Schedule, Notes/Handouts & Videos

The evolution of fluids is known (eg via experiments and numerical studies) to occur on multiple timescales. We discuss how this can be analyzed rigorously in two fundamental models of fluids: the 1D Burgers equation and the 2D Navier-Stokes equation. For Burgers equation, we provide a complete geometric explanation involving invariant manifolds in the phase space of the evolution. For Navier-Stokes on the 2D torus, we discuss two complementary approaches. The first involves the theory of hypocoercive operators, and the second involves invariant manifolds and geometric singular perturbation theory in the Fourier phase space.

In this talk, we discuss the connection between various diffusion processes and Gibbs measures for Hamiltonian PDEs. We analyze various examples of this connection, and discuss some recent results.

In this talk we discuss new negative energy blowup results for the Hartree hierarchy, an infinitely coupled system of PDEs arising from the study of many-body quantum mechanics.  We obtain results both with and without an assumption of finite variance on the initial data.  The key tools involved are virial identities for the Hartree hierarchy, together with localized variants of these identities.  The most delicate case of the analysis is the proof without finite variance -- here, we use a suitable quantum de Finetti theorem and a carefully chosen truncation lemma allowing for the control of additional terms appearing from the localization procedure.

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients, and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman-Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients, and  the a priori regularity of the weak solutions. This is joint work with Charles Epstein

We study the focusing nonlinear Schrodinger equation with finite energy and finite variance initial data. While considering the mass-supercritical regime we investigate solutions above the energy (or mass-energy) threshold, i.e., when the nergy of the solution exceeds the energy of the so-called ground state. We extend the known scattering versus blow-up dichotomy above the energy threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, characterizing invariant sets of solutions (with either scattering or blow-up in finite time behavior) possibly with arbitrary large mass and energy. We investigate other dispersive equations in a similar manner.

We propose a stochastic model associated with inhomogeneous scaling of left invariant metrics on a Lie group. We discuss reduction to slow variable, convergence of singular perturbation on manifolds and classification of limiting operators.

Moderator: Sevetlana