论文标题
半线的较高分散kDV方程
A higher dispersion KdV equation on the half-line
论文作者
论文摘要
使用FOKAS UNIFIED TRONSSION方法（UTM）研究了$ m $ ther订单分散量的初始符合条件问题（IBVP），在半线上具有粗糙的数据和解决方案，并在有限的波尔加因空间中使用粗糙的数据和解决方案进行了粗糙的数据和解决方案。因此，这项工作推进了fokas方法的实现，该方法在半线上较早用于KDV，并在经典的Hadamard空间中使用平滑的数据和解决方案，由时间连续的功能和空间变量中的Sobolev连续，与更一般的Bourgain空间框架的分配方程式有关，该方程在半线上进行了粗糙的数据。所需的空间和所需的估计值在线性级别出现，尤其是在线性纯IBVP的估计中，该线性纯IBVP具有强迫和初始数据为零但非零边界数据。使用由该地图引入的修改的波尔加因空间中的双线性估计的FOKAS解决方案定义的迭代图，该估计是在此图中建立了非线性IBVP的良好性，以使其在同一初始价值中的sobolev sobolev sobolev sobolev sobolev sobolev空间中的粗糙初始数据和边界数据都在此方程式上属于该方程的同一最佳空间。
The initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method (UTM). Thus, this work advances the implementation of the Fokas method, used earlier for the KdV on the half-line with smooth data and solution in the classical Hadamard space, consisting of function that are continuous in time and Sobolev in the spatial variable, to the more general Bourgain spaces framework of dispersive equations with rough data on the half-line. The spaces needed and the estimates required arise at the linear level and in particular in the estimation of the linear pure ibvp, which has forcing and initial data zero but non-zero boundary data. Using the iteration map defined by the Fokas solution formula of the forced linear ibvp in combination with the bilinear estimates in modified Bourgain spaces introduced by this map, well-posedness of the nonlinear ibvp is established for rough initial and boundary data belonging in Sobolev spaces of the same optimal regularity as in the case of the initial value problem for this equation on the whole line.
