标题： 关于某些耦合KdV方程组的局部适定性 显示英文标题

标题： On The Local Well-Posedness for Some Systems of Coupled KdV Equations

摘要： 利用Kenig、Ponce和Vega发展的理论，我们证明在Sobolev空间$H^s(\mathbb{R}) \times H^{s}(\mathbb{R})$中，Hirota-Satsuma系统对于$3/4<s\le1$是局部适定的。 我们引入了一些Bourgain型空间$X_{s,b}^a$，用于$a\not =0$，$s,b \in \mathbb{R}$，以在$H^s(\mathbb{R})\times H^s(\mathbb{R})$中获得Gear-Grimshaw系统的局部适定性，对于$s>-3/4$，通过建立涉及两个Bourgain型空间$X_{s,b}^{-\alpha_-}$和$X_{s,b}^{-\alpha_+}$的新混合双线性估计，这些空间分别适用于$\partial_t+\alpha_-\partial_x^3$和$\partial_t+\alpha_+\partial_x^3$，其中$|\alpha_+|=|\alpha_-|\not = 0$。 显示英文摘要

摘要： Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota-Satsuma system is locally well-posed in Sobolev spaces $H^s(\mathbb{R}) \times H^{s}(\mathbb{R})$ for $3/4<s\le1$. We introduce some Bourgain-type spaces $X_{s,b}^a$ for $a\not =0$, $s,b \in \mathbb{R}$ to obtain local well-posedness for the Gear-Grimshaw system in $H^s(\mathbb{R})\times H^s(\mathbb{R})$ for $s>-3/4$, by establishing new mixed-bilinear estimates involving the two Bourgain-type spaces $X_{s,b}^{-\alpha_-}$ and $X_{s,b}^{-\alpha_+}$ adapted to $\partial_t+\alpha_-\partial_x^3$ and $\partial_t+\alpha_+\partial_x^3$ respectively, where $|\alpha_+|=|\alpha_-|\not = 0$.