Prerequisite: some background in PDE. The goal of the course is to study the inviscid limit of the free-boundary Navier-Stokes system and recover at the limit the free-boundary Euler system. This will allow us to link together two important problems in Ffuid Mmchanics, namely free boundary problems and boundary layers. Grading: this course will be graded as a seminar course.

The first class will be on Monday September 12th.

The plan is to cover the following 3 parts :

I) Boundary layers

II) Free boundary problems

III) Inviscid limit of the free boundary Navier-Stokes sytem

For the first part :

1/ Derivation of some boundary layers systems :

Prandtl boundary layers,

Ekman layers

Knudsen boundary layers

Boundary layers in homogenization, ...

2/ Existence resutls for some boundary layers systems

3/ Convergence resutls

Books :

1)O. A. Oleinik and V. N. Samokhin, Mathematical models in boundary layer theory, Applied Mathematics and Mathematical Computation, vol. 15, Chapman & Hall/CRC, Boca Raton, FL, 1999.

Articles :

1)L. Prandtl, Uber flussigkeits-bewegung bei sehr kleiner reibung., Actes du 3`eme Congr'es in- ternational dse Math'ematiciens, Heidelberg, Teubner, Leipzig, 1904, pp. 484–491.

2)Gérard-Varet, David; Dormy, Emmanuel On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc. 23 (2010), no. 2, 591–609.

3) David G\'erard-Varet and Nader Masmoudi, Homogenization in polygonal domains (to appear in JEMS)

4) Maria Carmela Lombardo, Marco Cannone, and Marco Sammartino, Well-posedness of the boundary layer equations, SIAM J. Math. Anal. 35 (2003), no. 4, 987–1004 (electronic).

5) Lan Hong and John K. Hunter, Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations, Commun. Math. Sci. 1 (2003), no. 2, 293–316.

6) Iftimie, Dragoş; Sueur, Franck Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal. 199 (2011), no. 1, 145–175,

7)Grenier E and Masmoudi N 1997 Ekman layers of rotating fluids, the case of well prepared initial data Commun. Partial Diff. Eqns 22 953–75