This summer school focuses on the most modern aspects of the mathematical analysis of Partial Differential Equations in fluid mechanics. Its goal is a unified presentation of recently developed mathematical tools for the study of three fundamental problems in fluid dynamics:

• the behavior of anisotropic flows (boundary layers, shallow water flows)
• the dynamics of vortex structures
• the evolution of floating structures on waves.

• either a lack of regularity, which manifests itself for vorticity filaments (described by Dirac masses along curves), or at the contact line between a fluid and a floating structure ;
• or a singular dependence of the solution on a parameter, typically the Reynolds number (as in boundary layers).

The format will consists of eight lecture series, intended for a wide audience (Master and Ph.D. students), and more specialized talks, presenting some of the most recent developements in the field.

 A poster session will be organised and will remain on display during the two weeks.

The conference will take place at the Institut Fourier, in the campus of the Université Grenoble Alpes.

Week 1 (June 5-9, 2023)

1. Helena Nussenzveig Lopes (Federal University of Rio de Janeiro) "An introduction to mathematical analysis of incompressible fluid flow"
2. Thomas Alazard (CNRS, Ecole Normale Supérieure Paris-Saclay) "Introduction to the Theory of Water Waves"
   
3. Maria Colombo (Ecole Polytechnique Fédérale de Lausanne) "Instability and non-uniqueness for the Euler and Navier-Stokes equations"
   
4. Antoine Venaille (CNRS, Ecole Normale Supérieure Lyon) "Introduction to geophysical flows"
    

Week 2 (June 12-16, 2023)

1. Geoffrey Beck (INRIA, Rennes) and David Lannes (CNRS, université de Bordeaux) "Wave-structure interactions"
   
2. Michele Coti Zelati (Imperial College, London) "Enhanced Dissipation Timescales and Mixing Rates"
3. Tarek Elgindi (Duke University) "Singularity formation in the incompressible Euler equation in finite and infinite time"
   
4. Emmanuel Grenier (Ecole Normale Supérieure Lyon) and Toan Nguyen (Penn State University) "Instabilities in Boundary Layers"

• Ángel Castro (Instituto de Ciencias Matemáticas, Madrid) "Traveling waves near shear flows"
• Guido Cavallaro (Sapienza Università di Roma) "Concentrated vortex rings for Euler and Navier-Stokes equations"
• Noemi David (Université Claude Bernard Lyon 1) "On the incompressible limit for porous medium models of tumor growth"
• Michele Dolce (Ecole Polytechnique Fédérale de Lausanne) "On maximally mixed equilibria of two-dimensional perfect fluid"
• Marie Farge (CNRS-INSMI, LMD-ENS) "How to analyze, model and compute turbulent flows using wavelets?"
• Mei Ming (Yunnan University, China) "Water waves with contact angles and surface tension"
• Monica Musso (University of Bath, UK) "Long time behavior for vortex dynamics in the 2 dimensional Euler equations"

Thierry Gallay, David Gérard-Varet, Christophe Lacave, David Lannes and Evelyne Miot