Uniqueness of global conservative weak solutions for two-component higher-order shallow water system

Huijun He; Chunxia Guan; Zhaoyang Yin

doi:10.3934/dcdsb.2024097

Article Contents

2025, Volume 30, Issue 2: 474-495. Doi: 10.3934/dcdsb.2024097

This issue Previous Article Minimization with differential inequality and equality constraints applied to complete Lyapunov functions Next Article Spectral analysis and exponential stability of a generalized fractional Moore-Gibson-Thompson equation

Uniqueness of global conservative weak solutions for two-component higher-order shallow water system

1.
School of Mathematics and Statistics, Guangdong University of Technology, China
2.
School of Science, Shenzhen Campus of Sun Yat-sen University, China

^*Corresponding author: Chunxia Guan
^*Corresponding author: Chunxia Guan

Received: December 2023

Revised: July 2024

Early access: July 23, 2024

Published online: July 23, 2024

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We investigate the uniqueness of global-in-time conservative weak solutions for the generalized two-component higher-order shallow water system on $ \mathbb R $. The proof strategy relies on characteristics. Given a conservative weak solution, we introduce the characteristic equation to uniquely determine a characteristic curve through each initial point, transforming the Eulerian coordinate into a Lagrangian coordinate. We demonstrate that the Cauchy problem of the generalized two-component higher-order Camassa-Holm system, equipped with initial data $ z_0 = (u_0, \rho_0)\in (H^2(\mathbb R)\cap W^{2, p}(\mathbb R))\times(L^2(\mathbb R)\cap L^\infty(\mathbb R)) $, has a unique global conservative weak solution.

Keywords:

Generalized two-component higher-order Camassa-Holm system,
Characteristics,
Lagrangian coordinates,
Global conservative weak solution,
uniqueness.

Mathematics Subject Classification: Primary: 35G55, 35B35, 35Q53.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and a theorem of Stieltjes, Inverse Problems, 15 (1999), 1-4. doi: 10.1088/0266-5611/15/1/001.
[2]	A. Bressan, G. Chen and Q. Zhang, Uniqueness of Conservative solutions to the Camassa-Holm equation via characteristics, Discrete Contin. Dyn. Syst., 35 (2015), 25-42. doi: 10.3934/dcds.2015.35.25.
[3]	A. Bressan, G. Chen and Q. Zhang, Unique solutions to a variational wave equation, Arch. Rat. Mech. Anal., 217 (2015), 1069-1101. doi: 10.1007/s00205-015-0849-y.
[4]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.
[5]	A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.
[6]	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.
[7]	R. Camassa, D. 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.
[8]	R. Chen and S. Zhou, Well-posedness and persistence properties for two-component higher order Camassa-Holm systems with fractional inertia operator, Nonlinear Anal.: RWA, 33 (2017), 121-138. doi: 10.1016/j.nonrwa.2016.06.003.
[9]	G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2006), 1044-1069. doi: 10.1137/040616711.
[10]	G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Differential Equations, 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014.
[11]	A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85.
[12]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.
[13]	A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.
[14]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.
[15]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328.
[16]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.
[17]	A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.
[18]	A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Uni. Math. J., 47 (1998), 1527-1546. doi: 10.1512/iumj.1998.47.1466.
[19]	A. Constantin and J. Escher, On the blow-up rate and the blow-up of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.
[20]	A. Constantin and J. Escher, Particles trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.
[21]	A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.
[22]	A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res., 40 (2008), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004.
[23]	A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.
[24]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.
[25]	A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.
[26]	A. Constantin and W. A. 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.
[27]	A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x.
[28]	H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373.
[29]	R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988. doi: 10.57262/die/1356123175.
[30]	H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 194501-194504. doi: 10.1103/PhysRevLett.87.194501.
[31]	J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. A, 19 (2007), 493-513. doi: 10.3934/dcds.2007.19.493.
[32]	J. Escher and T. Lyons, Two-component higher order Camassa-Holm systems with fractionalinertia operator: A geometric approach, J. Geom. Mech., 7 (2015), 281-293. doi: 10.3934/jgm.2015.7.281.
[33]	J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010.
[34]	A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X.
[35]	C. Guan, H. He and Z. Yin, Global weak solutions for a generalized two-component shallow water system with higher-order inertia operators, work in progress.
[36]	C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002.
[37]	C. Guan and Z. Yin, Global weak solutions for a two-component Camassa-Holm shallow water system, J. Funct.Anal., 260 (2011), 1132-1154. doi: 10.1016/j.jfa.2010.11.015.
[38]	Z. Guo, X. Liu, L. Molinet and Z. Yin, Ill-posedness of the Camassa-Holm and related equations in the critical space, J. Differential Equations, 266 (2019), 1698-1707. doi: 10.1016/j.jde.2018.08.013.
[39]	H. He and Z. Yin, On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators, Discrete Contin. Dyn. Syst., 37 (2017), 1509-1537. doi: 10.3934/dcds.2017062.
[40]	H. He and Z. Yin, The global Gevrey regularity and analyticity of a two-component shallow water system with higher-order inertia operators, J. Differential Equations, 267 (2019), 2531-2559. doi: 10.1016/j.jde.2019.03.022.
[41]	H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation - A Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674.
[42]	R. I. Ivanov, Water waves and integrability, Philos. Trans. Roy. Soc. London A, 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.
[43]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid. Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.
[44]	B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philos. Trans. Roy. Soc. Lond. A, 365 (2007), 2333-2357. doi: 10.1098/rsta.2007.2012.
[45]	M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, Tsunami and Nonlinear Waves, Springer, Berlin, 2007, 31-49. doi: 10.1007/978-3-540-71256-5_2.
[46]	J. Li and Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143. doi: 10.1016/j.jde.2016.08.031.
[47]	M. Li and Q. Zhang, Uniqueness of Conservative solutions to the two-component Camassa-Holm system via characteristics, Commun. Math. Sci., 14 (2016), 1581-1597. doi: 10.4310/CMS.2016.v14.n6.a6.
[48]	Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683.
[49]	G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X.
[50]	W. Tan and Z. Yin, Global conservative solutions of a modified two-component Camassa-Holm shallow water system, J. Differential Equations, 251 (2011), 3558-3582. doi: 10.1016/j.jde.2011.08.010.
[51]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.
[52]	G. B. Whitham, Linear and Nonlinear Waves, J. Wiley & Sons, New York, 1980.
[53]	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.
[54]	Z. Yin, Well-posedness, global existence and blowup phenomena for an integrable shallow water equation, Discrete Contin. Dyn. Syst. Ser. A, 10 (2004), 393-411. doi: 10.3934/dcds.2004.11.393.
[55]	S. Zhou, Well-posedness, blow-up phenomena and analyticity for a two-component higher order Camassa–Holm system, Mathematische Nachrichten, 291 (2018), 1595-1619. doi: 10.1002/mana.201600469.