Prerequisite: some background in PDE. The course will be mostly about stability and instability results for 2D Euler. It will be a mixture of rigorous proofs and of some formal arguments. We will discuss in particular : (1) the Rayleigh criterion and its generalizations; (2) the Orr-Sommerfeld equation; (3) Different notions of stability and instability; (4) some boundary layers. Grading: this course will be graded as a seminar course.

The first class will be on Monday September 9th.

The plan is to cover :

I) Classical existence results about Euler system.

II) Classical stability results including the Rayleigh criterion and its generalizations

III) Some boundary layers

Books :

Schmid, Peter J.; Henningson, Dan S. Stability and transition in shear flows. Applied Mathematical Sciences, 142. Springer-Verlag, New York, 2001. xiv+556 pp.

Chemin, Jean-Yves Perfect incompressible fluids. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications, 14. The Clarendon Press, Oxford University Press, New York, 1998.

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) W.~Orr. The stability or instability of steady motions of a perfect liquid and of a viscous liquid, {Part I}: a perfect liquid. { Proc. Royal Irish Acad. Sec. A: Math. Phys. Sci.}, 27:9--68, 1907.

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.