# Interface propagation and mixing phenomena in fluids

**Time: 20.11.2019, 2-2:30pm**

**Room: 03.323 – Besprechungsraum** Lehrstuhl für Angewandte Analysis (Alexander von Humboldt-Professur)

**Interface propagation and mixing phenomena in fluids
Nicola De Nitti (Università degli Studi di Bari Aldo Moro)
**

This talk is devoted to presenting some recent works and research perspectives related to the

following topics:

- evolution of the free boundary for solutions to the thin-film equation;
- regularity propagation for the flow generated by a rough vector field.

The thin-film equation is a degenerate fourth-order parabolic equation that describes the surface-tension-driven evolution of the height of a viscous thin liquid film on a flat surface. Like the porous medium equation (its second-order analogue), the thin-film equation gives rise to a free boundary problem — the free boundary being the boundary of the liquid film. In the first part of the talk, we shall analyze the key ideas of some recent results obtained in collaboration with J. Fischer (IST Austria) on sharp criteria for the instantaneous front propagation in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is locally conserved by the thin-film equation. We shall also describe some open problems and works in progress on related issues.

In the second part of the talk, we shall consider the problem of propagation of regularity for the flow generated by a rough vector field. The classical Cauchy-Lipschitz theorem guarantees existence, uniqueness, and Lipschitz regularity of the flow associated with a Lipschitz continuous vector field. In the last few decades — in view of several applications to fluid dynamics and to the theory of conservation laws — much effort has been put into the study of ODEs driven by vector fields which are not necessarily Lipschitz continuous, but belong only to some class of weak differentiability. We shall present a regularity result for the flow associated with a nearly incompressible BV vector field that has been recently obtained in collaboration with S. Bianchini (SISSA, Italy). We shall conclude by presenting some open problems (including Bressan’s “mixing conjecture”) and also some issues and future directions of research related to the controllability of transport equations.