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The local criteria for blowup of the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 5, pp. 995-1012.

Nous étudions les critères d’explosion pour l’équation de Dullin-Gottwald-Holm et pour le système de Dullin-Gottwald-Holm à deux composantes. Nous établissons un nouveau critère d’explosion pour le cas général γ+c 0 α 2 0, impliquant des conditions locales en espace sur les données initiales.

We investigate wave breaking criteria for the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system. We establish a new blow-up criterion for the general case γ+c 0 α 2 0 involving local-in-space conditions on the initial data.

Publié le : 2016-11-13
DOI : https://doi.org/10.5802/afst.1519
@article{AFST_2016_6_25_5_995_0,
     author = {Duc-Trung Hoang},
     title = {The local criteria for blowup of the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {5},
     year = {2016},
     pages = {995-1012},
     doi = {10.5802/afst.1519},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2016_6_25_5_995_0/}
}
Duc-Trung Hoang. The local criteria for blowup of the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 5, pp. 995-1012. doi : 10.5802/afst.1519. https://afst.centre-mersenne.org/item/AFST_2016_6_25_5_995_0/

[1] Brandolese (L.).— Local-in-space criteria for blowup in shallow water and dispersive rod equations, Comm. Math. Phys, 330, p. 401-414 (2014).

[2] Brandolese (L.), Fernando Cortez (M.).— Blowup issues for a class of nonlinear dispersive wave equations J. Diff. Equ. 256 p. 3981-3998 (2014).

[3] Brandolese (L.), Fernando Cortez (M.).— On permanent and breaking waves in hyperelastic rods and rings, J. Funct. Anal. 266, 12 p. 6954-6987 (2014).

[4] Constantin (A.) and Escher (J.).— Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51, p. 475-504 (1998).

[5] Constantin (A.) and Molinet (L.).— Global weak solutions for a shallow water equation, Comm. Math. Phys., 211, p. 45-61 (2000).

[6] Constantin (A.) and Ivanov (R. I.).— On an integrable two-component Camassa-Holm shallow water system, Physics Letters A, vol. 372, no. 48, p. 7129-7132 (2008).

[7] Camassa (R.) and Holm (D. D.).— An integrable shallow water equation with peaked solitons, Physical Review Letters, vol. 71, no. 11, p. 1661-1664 (1993).

[8] Dullin (H.), Gottwald (G.), Holm (D.).— An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett. 87 p. 1945-1948 (2001).

[9] Escher (J.), Lechtenfeld (O.), and Yin (Z.).— Well-posedness and blowup phenomena for the 2-component Camassa-Holm equation, Discrete and Continuous Dynamical Systems A, vol. 19, no. 3, p. 493-513 (2007).

[10] Guo (Z.) and Zhou (Y.).— Wave breaking and persistence properties for the dispersive rod equation, SIAM Journal on Mathematical Analysis, vol. 40, no. 6, p. 2567-2580 (2009).

[11] Guo (F.), Gao (H.), and Liu (Y.).— On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system, J. of the London Math. Soc., vol. 86, no. 3, p. 810-834 (2012).

[12] Himonas (A. A.) and Misiolek (G.).— The Cauchy problem for an integrable shallow-water equation, Differential and Integral Equations, vol. 14, no. 7, p. 821-831 (2001).

[13] Kato (T.).— Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, vol. 4487 of Lecture Notes in Mathematics, p. 25-70, Springer, Berlin, Germany, (1975).

[14] Li (Y.) and Olver (P.).— Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Diff. Eq., 162, p. 27-63 (2000).

[15] Liu (Y.).— Global existence and blow-up solutions for a nonlinear shallow water equation., Math. Annalen, vol. 335, no. 3, p. 717-735 (2006).

[16] Lv (W.), Zhu (W.).— Wave breaking for the modified two-component Camassa-Holm system, Abstract and Applied Analysis, vol. 2014, Article ID 520218 (2014). 4pp.

[17] Mckean (H.P.).— Breakdown of a shallow water equation, Asian J. Math. 2(4) p. 867-874 (1998).

[18] Misiolek (G.).— Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12, p. 1080-1104 (2002).

[19] Shkoller (S.).— Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics, J. Funct. Anal., 160, p. 337-365 (1998).

[20] Tian (L.X.), Gui (G.), Liu (Y.).— On the Cauchy problem and the scattering problem for the Dullin-Gottwald-Holm equation, Comm. Math. Phys., vol. 257, no. 3, p. 667-701 (2005).

[21] Xin (Z.) and Zhang (P.).— On the weak solution to a shallow water equation, Comm. Pure Appl. Math., 53, p. 1411-1433 (2000).

[22] Zhai (P.), Guo (Z.), and Wang (W.).— Blow-up phenomena and persistence properties of solutions to the two-component DGH equation, Abstract and Applied Analysis, vol. 2013, Article ID 750315 (2013). 26pp.

[23] Zhou (Y.).— Blow-up of solutions to the DGH equation, Journal of Functional Analysis, vol. 250, no. 1, p. 227, 248 (2007).

[24] Zhu (M.), Xu (J.).— Wave-breaking phenomena and global solutions for periodic two-component Dullin-Gottwald-Holm Systems, Electronic Journal of Differential Equations, Vol. 2013, No. 44, p. 1-27 (2013).

[25] Zhu (M.) and Xu (J.).— On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system, Journal of Mathematical Analysis and Applications, vol. 391, no. 2, p. 415-428 (2012).

[26] Zhu (M.), Jin (L.), and Jiang (Z.).— A New blow-up criterion for the DGH equation, Abstract and Applied Analysis, vol. 2012, Article ID 515948 (2012). 10pp.