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

TL;DR

This paper introduces a rigorous spectral Fourier-Chebyshev method for solving semilinear parabolic PDEs, including the 2D Navier-Stokes equations, with explicit error bounds and proof of global existence.

Contribution

It develops a new spectral approach combining Fourier and Chebyshev expansions, with theoretical decay estimates and a Newton-Kantorovich framework for PDEs.

Findings

Explicit decay estimate for the inverse linear operator

Construction of an approximate inverse for the Fréchet derivative

Proof of global existence for 2D Navier-Stokes initial value problem

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 Fr\'echet 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.