论文标题
半线的Majda-Biello系统
The Majda-Biello system on the half-line
论文作者
论文摘要
Majda-Biello系统建模了Rossby Waves的相互作用。它由两个耦合的KDV方程组成，其中一个具有参数$α$作为其分散系数。这项工作与Robin，Neumann和Dirichlet边界数据一起在半线上研究了该系统。它表明，$ 0 <α<1 $或$ 1 <α<4 $ $所有这些问题对于Sobolev Spaces $ H^S $，$ S \ ge 0 $的初始数据都很好。对于$α= 1 $或$α> 4 $ $ a $ appophits的数据，如果$ s> -3/4 $，而对于Neumann和Robin数据，则依赖于数据中涉及的参数的符号。对于$α= 4 $适当的所有问题的适合度，$ s \ ge 3/4 $。罗宾和诺伊曼边界数据以$ h^{s/3} $为单位，而dirichlet边界数据则以$ h^{（s+1）/3} $为单位。这些与相应的线性系统的Cauchy问题的时间规律性一致。该证明是基于通过使用强制线性系统的FOKAS解决方案和耦合非线性建议的适当的双线性估计来得出的波尔加因空间中的线性估计。这些表明，通过FOKAS公式定义的迭代图是适当的解决方案空间中的收缩。此处获得的所有适合性结果都是最佳的。
The Majda-Biello system models the interaction of Rossby waves. It consists of two coupled KdV equations one of which has a parameter $α$ as coefficient of its dispersion. This work studies this system on the half line with Robin, Neumann, and Dirichlet boundary data. It shows that for $0<α<1$ or $1<α<4$ all these problems are well-posed for initial data in Sobolev spaces $H^s$, $s\ge 0$. For $α=1$ or $α>4$ well-posedness holds for Dirichlet data if $s>-3/4$, while for Neumann and Robin data it depends on the sign of the parameters involved in the data. For $α=4$ well-posedness of all problems holds for $s\ge 3/4$. The Robin and Neumann boundary data are in $H^{s/3}$ while the Dirichlet boundary data are in $H^{(s+1)/3}$. These are consistent with the time regularity of the Cauchy problem for the corresponding linear system. The proof is based on linear estimates in Bourgain spaces derived by utilizing the Fokas solution formula for the forced linear system, and appropriate bilinear estimates suggested by the coupled nonlinearities. These show that the iteration map defined via the Fokas formula is a contraction in appropriate solution spaces. All the well-posedness results obtained here are optimal.
