Mathematics

The dynamics of homogeneous incompressible fluids are described by the Navier-Stokes equations or, in the case of zero viscosity, by the Euler equations. In both cases, the problem’s unknowns are the velocity and pressure fields. However, when studying the behavior of two or more incompressible fluids, the density can no longer be assumed constant, introducing an additional unknown to the system. This added complexity increases the instability of the problem, which manifests as anomalous energy dissipation. A notable example of such instability is the Rayleigh-Taylor instability, where anomalous energy dissipation arises due to two mechanisms: the irregularity of the velocity field and the mixing of densities. In this talk, we will analyze these two mechanisms within a critical regularity framework. A key focus of the analysis will be the role of incompressibility in driving these phenomena.