Energy of the finite-volume ground state of a random Schroedinger operator is studied in the limit as the volume increases. We relate its fluctuations to a classical probability problem---extreme statistics of IID random variables---and describe the detailed behavior of its distribution. Surprisingly, the distributions do not converge---presence of two scales in the system leads to a chaotic volume dependence. A possible application to a sharp estimate of the Lifshits tail will be mentioned. The work presented is done jointly with Michael Bishop.

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the immediate loss of smoothness and ill-posedness of solutions of the three-dimensional Euler equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface. This is very different from what has been proven for the two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture.

This is a joint work with Claude Bardos.

We study the asymptotic analysis of solutions of a sequence of PDE with gradient constraint arising in an option pricing model with transaction costs. The limit of these solutions satisfies a nonlinear Black Scholes type equation. An interesting feature of this
work is that the nonlinearity in the Black Scholes type equation comes about as a solution of a nonstandard eigenvalue PDE problem.