Dynamical Systems Weekend
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Abstract: Geometric optics studies the behavior of special solutions of PDEs in the form of oscillatory waves propagating along characteristics. In the context of incompressible fluid dynamics these solutions have been used to describe shortwave instabilities in the linear approximation and the essential spectrum of the linearized systems. In this talk we describe some of our recent attempts to develop the weakly and fully nonlinear versions of geometric optics for the Euler equation. We discuss applications to the problem of justification of the linearization method and L2-nonlinear instability of certain flows with uniform strain rates. A part of this work is joint with A. Shnirelman and Y. Latushkin.