Computer-Assisted Proofs in Fluid PDEs using Interval Arithmetic

Abstract: The formation of singularities in fluid PDEs is one of the central and most challenging problems in mathematical analysis. This talk describes a modern approach that combines classical PDE analysis with rigorous numerical computation, specifically interval arithmetic, to produce genuine mathematical proofs of self-similar blow-up to fluid equations.

Interval arithmetic provides a framework in which floating-point computations carry certified error bounds — transforming numerical evidence into rigorous mathematical fact. We describe how this philosophy is being deployed in the setting of nonlinear fluid PDEs, focusing on recent work on self-similar blow-up profiles for fluid equations, where high-precision numerical solutions serve as the starting point for computer-assisted proofs.