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Recent advances about the rigorous integration of parabolic PDEs via fully spectral Fourier-Chebyshev expansions

Matthieu Cadiot, Jean-Philippe Lessard

TL;DR

This work develops a rigorous, computer-assisted framework for solving semilinear parabolic PDEs using fully spectral Fourier-Chebyshev expansions. By recasting the Cauchy problem as a zero of a Fourier-Chebyshev coefficient map $F$, it combines an explicit decay estimate for the linear inverse with a Newton-Kantorovich scheme and a carefully constructed approximate inverse to obtain existence with explicit error bounds and the ability to use larger time steps. The approach is then demonstrated on the 2D Navier–Stokes equations (including a global existence result for a nontrivial initial condition) and the Swift–Hohenberg equation, with explicit computable bounds and numerical verifications provided; the techniques are supported by detailed tail estimates and a rigorous analysis of the infinite-dimensional operators. Overall, the paper provides a robust, scalable method for guaranteed existence and error control in high-dimensional dissipative PDEs via fully spectral methods, accompanied by publicly available code.

Abstract

This paper presents a novel approach to rigorously solving initial value problems for semilinear parabolic partial differential equations (PDEs) using fully spectral Fourier-Chebyshev expansions. By reformulating the PDE as a system of nonlinear ordinary differential equations and leveraging Chebyshev series in time, we reduce the problem to a zero-finding task for Fourier-Chebyshev coefficients. A key theoretical contribution is the derivation of an explicit decay estimate for the inverse of the linear part of the PDE, enabling larger time steps. This allows the construction of an approximate inverse for the Fréchet derivative and the application of a Newton-Kantorovich theorem to establish solution existence within explicit error bounds. Building on prior work, our method is extended to more complex partial differential equations, including the 2D Navier-Stokes equations, for which we establish global existence of the solution of the IVP for a given nontrivial initial condition.

Recent advances about the rigorous integration of parabolic PDEs via fully spectral Fourier-Chebyshev expansions

TL;DR

This work develops a rigorous, computer-assisted framework for solving semilinear parabolic PDEs using fully spectral Fourier-Chebyshev expansions. By recasting the Cauchy problem as a zero of a Fourier-Chebyshev coefficient map , it combines an explicit decay estimate for the linear inverse with a Newton-Kantorovich scheme and a carefully constructed approximate inverse to obtain existence with explicit error bounds and the ability to use larger time steps. The approach is then demonstrated on the 2D Navier–Stokes equations (including a global existence result for a nontrivial initial condition) and the Swift–Hohenberg equation, with explicit computable bounds and numerical verifications provided; the techniques are supported by detailed tail estimates and a rigorous analysis of the infinite-dimensional operators. Overall, the paper provides a robust, scalable method for guaranteed existence and error control in high-dimensional dissipative PDEs via fully spectral methods, accompanied by publicly available code.

Abstract

This paper presents a novel approach to rigorously solving initial value problems for semilinear parabolic partial differential equations (PDEs) using fully spectral Fourier-Chebyshev expansions. By reformulating the PDE as a system of nonlinear ordinary differential equations and leveraging Chebyshev series in time, we reduce the problem to a zero-finding task for Fourier-Chebyshev coefficients. A key theoretical contribution is the derivation of an explicit decay estimate for the inverse of the linear part of the PDE, enabling larger time steps. This allows the construction of an approximate inverse for the Fréchet derivative and the application of a Newton-Kantorovich theorem to establish solution existence within explicit error bounds. Building on prior work, our method is extended to more complex partial differential equations, including the 2D Navier-Stokes equations, for which we establish global existence of the solution of the IVP for a given nontrivial initial condition.

Paper Structure

This paper contains 15 sections, 19 theorems, 236 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Set-up, Newton-Kantorovich bounds and Regularity
  3. A proof of existence using a Newton-Kantorovich approach
  4. Derivation of the bounds for the Newton-Kantorovich Theorem
  5. A posteriori regularity
  6. Computation of the functions $C_0 ~$ and $C_{1,n}$
  7. Computation for $\mu \in [0,\mu^*]$
  8. Computation for $\mu \in (\mu^*,\infty)$
  9. Analysis of the tridiagonal infinite matrix $T$
  10. Explicit estimates for the entries of $T(n)^{-1}$
  11. Computation of explicit estimates
  12. Applications
  13. The 2D Navier-Stokes equations
  14. Global Existence
  15. The Swift-Hohenberg PDE

Key Result

Theorem 2.2

Let $A$ be an injective linear operator such that $AF : X \to X$ is Fréchet differentiable. Let $Y, {Z}_1, r^*$ be non-negative constants and ${Z}_2 : [0,r^*] \to [0,\infty)$ be a non-negative function such that for all $0 \leq r \leq r^{*}$ If there exists $0 \leq r \leq r^*$ such that then there exists a unique zero of $F$ in $\overline{B_r(\overline{U})}$.

Figures (4)

  • Figure 1: The operator $\mathcal{L}_k$ and its approximate inverse operator $A_k$ (figure taken from jacek_integration_cheb)
  • Figure 2: Representation of the functions $C_0$ (left) and $C_1$ (right).
  • Figure 3: Numerical approximation for the 2D Navier-Stokes initial value problem at times $t=0$ (up-left), $t=0.96$ (up-right), $t=1.92$ (bottom-left) and $t=2.91$ (bottom-right).
  • Figure :

Theorems & Definitions (44)

  • Remark 2.1
  • Theorem 2.2: Newton-Kantorovich Theorem
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 34 more