2025/2026 Seminars

Abstract: We consider a model for an electrically charged liquid drop laying on a solid surface. The surface tension of the droplet is represented by the classical De Giorgi perimeter with a capillarity modification whereas the electric charge is modeled by the Riesz energy. We study the resulting functional under convexity constraint in dimension 2, focusing on existence and regularity of minimizers. In the end we show the validity of Young's law, describing the contact angle between the droplet and the supporting surface.  

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.