Emmanuel Grenier

TL;DR: In this article, the authors use anisotropic spaces that enable them to prove existence theorems for less regular initial data than usual, and prove Strichartz-type, dispersive estimates in the whole space.

Abstract: In this chapter, we focus on multi-fluid models including compressible fluids in the isentropic or isothermal case. The main objective of these notes is to present at the level of beginners an introduction to mesoscopic multi-fluid models. The guideline is to show how we derive mathematically some multi-fluid systems starting from mono-fluid models and how we use the mathematical structures of multi-fluid PDEs to prove well-posedness (local strong and global weak solutions). We hope by this chapter to motivate young researchers to work on such a difficult topic important for applications in industry (turbulent mixing in D. Bresch is partially supported by the ANR13-BS01-0003-01 project DYFICOLTI. M. Hillairet is partially supported by the ANR13-BS01-0003-01 project DYFICOLTI. D. Bresch ( ) LAMA UMR 5127 CNRS Batiment le Chablais, Université de Savoie Mont-Blanc, Le Bourget du Lac, France e-mail: didier.bresch@univ-smb.fr; didier.bresch@univ-savoie.fr B. Desjardins Fondation Mathématique Jacques Hadamard, CMLA, ENS Cachan, CNRS and Modélisation Mesures et Applications S.A., Paris, France e-mail: Benoit.Desjardins@mines.org J.-M. Ghidaglia CMLA, ENS Cachan, CNRS, Université Paris-Saclay, Cachan, France e-mail: jmg@cmla.ens-cachan.fr E. Grenier Unité de Mathématiques Pures et Appliquées, ENS Lyon, Lyon Cedex 07, France e-mail: egrenier@umpa.ens-lyon.fr; emmanuel.grenier@ens-lyon.fr M. Hillairet Institut Montpelliérain Alexander Grothendiek, UMR5149 CNRS, Université de Montpellier, Montpellier, France e-mail: matthieu.hillairet@univ-montp2.fr © Springer International Publishing AG 2017 Y. Giga, A. Novotny (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, DOI 10.1007/978-3-319-10151-4_74-1 1