On the local analyticity for the Euler equations

TL;DR

This paper introduces the diamond-analyticity framework to study the existence and uniqueness of Euler equation solutions with mixed boundary regularity, extending analyticity persistence results.

Contribution

It presents a novel diamond-analyticity approach and a nonstandard mollification technique to analyze analyticity near boundaries in Euler equations.

Findings

Extended analyticity persistence beyond traditional constraints

Introduced the diamond-analyticity framework for structured decay analysis

Demonstrated control over the evolution of solutions with boundary effects

Abstract

In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the introduction of the diamond-analyticity framework, which captures the spatial decay of the analyticity radius in a structured manner, improving upon uniform analyticity approaches. We employ the Leray projection and a nonstandard mollification technique to demonstrate that the quotient between the imaginary and real parts of the analyticity radius remains unrestricted, thus extending the analyticity persistence results beyond traditional constraints. Our methodology combines analytic-Sobolev estimates with an iterative scheme which is nonstandard in the Cauchy-Kowalevskaya framework, ensuring rigorous control over the evolution of the solution. These results contribute to a deeper understanding of the interplay between analyticity and boundary effects in fluid equations. They might have implications for the study of the inviscid limit of the Navier-Stokes equations and the role of complex singularities in fluid dynamics.