In this talk, I will present the recent developments in the topic of the existence of solutions to the two-fluid systems. The compensated compactness technique of P.-L. Lions and E. Feireisl for single-component fluids has certain limitations, distinctly in the context of multi-component flow models. A particular example of such a model is the two-fluids Stokes system with a single velocity field and two densities, and with an algebraic pressure law closure. The first result that I will present is the existence of weak solutions for such a system, using the compactness criterion introduced recently by D. Bresch and P.-E. Jabin. I will also outline an innovative construction of solutions relying on the G. Crippa and C. DeLellis stability estimates for the transport equation. In the last part of my talk, I will relate to a couple of more recent results: the existence of solutions to the one-dimensional system, non-uniqueness of solutions to the inviscid system, and I will comment on issues around weak-strong uniqueness.
Ewelina Zatorska obtained her PhD in 2013 from the University of Warsaw. She then held two postdoctoral positions: at Ecole Polytechnique Paris, and as the Chapman Fellowship of Imperial College London, along with Assistant Professorship at the University of Warsaw, and three visiting positions: Invited Professor at the Universite Paris Dauphine, Visiting Professor of the Waseda University in Tokyo, and the Fellow of the Institut Mittag-Leffler in Stockholm. In 2017 Ewelina became a Lecturer at the University College London, and in 2020 she moved to Imperial College London, where she works until today as a Senior Lecturer in the Applied and Numerical Analysis.