Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation

TL;DR

This paper establishes a lower bound on the decay rate of the spatial analyticity radius for solutions to the KdV equation, showing it cannot decay faster than |t|^{-4/3}, refining previous results.

Contribution

It improves the known decay rate bounds for the radius of spatial analyticity of KdV solutions, using almost conservation laws and bilinear estimates.

Findings

The radius of spatial analyticity cannot decay faster than |t|^{-4/3}.

The decay rate bound is sharper than previous results with an epsilon margin.

The proof employs almost conservation laws and dyadic bilinear estimates.

Abstract

It is shown that the uniform radius of spatial analyticity $\sigma(t)$ of solutions at time $t$ to the KdV equation cannot decay faster than $|t|^{-4/3}$ as $|t| \to \infty$ given initial data that is analytic with fixed radius $\sigma_0$. This improves a recent result of Selberg and Da Silva, where they proved a decay rate of $|t|^{-(4/3 + \varepsilon) }$ for arbitrarily small positive $\varepsilon$. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear $L^2$ estimates associated with the KdV equation.