The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, p. 443-469
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

In this paper, we are concerned with the Cauchy problem for the modified two-component Camassa–Holm system in the Besov space with data having critical regularity. The key elements in our paper are the real interpolations and logarithmic interpolation among inhomogeneous Besov space and Lemma 5.2.1 of [7] which is also called Osgood Lemma and the Fatou Lemma. The new ingredient that we introduce in this paper can be seen on pages 453–457.

DOI : https://doi.org/10.1016/j.anihpc.2014.01.003
Classification:  35G25,  35L05,  35R25
Keywords: Cauchy problem, Modified two-component Camassa–Holm system, Critical Besov space, Osgood Lemma
@article{AIHPC_2015__32_2_443_0,
author = {Yan, Wei and Li, Yongsheng},
title = {The Cauchy problem for the modified two-component Camassa--Holm system in critical Besov space},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {32},
number = {2},
year = {2015},
pages = {443-469},
doi = {10.1016/j.anihpc.2014.01.003},
zbl = {1336.35121},
mrnumber = {3325245},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2015__32_2_443_0}
}

Yan, Wei; Li, Yongsheng. The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, pp. 443-469. doi : 10.1016/j.anihpc.2014.01.003. http://www.numdam.org/item/AIHPC_2015__32_2_443_0/

[1] H. Bahouri, J.Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg (2011), http://dx.doi.org/10.1007/978-3-642-16830-7 | MR 2768550 | Zbl 1227.35004

[2] G. Blanco, On the Cauchy problem for the Camassa–Holm equation, Nonlinear Anal. 46 (2001), 309 -327 | MR 1851854 | Zbl 0980.35150

[3] A. Bressan, A. Constantin, Global conservative solutions of the Camassa–Holm equation, Arch. Ration. Mech. Anal. 183 (2007), 215 -239 | MR 2278406 | Zbl 1105.76013

[4] A. Bressan, A. Constantin, Global dissipative solutions of the Camassa–Holm equation, Appl. Anal. 5 (2007), 1 -27 | MR 2288533 | Zbl 1139.35378

[5] J. Chemin, Localization in Fourier space and Navier–Stokes, Phase Space Analysis of Partial Differential Equations, CRM Ser. , Scuola Norm. Sup., Pisa (2004), 53 -136 | MR 2144406 | Zbl 1081.35074

[6] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661 -1664 | MR 1234453 | Zbl 0972.35521

[7] J. Chemin, Perfect Incompressible Fluids, Oxford Lect. Ser. Math. Appl. vol. 14 , The Clarendon Press, Oxford University Press, New York (1998) | MR 1688875

[8] A. Constantin, The Hamiltonian structure of the Camassa–Holm equation, Expo. Math. 15 (1997), 53 -85 | MR 1438436

[9] A. Constantin, On the scattering problem for the Camassa–Holm equation, Proc. R. Soc. Lond. Ser. A 457 (2001), 953 -970 | MR 1875310 | Zbl 0999.35065

[10] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble) 50 (2000), 321 -362 | Numdam | MR 1775353 | Zbl 0944.35062

[11] A. Constantin, J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa 26 (1998), 303 -328 | Numdam | MR 1631589 | Zbl 0918.35005

[12] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998), 229 -243 | MR 1668586 | Zbl 0923.76025

[13] A. Constantin, L. Molinet, Global weak solutions for a shallow water equation, Commun. Math. Phys. 211 (1998), 45 -61 | MR 1757005 | Zbl 1002.35101

[14] A. Constantin, J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J. 47 (1998), 1527 -1545 | MR 1687106 | Zbl 0930.35133

[15] A. Constantin, W. Strauss, Stability of solitons, Commun. Pure Appl. Math. 53 (2000), 603 -610 | MR 1737505 | Zbl 1049.35149

[16] A. Constantin, W. Strauss, Stability of the Camassa–Holm solitons, J. Nonlinear Sci. 12 (2002), 415 -422 | MR 1915943 | Zbl 1022.35053

[17] A. Constantin, B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv. 78 (2003), 787 -804 | MR 2016696 | Zbl 1037.37032

[18] A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2007), 165 -186 | MR 2481064 | Zbl 1169.76010

[19] R. Danchin, A few remarks on the Camassa–Holm equation, Differ. Integral Equ. 14 (2001), 953 -988 | MR 1827098 | Zbl 1161.35329

[20] R. Danchin, A note on well-posedness for Camassa–Holm equation, J. Differ. Equ. 192 (2003), 429 -444 | MR 1990847 | Zbl 1048.35076

[21] R. Danchin, Fourier analysis method for PDEs, Lecture Notes, 14 November, 2005.

