Some remarks on the radius of spatial analyticity for the Euler equations

TL;DR

This paper establishes lower bounds on the spatial analyticity radius of solutions to the Euler equations on a torus, using a new Sobolev space-based method, refining previous results on analyticity decay over time.

Contribution

It introduces a novel inductive Sobolev space approach to derive more precise bounds on the analyticity radius evolution for Euler equations.

Findings

Lower bounds for the analyticity radius over time.

Method aligns with previous results but improves precision.

Provides a new technique for analyzing analyticity in PDEs.

Abstract

We consider the Euler equations on $\mathbb{T}^d$ with analytic data and prove lower bounds for the radius of spatial analyticity $\epsilon(t)$ of the solution using a new method based on inductive estimates in standard Sobolev spaces. Our results are consistent with similar previous results proved by Kukavica and Vicol, but give a more precise dependence of $\epsilon(t)$ on the radius of analyticity of the initial datum.