Asymptotic Behavior of the Einstein-Yang-Mills-Dilaton System for a Closed Friedmann-Lemaitre Universe

We study the coupled Einstein-Yang-Mills-Dilaton (EYMD) equations for a Fried\-mann-Le\-mai\-tre universe with constant curvature $k=1$. Our detailed analysis is restricted to the case where the dilaton potential and the cosmological constant vanish. Also assuming a static gauge field, we present analytical and numerical results on the behavior of solutions of the EYMD equations. For different values of the dilaton coupling constant we analyze the phase portrait for the time evolution of the dilaton field and give the behavior of the scale factor. It turns out that there are no inflationary stages in this model.Comment: 18 pages, Uuencoded gzip compressed tar file containing a latex file and 12 figures. The epsfig.sty is neede

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

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].Fundação para a Ciência e a Tecnologia (FCT) - POCI 2010/FEDER, bolsa SFRH/BPD/22018/2005Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP