear model if its initial value belongs to space $ L^2(\mathbb{R}) $. Utilizing the bounded property leads to several estimates about its solutions. The viscous approximation technique is employed to establish the well-posedness of entropy weak solutions.
- Entropy solution,
- shallow water wave model,
- existence and uniqueness
Citation: Mingming Li, Shaoyong Lai. The entropy weak solution to a nonlinear shallow water wave equation including the Degasperis-Procesi model[J]. AIMS Mathematics, 2024, 9(1): 1772-1782. doi: 10.3934/math.2024086

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Abstract

A nonlinear model, which characterizes motions of shallow water waves and includes the famous Degasperis-Procesi equation, is considered. The essential step is the derivation of the $ L^2(\mathbb{R}) $ uniform bound of solutions for the nonlinear model if its initial value belongs to space $ L^2(\mathbb{R}) $. Utilizing the bounded property leads to several estimates about its solutions. The viscous approximation technique is employed to establish the well-posedness of entropy weak solutions.

References

[1]	A. Bressan, A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. An., 183 (2007), 215–239. http://doi.org/10.1007/s00205-006-0010-z doi: 10.1007/s00205-006-0010-z
[2]	R. Camassa, D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661–1664. https://doi.org/10.1103/PhysRevLett.71.1661 doi: 10.1103/PhysRevLett.71.1661
[3]	G. M. Coclite, K. H. Karlsen, Periodic solutions of the Degasperis-Procesi equation: Well-posedness and asymptotics, J. Funct. Anal., 268 (2015), 1053–1077. https://doi.org/10.1016/j.jfa.2014.11.008 doi: 10.1016/j.jfa.2014.11.008
[4]	G. M. Coclite, K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60–91. https://doi.org/10.1016/j.jfa.2005.07.008 doi: 10.1016/j.jfa.2005.07.008
[5]	G. M. Coclite, K. H. Karlsen, Bounded solutions for the Degasperis-Procesi equation, Boll. Unione Mat. Ital., 9 (2008), 439–453.
[6]	G. M. Coclite, H. Holden, K. H. Karlsen, Wellposedness for a parabolic-elliptic system, Discrete Cont. Dyn. Syst., 13 (2005), 659–682. https://doi.org/10.3934/dcds.2005.13.659 doi: 10.3934/dcds.2005.13.659
[7]	A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229–243. https://doi.org/10.1007/BF02392586 doi: 10.1007/BF02392586
[8]	A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. An., 192 (2009), 165–186. https://doi.org/10.1007/s00205-008-0128-2 doi: 10.1007/s00205-008-0128-2
[9]	A. Constantin, J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation, Commun. Pur. Appl. Math., 51 (1998), 475–504. https://doi.org/10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5 doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[10]	A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. A, 457 (2001), 953–970. https://doi.org/10.1098/rspa.2000.0701 doi: 10.1098/rspa.2000.0701
[11]	A. Constantin, R. I. Ivanov, J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559–2575. http://doi.org/10.1088/0951-7715/23/10/012 doi: 10.1088/0951-7715/23/10/012
[12]	A. Degasperis, M. Procesi, Asymptotic integrability, In: Symmetry and Perturbation Theory (A. Degasperis and G. Gaeta, eds.), World Scientific, Singapore, 1 (1999), 23–37. https://doi.org/10.1142/9789812833037
[13]	A. Degasperis, D. D. Holm, A. N. W. Hone, Integrable and non-integrable equations with peakons, World Scientific Publishing, 2003, 37–43. https://doi.org/10.1142/9789812704467_0005
[14]	J. Escher, Y. Liu, Z. Y. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457–485. https://doi.org/10.1016/j.jfa.2006.03.022 doi: 10.1016/j.jfa.2006.03.022
