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The Amplitude Equation for the Space-Fractional Swift-Hohenberg Equation

Christian Kuehn, Sebastian Throm

Abstract

Non-local reaction-diffusion partial differential equations (PDEs) involving the fractional Laplacian have arisen in a wide variety of applications. One common tool to analyse the dynamics of classical local PDEs near instability is to derive local amplitude/modulation approximations, which provide local normal forms classifying a wide variety of pattern-formation phenomena. In this work, we study amplitude equations for the space-fractional Swift-Hohenberg equation. The Swift-Hohenberg equation is a basic model problem motivated by pattern formation in fluid dynamics and has served as one of the main PDEs to develop general techniques to derive amplitude equations. We prove that there exists near the first bifurcation point an approximation by a (real) Ginzburg-Landau equation. Interestingly, this Ginzburg-Landau equation is a local PDE, which provides a rigorous justification of the physical conjecture that suitably localized unstable modes can out-compete superdiffusion and re-localize a PDE near instability. Our main technical contributions are to provide a suitable function space setting for the approximation problem, and to then bound the residual between the original PDE and its amplitude equation.

The Amplitude Equation for the Space-Fractional Swift-Hohenberg Equation

Abstract

Non-local reaction-diffusion partial differential equations (PDEs) involving the fractional Laplacian have arisen in a wide variety of applications. One common tool to analyse the dynamics of classical local PDEs near instability is to derive local amplitude/modulation approximations, which provide local normal forms classifying a wide variety of pattern-formation phenomena. In this work, we study amplitude equations for the space-fractional Swift-Hohenberg equation. The Swift-Hohenberg equation is a basic model problem motivated by pattern formation in fluid dynamics and has served as one of the main PDEs to develop general techniques to derive amplitude equations. We prove that there exists near the first bifurcation point an approximation by a (real) Ginzburg-Landau equation. Interestingly, this Ginzburg-Landau equation is a local PDE, which provides a rigorous justification of the physical conjecture that suitably localized unstable modes can out-compete superdiffusion and re-localize a PDE near instability. Our main technical contributions are to provide a suitable function space setting for the approximation problem, and to then bound the residual between the original PDE and its amplitude equation.
Paper Structure (18 sections, 19 theorems, 99 equations)

This paper contains 18 sections, 19 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Fractional Laplacian and Bessel potential spaces
  3. Main result
  4. Abstract approximation result
  5. Strategy and preparation
  6. The function spaces
  7. The mode filters
  8. Semi-group estimates
  9. The approximation
  10. Estimating the non-linearities
  11. The fractional Swift-Hohenberg operator
  12. Approximation of the fractional Swift-Hohenberg operator
  13. Scaling properties
  14. The residuum
  15. Computing the residuum
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $\mathfrak{s}\in[1,2)$, $\theta\geq 1$ and let $A\in C^1([0,T_*],H^{\theta+3})$ be a solution to eq:GL. Let $\psi$ be given by eq:first:approx. There exist solutions $u$ to eq:SH:qc such that

Theorems & Definitions (37)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 27 more