Local asymptotics for the nonlocal Swift-Hohenberg equation

TL;DR

This paper investigates the transition from nonlocal to local formulations of the Swift-Hohenberg equation, establishing well-posedness and asymptotic behavior crucial for understanding pattern formation in PDEs.

Contribution

It proves well-posedness and analyzes nonlocal-to-local asymptotics for the Swift-Hohenberg equation with multiple nonlocal terms under Neumann boundary conditions.

Findings

Established well-posedness of the nonlocal Swift-Hohenberg equation.

Derived energy estimates for nonlocal-to-local asymptotics.

Analyzed effects of one and two nonlocal contributions.

Abstract

The nonlocal-to-local asymptotics investigation for evolutionary problems is a central topic both in the theory of PDEs and in functional analysis. More recently, it became the main core of the mathematical analysis of phase-separation models. In this paper we focus on the Swift-Hohenberg equations which are key benchmark models in pattern formation problems and amplitude equations. We prove well-posedness of the nonlocal Swift-Hohenberg equation, and study the nonlocal-to-local asymptotics with one and two nonlocal contributions under homogeneous Neumann boundary conditions using suitable energy estimates on the nonlocal problems.