Nous obtenons lʼexistence globale en temps de solutions faibles pour le problème de Cauchy dʼune équation modifiée Camassa–Holm à deux composantes. La solution faible globale est obtenue comme une limite de par approximation visqueuse. Les éléments clé dans notre analyse sont le théorème de Helly et certaines estimations a priori de supernorme dʼun seul côté et dʼintégrabilité dans lʼespace-temps des dérivées premières des solutions approchées.
We obtain the existence of global-in-time weak solutions for the Cauchy problem of a modified two-component Camassa–Holm equation. The global weak solution is obtained as a limit of viscous approximation. The key elements in our analysis are the Helly theorem and some a priori one-sided supernorm and space–time higher integrability estimates on the first-order derivatives of approximation solutions.
Mots-clés : A modified two-component Camassa–Holm equation, Well-posedness, Blow-up scenario, Strong solution, Global weak solution
@article{AIHPC_2011__28_4_623_0, author = {Guan, Chunxia and Yin, Zhaoyang}, title = {Global weak solutions for a modified two-component {Camassa{\textendash}Holm} equation}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {623--641}, publisher = {Elsevier}, volume = {28}, number = {4}, year = {2011}, doi = {10.1016/j.anihpc.2011.04.003}, zbl = {1241.35159}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.04.003/} }
TY - JOUR AU - Guan, Chunxia AU - Yin, Zhaoyang TI - Global weak solutions for a modified two-component Camassa–Holm equation JO - Annales de l'I.H.P. Analyse non linéaire PY - 2011 SP - 623 EP - 641 VL - 28 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2011.04.003/ DO - 10.1016/j.anihpc.2011.04.003 LA - en ID - AIHPC_2011__28_4_623_0 ER -
%0 Journal Article %A Guan, Chunxia %A Yin, Zhaoyang %T Global weak solutions for a modified two-component Camassa–Holm equation %J Annales de l'I.H.P. Analyse non linéaire %D 2011 %P 623-641 %V 28 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2011.04.003/ %R 10.1016/j.anihpc.2011.04.003 %G en %F AIHPC_2011__28_4_623_0
Guan, Chunxia; Yin, Zhaoyang. Global weak solutions for a modified two-component Camassa–Holm equation. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 623-641. doi : 10.1016/j.anihpc.2011.04.003. http://www.numdam.org/articles/10.1016/j.anihpc.2011.04.003/
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