In this talk, we are interested in the well-posedness theory for a system of PDEs describing the dynamics of an incompressible fluidwhich presents non-dissipative viscosity effects. At the level of the mathematical model, the non-dissipative nature of the viscosity is encoded by a skew-symmetric term, dubbed odd viscosity tensor. Differently from classical viscosity, the odd viscosity term does not provide any gain of regularity; on the contrary, it is responsible for a loss of derivatives in the a priori estimates.
In this talk, we show how to circumvent such a loss of derivatives and establish a well-posedness result in the framework of Sobolev (or, more generally, Besov) spaces of high enough regularity. The key is the identification of a suitable effective velocity in the model, which allows to recast the system as a system of transport equations.
The talk is based on joint works with Rafael Granero-Belinchón (Universidad de Cantabria), Stefano Scrobogna (Università degli Studi di Trieste) and Alexis Vasseur (University of Texas Austin).