# zbMATH — the first resource for mathematics

The global weak solution for a generalized Camassa-Holm equation. (English) Zbl 1282.35336
Summary: A nonlinear generalization of the famous Camassa-Holm model is investigated. Provided that initial value $$u_0 \in H^s(\mathbb{R})(1 \leq s \leq 3/2)$$ and $$(1 - \partial^2_x)u_0$$ satisfies an associated sign condition, it is shown that there exists a unique global weak solution to the equation in space $$u(t, x) \in L^2([0, +\infty), H^s(\mathbb{R}))$$ in the sense of distribution, and $$u_x \in L^\infty([0, +\infty) \times \mathbb{R})$$.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35D30 Weak solutions to PDEs
Full Text:
##### References:
 [1] Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Physical Review Letters, 71, 11, 1661-1664, (1993) · Zbl 0972.35521 [2] Johnson, R. S., Camassa-Holm, Korteweg-de Vries and related models for water waves, Journal of Fluid Mechanics, 455, 1, 63-82, (2002) · Zbl 1037.76006 [3] Johnson, R. S., On solutions of the Camassa-Holm equation, Proceedings of the Royal Society A, 459, 2035, 1687-1708, (2003) · Zbl 1039.76006 [4] Fokas, A.; Fuchssteiner, B., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4, 1, 47-66, (1981) · Zbl 1194.37114 [5] Constantin, A.; Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Archive for Rational Mechanics and Analysis, 192, 1, 165-186, (2009) · Zbl 1169.76010 [6] Constantin, A., On the scattering problem for the Camassa-Holm equation, Proceedings of the Royal Society A, 457, 2008, 953-970, (2001) · Zbl 0999.35065 [7] Lenells, J., Conservation laws of the Camassa-Holm equation, Journal of Physics A, 38, 4, 869-880, (2005) · Zbl 1076.35100 [8] McKean, H. P., Fredholm determinants and the Camassa-Holm hierarchy, Communications on Pure and Applied Mathematics, 56, 5, 638-680, (2003) · Zbl 1047.37047 [9] Constantin, A.; Escher, J., Global existence and blow-up for a shallow water equation, Annali della Scuola Normale Superiore di Pisa, 26, 2, 303-328, (1998) · Zbl 0918.35005 [10] Constantin, A., Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Annales de l’Institut Fourier, 50, 2, 321-362, (2000) · Zbl 0944.00010 [11] Constantin, A., On the inverse spectral problem for the Camassa-Holm equation, Journal of Functional Analysis, 155, 2, 352-363, (1998) · Zbl 0907.35009 [12] Constantin, A.; Escher, J., Global weak solutions for a shallow water equation, Indiana University Mathematics Journal, 47, 4, 1527-1545, (1998) · Zbl 0930.35133 [13] Constantin, A.; Kolev, B., Geodesic flow on the diffeomorphism group of the circle, Commentarii Mathematici Helvetici, 78, 4, 787-804, (2003) · Zbl 1037.37032 [14] Constantin, A.; Kappeler, T.; Kolev, B.; Topalov, P., On geodesic exponential maps of the Virasoro group, Annals of Global Analysis and Geometry, 31, 2, 155-180, (2007) · Zbl 1121.35111 [15] Lai, S. Y.; Wu, Y. H., The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation, Journal of Differential Equations, 248, 8, 2038-2063, (2010) · Zbl 1187.35179 [16] Lai, S. Y.; Wu, Y. H., A model containing both the Camassa-Holm and Degasperis-Procesi equations, Journal of Mathematical Analysis and Applications, 374, 2, 458-469, (2011) · Zbl 1202.35231 [17] Lai, S. Y.; Wu, Y. H., Existence of weak solutions in lower order Sobolev space for a Camassa-Holm-type equation, Journal of Physics A, 43, 9, (2010) · Zbl 1190.35060 [18] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Archive for Rational Mechanics and Analysis, 183, 2, 215-239, (2007) · Zbl 1105.76013 [19] Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5, 1, 1-27, (2007) · Zbl 1139.35378 [20] Li, N.; Lai, S. Y.; Li, S.; Wu, M., The local and global existence of solutions for a generalized Camassa-Holm equation, Abstract and Applied Analysis, 2012, (2012) · Zbl 1237.35139 [21] Kolev, B., Poisson brackets in hydrodynamics, Discrete and Continuous Dynamical Systems A, 19, 3, 555-574, (2007) · Zbl 1139.53040
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.