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Rigorous $C_1$ integration of dissipative PDEs

Jakub Banaśkiewicz

Abstract

We introduce a new $C^1$ algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with respect to initial data. As applications, we establish the existence of locally attracting periodic orbits for initial and boundary value problems for two non-autonomous dissipative PDEs: the Chafee-Infante equation and the Burgers equation with a fractional Laplacian.

Rigorous $C_1$ integration of dissipative PDEs

Abstract

We introduce a new algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with respect to initial data. As applications, we establish the existence of locally attracting periodic orbits for initial and boundary value problems for two non-autonomous dissipative PDEs: the Chafee-Infante equation and the Burgers equation with a fractional Laplacian.

Paper Structure

This paper contains 32 sections, 39 theorems, 206 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Algorithm for rigorous C1 integration of dissipative PDEs
  3. The non-autonomous Chafee--Infante problem
  4. The non-autonomous forced Burgers equation with fractional Laplacian
  5. Application of C1 algorithm in benchmark problems
  6. The further possible applications of C1 algorithm
  7. Structure of the article
  8. Notation
  9. Nonlinear differential equations in Banach spaces
  10. Abstract problem
  11. The non-autonomous Chafee–Infante equation
  12. The periodically forced Burgers equation with fractional Laplacian
  13. Variational equations
  14. Variational equation for non-autonomous Chafee–-Infante PDE
  15. Variational equation for Burgers equation with fractional Laplacian
  16. ...and 17 more sections

Key Result

Theorem 1.1

The Chafee--Infante equation eq:Chafee-InfateNonautonomous for parameters has the periodic orbit $u^*(t,.):\mathbb{R}\to C_0(0,\pi)$ with a period $1.$ Moreover there exist $\delta>0,$$D>0$ and $\kappa>0$ such that for every initial time $t_0\in\mathbb{R}$ and initial data $u^0\in C_0(0,\pi)$ satisfying $\left\|{u^*(t_0)-u^0}\right\|_{C_0(0,\pi)}\leq\delta$ there exists a $\blacktriangleleft$$\bl

Figures (3)

  • Figure 1: The approximation of evolution of Fourier modes for the periodic solution $u^*(t)$ from Theorem \ref{['th:Chaffe-InfanteAtractingOrbit1']} for $i=1,3,5,7.$
  • Figure 2: The approximation of evolution of Fourier modes for the periodic solution $u^*(t)$ from Theorem \ref{['th:Chaffe-InfanteAtractingOrbit2']} for $i=1,3,5,7.$
  • Figure 3: The approximation of evolution of Fourier modes for the periodic solution $u^*(t)$ from Theorem \ref{['th:Chaffe-InfanteAtractingOrbit3 ']} for $i=1,3,5,7.$

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 44 more