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Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods

Matthieu Cadiot, Jean-Philippe Lessard, Jean-Christophe Nave

TL;DR

A numerical method to rigorously prove existence of strong solutions to a large class of semi-linear PDEs in a Hilbert space via computer-assisted proofs and introduces a finite-dimensional trace theorem from which to build smooth functions with support on a hypercube.

Abstract

In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space $H^{l}\subset H^{s}(\mathbb{R}^{m})$ ($s\geq1$) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in $H^{l}$ as well as bounded linear operators from $L^{2}$ to $H^{l}$. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in $H^{4}(\mathbb{R})$ as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.

Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods

TL;DR

A numerical method to rigorously prove existence of strong solutions to a large class of semi-linear PDEs in a Hilbert space via computer-assisted proofs and introduces a finite-dimensional trace theorem from which to build smooth functions with support on a hypercube.

Abstract

In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space () via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in as well as bounded linear operators from to . In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.
Paper Structure (31 sections, 31 theorems, 219 equations, 1 figure, 1 table)

This paper contains 31 sections, 31 theorems, 219 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Set-up of the problem
  3. Notations on Sobolev spaces
  4. Formulation of the problem
  5. Linearization operators
  6. Periodic Sobolev spaces
  7. Approximation of the inverse of $D\mathbb{F}(u_0)$
  8. Preliminary results
  9. Construction of the approximate inverse
  10. Upper bounds for $\|I_d - \mathbb{A}D\mathbb{F}(u_0)\|_l$ and $\|D\mathbb{F}(u_0)^{-1}\|_{2,l}$
  11. Computation of the bound $\mathcal{Z}_{u}$
  12. Existence of solutions via computer-assisted analysis
  13. Construction of the approximate solution $u_0 \in H^l_{\Omega_0}$
  14. Trace theorem for periodic functions
  15. Finite dimensional trace theorem
  16. ...and 16 more sections

Key Result

Lemma 2.4

Suppose that Assumptions ass:A(1) and ass : LinvG in L1 are satisfied. Then, For each $i \in \{2,\dots,N_{\mathbb{G}}\}$, let $\kappa_i >0$ satisfying Then, for all $u \in H^l$,

Figures (1)

  • Figure 1: Numerical approximation $u_0$ on $\Omega_0$.

Theorems & Definitions (79)

  • Remark 2.2
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • proof
  • Remark 2.9
  • ...and 69 more