We are back! Our first talk is on the 18/11/2025 at 17:00, room 2AB40.

Speaker: Alessandro Violini (University of Basel)

Title: From the heat equation to Navier-Stokes: A dynamic approach to regularity

Abstract: We will study the evolution of the motion of a fluid surrounded by vacuum. This evolution is described by the two-dimensional incompressible Navier–Stokes equations.

The regularity of the motion depends both on the smoothness of the initial velocity field of the fluid and on the geometry of the region Ω initially occupied by it. In particular, we are interested in the case where Ω is a Lipschitz domain. We will show that, under a mild regularity assumption on the initial velocity field (belonging to a critical Besov space), the evolved region Ω_t remains Lipschitz.

The proof relies on Dynamic Interpolation, a time-dependent version of the classical Real Interpolation method for Banach spaces. To introduce this technique and the role of Besov spaces, we will first discuss the heat equation as a simpler toy model for the Navier–Stokes system.