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Local asymptotics for the nonlocal Swift-Hohenberg equation

Elisa Davoli, Christian Kuehn, Luca Scarpa, Lara Trussardi

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.

Local asymptotics for the nonlocal Swift-Hohenberg equation

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.

Paper Structure

This paper contains 13 sections, 4 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Swift-Hohenberg with one nonlocal contribution
  3. Setting of the problem
  4. Assumptions and main results
  5. Well-posedness of the nonlocal problem and nonlocal-to-local convergence
  6. Uniform estimates
  7. Passage to the limit
  8. Swift-Hohenberg with two nonlocal contributions
  9. Setting of the problem
  10. Assumptions and main results
  11. Well-posedness of the nonlocal problem and nonlocal-to-local convergence
  12. Uniform estimates
  13. Passage to the limit

Key Result

Theorem 2.1

For every $\varepsilon>0$ there exists a unique such that $u_\varepsilon(0)=u_{0,\varepsilon}$ and

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 3.1
  • Theorem 3.1
  • Theorem 3.2