Normal form smoothing and spatial analyticity for the dispersion-generalized periodic KdV equation

Abstract

This paper investigates the persistence of spatial analyticity for solutions to the dispersion-generalized periodic Korteweg-de Vries equation $${\partial }_{t}u-{\partial }_x|D_x{|}^{\alpha }u+u{\partial }_{x}u=0,\alpha \ge 2.$$ For real-valued initial data $u_0\in G^{{\sigma }_0,s}(\mathbb{T})$, with ${\sigma }_0>0$ and $s\ge -\alpha /4$, we prove that the corresponding global solution satisfies $$u\in C([-T,T];G^{\sigma (T),s}(\mathbb{T})),\sigma (T)=\min \left\{{\sigma }_0,\frac{c}{T}\right\},$$ for every $T>0$. As a consequence, our result improves the lower bound for the radius of spatial analyticity established by Himonas, Kalisch, and Selberg (2017) for $2\le \alpha <4$, including the classical periodic KdV equation, and establishes a unified $T^{-1}$ lower bound for the analyticity radius for every $\alpha \ge 2$. The main novelties are twofold. First, we introduce a normal form reduction at the level of the Gevrey energy, which reveals an additional cancellation in the cubic Gevrey commutator and separates it into a non-accumulating boundary correction and a quartic remainder. Second, we establish a global-in-time periodic $L_{t,x}^4$ estimate for the non-polynomial phase $k|k{|}^{\alpha }$, valid for every real $\alpha \ge 2$. Together, these ingredients yield a refined almost conservation law and the $T^{-1}$ lower bound for the radius of spatial analyticity.