In this talk, I will present a computer-assisted approach for constructing solutions to initial value problems (IVPs) in parabolic PDEs on the n-torus. The method is fully spectral and relies on a Chebyshev–Fourier (time–space) expansion of the solution. Starting from a numerical approximation, we rigorously prove the existence of a true solution to the PDE in a neighborhood of this approximation, within a well-chosen Banach space. The proof is based on the Banach fixed-point theorem, whose assumptions are verified through a combination of analytical estimates and rigorous numerics.

As an application, I will demonstrate the resolution of an IVP for the 2D Navier–Stokes equations with a fixed nontrivial initial data. Furthermore, using a Gronwall-type argument, we rigorously construct a trapping region that encloses initial conditions evolving to zero as time tends to infinity. This allows us to prove global existence for the IVP corresponding to the selected initial data.