论文标题
KDV方程的未经过滤的低调集成器，解决方案以下$ {\ bf H^1} $
An unfiltered low-regularity integrator for the KdV equation with solutions below ${\bf H^1}$
论文作者
论文摘要
本文涉及对低规范解决方案$ h^1 $的KDV方程的新时间离散化的构建和分析。建立了新的谐波分析工具，包括针对指数相函数的新平均近似值，新的频率分解技术和KDV运算符的新三线性估计，用于在低规范性条件下构建和分析具有较高收敛订单的时间离散。此外，引入了新技术，以在低常规条件下建立时间离散量的稳定性估计，而无需在能量技术失败时使用过滤器。事实证明，在规律性条件$ u \ in c（[0，t]; h^γ）$的$γ\ in（0,1] $的规律性条件$ u \下，该方法在$ l^2 $中以$ l^2 $的收敛为收敛。
This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below $H^1$. New harmonic analysis tools, including new averaging approximations to the exponential phase functions, new frequency decomposition techniques, and new trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new techniques are introduced to establish stability estimates of time discretizations under low-regularity conditions without using filters when the energy techniques fail. The proposed method is proved to be convergent with order $γ$ (up to a logarithmic factor) in $L^2$ under the regularity condition $u\in C([0,T];H^γ)$ for $γ\in(0,1]$.
