The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space

Yan, Wei; Li, Yongsheng

doi:10.1016/j.anihpc.2014.01.003

Yan, Wei ; Li, Yongsheng

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 443-469.

Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2014.01.003

Classification : 35G25, 35L05, 35R25
Mots clés : 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{\textendash}Holm} system in critical {Besov} space},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {443--469},
     publisher = {Elsevier},
     volume = {32},
     number = {2},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.01.003},
     mrnumber = {3325245},
     zbl = {1336.35121},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.01.003/}
}

TY  - JOUR
AU  - Yan, Wei
AU  - Li, Yongsheng
TI  - The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 443
EP  - 469
VL  - 32
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2014.01.003/
DO  - 10.1016/j.anihpc.2014.01.003
LA  - en
ID  - AIHPC_2015__32_2_443_0
ER  -

%0 Journal Article
%A Yan, Wei
%A Li, Yongsheng
%T The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 443-469
%V 32
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2014.01.003/
%R 10.1016/j.anihpc.2014.01.003
%G en
%F 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, Tome 32 (2015) no. 2, pp. 443-469. doi : 10.1016/j.anihpc.2014.01.003. http://www.numdam.org/articles/10.1016/j.anihpc.2014.01.003/

Bibliographie
Cité par

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

[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 | EuDML | Numdam | MR | Zbl

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

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

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

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

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

[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

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

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

[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

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

[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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

Cité par Sources :