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Global existence for long wave Hopf unstable spatially extended systems with a conservation law

Nicole Gauss, Anna Logioti, Guido Schneider, Dominik Zimmermann

Abstract

We are interested in reaction-diffusion systems, with a conservation law, exhibiting a Hopf bifurcation at the spatial wave number $k = 0$. With the help of a multiple scaling perturbation ansatz a Ginzburg-Landau equation coupled to a scalar conservation law can be derived as an amplitude system for the approximate description of the dynamics of the original reaction-diffusion system near the first instability. We use the amplitude system to show the global existence of all solutions starting in a small neighborhood of the weakly unstable ground state for original systems posed on a large spatial interval with periodic boundary conditions.

Global existence for long wave Hopf unstable spatially extended systems with a conservation law

Abstract

We are interested in reaction-diffusion systems, with a conservation law, exhibiting a Hopf bifurcation at the spatial wave number . With the help of a multiple scaling perturbation ansatz a Ginzburg-Landau equation coupled to a scalar conservation law can be derived as an amplitude system for the approximate description of the dynamics of the original reaction-diffusion system near the first instability. We use the amplitude system to show the global existence of all solutions starting in a small neighborhood of the weakly unstable ground state for original systems posed on a large spatial interval with periodic boundary conditions.
Paper Structure (18 sections, 7 theorems, 158 equations, 1 figure)

This paper contains 18 sections, 7 theorems, 158 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Analysis of the amplitude system
  3. Some preparations
  4. The mode-filters
  5. The normal form transformation
  6. The Ginzburg-Landau manifold
  7. Derivation of the amplitude system
  8. Construction of a higher order approximation
  9. The global existence and uniqueness result
  10. Discussion
  11. The analytic set-up
  12. Proof of the attractivity theorem \ref{['thatt']}
  13. The first attractivity step
  14. The second attractivity step
  15. The attractivity induction steps
  16. ...and 3 more sections

Key Result

Theorem 1.2

Consider the amplitude system as1intro-as2intro with periodic boundary conditions ABper and assume that the coefficients $a_0 ,\ldots , b_1$ satisfy the condition (Coeff). Then for all $s \geq 0$ there exists a $C_R > 0$ such that for all $C_1 > 0$ there exists a $T_0 > 0$ such that to a given init

Figures (1)

  • Figure 1: The relevant spectral curves of the linearization around the trivial solution plotted as a function over the Fourier wave numbers for $\widetilde{\alpha} - \widetilde{\alpha}_c = \varepsilon^2 > 0$. The left panel shows the real part of the eigenvalue curves $\lambda_0$ (in blue), $\lambda_1$, and $\lambda_2$ (both in red), the right panel shows the imaginary part.

Theorems & Definitions (35)

  • Example 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 2.1
  • ...and 25 more