[22] R. Danchin, On the well-posedness of the incompressible density-dependent Euler equations in the ${L}^{p}$ framework, J. Differ. Equ. 248 (2010), 2130 -2170 | MR 2595717 | Zbl 1192.35137

[23] J. Escher, Z. Yin, Initial boundary value problems of the Camassa–Holm equation, Commun. Partial Differ. Equ. 33 (2008), 377 -395 | MR 2398234 | Zbl 1145.35031

[24] J. Escher, Z. Yin, Initial boundary value problems for nonlinear dispersive equations, J. Funct. Anal. 256 (2009), 479 -508 | MR 2476950 | Zbl 1193.35108

[25] A. Fokas, B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditary symmetries, Physica D 4 (1981), 47 -66 | MR 636470 | Zbl 1194.37114

[26] T.M. Fleet, Differential Analysis, Cambridge University Press (1980) | MR 561908

[27] C. Guan, Z. Yin, Global weak solutions for a modified two-component Camassa–Holm equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28 (2011), 623 -641 | Numdam | MR 2823888 | Zbl 1241.35159

[28] Z. Guo, M. Jiang, Z. Wang, G. Zheng, Global weak solutions to the Camassa–Holm equation, Discrete Contin. Dyn. Syst. 21 (2008), 883 -906 | MR 2399441 | Zbl 1157.35021

[29] Z. Guo, M. Zhu, Wave breaking for a modified two-component Camassa–Holm system, J. Differ. Equ. 252 (2012), 2759 -2770 | MR 2860639 | Zbl 1230.37093

[30] C. Guan, K. Karlsen, Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa–Holm equation, Proceedings of the 2008–2009 Special Year in Nonlinear Partial Differential Equations, Contemp. Math. vol. 526 , Amer. Math. Soc. (2010), 199 -220 | Zbl 1213.35133

[31] D. Henry, Compactly supported solutions of the Camassa–Holm equation, J. Nonlinear Math. Phys. 12 (2005), 342 -347 | MR 2160386 | Zbl 1086.35079

[32] A. Himonas, G. Misiolek, The Cauchy problem for an integrable shallow water equation, Differ. Integral Equ. 14 (2001), 821 -831 | MR 1828326 | Zbl 1009.35075

[33] D. Holm, L. Naraigh, C. Tronci, Singular solution of a modified two-component Camassa–Holm equation, Phys. Rev. E 79 (2009), 1 -13 | MR 2552212

[34] Z. Jiang, L. Ni, Y. Zhou, Wave-breaking of the Camassa–Holm equation, J. Nonlinear Sci. 22 (2012), 235 -245 | MR 2912327 | Zbl 1247.35104

[35] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations, Lect. Notes Math. vol. 448 , Springer-Verlag, Berlin (1975), 25 -70 | MR 407477

[36] 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. R. Soc. Lond. Ser. A 365 (2007), 2333 -2357 | MR 2329152 | Zbl 1152.37344

[37] J. Lenells, Stability of periodic peakons, Int. Math. Res. Not. 10 (2004), 485 -499 | MR 2039134 | Zbl 1075.35052

[38] J. Lenells, The correspondence between KdV and Camassa–Holm, Int. Math. Res. Not. 71 (2004), 3797 -3811 | MR 2104474 | Zbl 1082.35134

[39] J. Lenells, Travelling wave equations of the Camassa–Holm equation, J. Differ. Equ. 217 (2005), 393 -430 | MR 2168830 | Zbl 1082.35127

[40] 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 | MR 1741872 | Zbl 0958.35119

[41] H. Mckean, Breakdown of a shallow water equation, Asian J. Math. 2 (1998), 867 -874 | MR 1734131 | Zbl 0959.35140

[42] M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal. 145 (1998), 197 -214 | MR 1664597 | Zbl 0926.35123

[43] Z. Xin, P. Zhang, On the weak solutions to a shallow water equation, Commun. Pure Appl. Math. 53 (2000), 1411 -1433 | MR 1773414 | Zbl 1048.35092

[44] Z. Xin, P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Commun. Partial Differ. Equ. 27 (2002), 1815 -1844 | MR 1941659 | Zbl 1034.35115

[45] W. Yan, L. Tian, M. Zhu, Local well-posedness and blow-up phenomenon for a modified two-component Camassa–Holm system in Besov spaces, Int. J. Nonlinear Sci. 13 (2012), 99 -104 | MR 2904026 | Zbl 1260.35192