## On the Cauchy problem for the Camassa-Holm equation.(English)Zbl 0980.35150

From the introduction: We study local and global well-posedness with respect to the real-valued solutions of the Cauchy problem associated to the Camassa-Holm equation $u_t- u_{xxt}+ 3uu_x= 2u_x u_{xx}+ u_{xxx},\quad x\in \mathbb{R},\;t\in \mathbb{R},$ that was derived recently as a new model for dispersive shallow water waves.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text:

### References:

 [1] Camassa, R.; Holm, D.D., An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 11, 1661-1664, (1993) · Zbl 0972.35521 [2] Constantin, A., On the Cauchy problem for the periodic camassa – holm equation, J. differential equations, 141, 2, 218-235, (1997) · Zbl 0889.35022 [3] Constantin, A., The Hamiltonian structure of the camassa – holm equation, Exposition. math., 15, 1, 53-85, (1997) [4] Constantin, A.; Escher, J., Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. pure appl. math., 51, 5, 475-504, (1998) · Zbl 0934.35153 [5] A. Constantin, J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., to appear. · Zbl 0930.35133 [6] Iório, R.J., On the Cauchy problem for the benjamin – ono equation, Comm. partial differential equations, 11, 10, 1031-1081, (1986) · Zbl 0608.35030 [7] R.J. Iório, Jr., On Kato’s theory of quasilinear equations, II Jornada de EDP e Análise Numérica Em homenagem ao Professor Luis Adauto Justa Medeiros, Publicação do IMUFRJ, 1997, pp. 153-178. [8] R.J. Iório, Jr., N.W. Vieira, Introdução às Equações de Evolução não Lineares, XVIII Colóquio Brasileiro de Matemática, 1993. [9] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations. in: Spectral theory and differential equations, Proceedings of the Symposium Dundee, 1974, dedicated to Konrad Jrgens, Lecture Notes in Math, Vol. 448, Springer, Berlin, 1975, pp. 25-70. [10] Kato, T., On the korteweg – de-Vries equation, Manuscripta math., 28, 1-3, 89-99, (1979) · Zbl 0415.35070 [11] T. Kato, On the Cauchy problem for the (generalized) Korteweg – de Vries equation. Studies in Applied Mathematics, Adv. Math. Suppl. Stud., Vol. 8, Academic Press, New York, 1983, pp. 93-128. [12] T. Kato, Abstract evolution equations, linear and quasilinear, revisited. in: Functional analysis and Related Topics, 1991 Kyoto, Lecture Notes in Math, Vol. 1540, Springer, Berlin, 1993, pp. 103-125. [13] Kato, T.; Ponce, G., Commutator estimates and the Euler and navier – stokes equations, Comm. pure appl. math., 41, 7, 891-907, (1988) · Zbl 0671.35066 [14] Y.A. Li, P.J. Olver, Well-posedness and blow – up solutions for an integrable nonlinearly dispersive model wave equation, preprint IMA No, 1527, 1997. · Zbl 0958.35119 [15] Schiff, J., Zero curvature formulations of dual hierarchies, J. math. phys., 37, 4, 1928-1938, (1996) · Zbl 0863.35093 [16] Taylor, M.E., Partial differential equations III. nonlinear equations, Appl. math. sci., Vol. 117, (1996), Springer Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.