[15]	I. L. Freire, Conserved quantities, continuation and compactly supported solutions of some shallow water models, J. Phys. A-Math. Theor., 54 (2020), 015207. https://doi.org/10.1088/1751-8121/abc9a2 doi: 10.1088/1751-8121/abc9a2
[16]	G. L. Gui, Y. Liu, P. J. Olver, C. Z. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Commun. Math. Phys., 319 (2013), 731–759. https://doi.org/10.1007/s00220-012-1566-0 doi: 10.1007/s00220-012-1566-0
[17]	Z. G. Guo, X. G. Li, C. Xu, 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
[18]	A. A. Himonas, C. Holliman, The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differential Equ., 19 (2014), 161–200. https://doi.org/10.57262/ade/1384278135 doi: 10.57262/ade/1384278135
[19]	A. A. Himonas, C. Holliman, C. Kenig, Construction of 2-peakon solutions and ill-posedness for the Novikov equation, SIAM J. Math. Anal., 50 (2018), 2968–3006. https://doi.org/10.1137/17M1151201 doi: 10.1137/17M1151201
[20]	S. N. Kru$\check{z}$kov, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217–243. https://doi.org/10.1070/SM1970v010n02ABEH002156 doi: 10.1070/SM1970v010n02ABEH002156
[21]	S. Y. Lai, Y. H. Wu, A model containing both the Camassa-Holm and Degasperis-Procesi equations, J. Math. Anal. Appl., 374 (2011), 458–469. https://doi.org/10.1016/j.jmaa.2010.09.012 doi: 10.1016/j.jmaa.2010.09.012
[22]	Y. Liu, Z. Y. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Commun. Math. Phys., 267 (2006), 801–820. https://doi.org/10.1007/s00220-006-0082-5 doi: 10.1007/s00220-006-0082-5
[23]	H. Lundmark, J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Probl., 19 (2003), 1241–1245. http://doi.org/10.1088/0266-5611/19/6/001 doi: 10.1088/0266-5611/19/6/001
[24]	F. Y. Ma, Y. Liu, C. Z. Qu, Wave-breaking phenomena for the nonlocal Whitham-type equations, J. Differ. Equations, 261 (2016), 6029–6054. https://doi.org/10.1016/j.jde.2016.08.027 doi: 10.1016/j.jde.2016.08.027
[25]	Y. Matsuno, Multisoliton solutions of the Deagsperis-Procesi equation and their peakon limit, Inverse Probl., 21 (2005), 1553–1570. http://doi.org/10.1088/0266-5611/21/5/004 doi: 10.1088/0266-5611/21/5/004
[26]	F. Murat, L${'}$ injection du c$\hat{o}$ne positif de $H^{-1}$ dans $W^{-1, q}$ est compacte pour tout $q < 2$, J. Math. Pures Appl., 60 (1981), 309–322. https://doi.org/10.1080/00263209808701214 doi: 10.1080/00263209808701214
[27]	M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Commun. Part. Diff. Eq., 7 (1982), 959–1000. https://doi.org/10.1080/03605308208820242 doi: 10.1080/03605308208820242
[28]	P. L. Silva, I. L. Freire, Existence, persistence, and continuation of solutions for a generalized 0-Holm-Staley equation, J. Differ. Equations, 320 (2022), 371–398. https://doi.org/10.1016/j.jde.2022.02.058 doi: 10.1016/j.jde.2022.02.058
[29]	L. Tartar, Compensated compactness and applications to partial differential equations, In: Heriot-Watt Symposium, Nonlinear analysis and mechanics, Pitman Boston, Mass., IV, 1979.
[30]	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
[31]	Z. 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
[32]	S. Zhou, C. Mu, The properties of solutions for a generalized b-family equation with peakons, J. Nonlinear Sci., 23 (2013), 863–889. https://doi.org/10.1007/s00332-013-9171-8 doi: 10.1007/s00332-013-9171-8
[33]	Y. Zhou, On solutions to the Holm-Staley b-family of equations, Nonlinearity, 23 (2010), 369–381. https://doi.org/10.1088/0951-7715/23/2/008 doi: 10.1088/0951-7715/23/2/008

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