On the cauchy problem for a coupled system of kdv equations : critical case

• Autor(a) principal: Panthee, Mahendra Prasad
• Data de Publicação: 2008
• Outros Autores: Scialom, Marcia
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/1822/11600
• topic: Cauchy problem; KdV equation; Well-posedness

We investigate some well-posedness issues for the initial value problem associated to the system \begin{equation*} \begin{cases} u_{t}+\partial_x^3u+\partial_x(u^2v^3) =0,\\ v_{t}+\partial_x^3v+\partial_x(u^3v^2)=0, \end{cases} \end{equation*} for given data in low order Sobolev spaces $H^s(\mathbb{R})\times H^s(\mathbb{R})$. We prove local and global well-posedness results utilizing the sharp smoothing estimates associated to the linear problem combined with the contraction mapping principle. For data with small Sobolev norm we obtain global solution whenever $s\geq 0$ by using global smoothing estimates. In particular, for data satisfying $\delta<\|(u_0, v_0)\|_{L^2\times L^2} < \|(S, S)\|_{L^2\times L^2}$, where $S$ is solitary wave solution, we get global solution whenever $s>3/4$. To prove this last result, we apply the splitting argument introduced by Bourgain [5] and further simplified by Fonseca, Linares and Ponce [6, 7].