MIT PDE/Analysis Seminar

Abstract: The problem of Sobolev norm growth—namely, the transfer of energy from low to arbitrarily high modes—has been extensively studied in Hamiltonian systems with a deterministic structure, such as the cubic nonlinear Schrödinger equation. In this talk, I will present an extension of this problem to Hamiltonian systems dominated by random nonlinear interactions. I will introduce analytic solutions describing three types of energy cascades that lead either to unbounded growth or to finite-time blow-up of Sobolev norms. After, I will present numerical simulations demonstrating the rapid emergence of these dynamics from incoherent initial conditions. Taken together, these results demonstrate coherent energy cascades as robust mechanisms of energy transfer in some systems with random structures.

Abstract: The Radiative Vlasov-Maxwell equations model the radiative kinetics of collisionless relativistic plasma. In them the Lorentz force is modified by the addition of radiation reaction forces. The radiation forces produce damping of particle energy but these forces are not divergence-free in momentum space, which has an effect of concentration near zero momentum. We prove unconditional global regularity of solutions for a class of Radiative Vlasov-Maxwell equations with large initial data.

This is joint work with Peter Constantin.

Abstract: A celebrated conjecture due to Stieltjes (1890, in a letter to Hermite) states that no two Legendre polynomials can share a non-zero root. We discuss the spectral geometric origins of this problem and we prove that given any root of a Legendre polynomial there are at most finitely many Legendre polynomials sharing that root. We also give quantitative bounds. In particular, we confirm Stieltjes conjecture for several families of roots.

The talk is based on a joint work in progress with Borys Kadets and a joint work with Adi Weller Weiser.

Abstract: For a convex cocompact discrete subgroup of $SL(2,R)$, the resonance spectrum serves as the fundamental spectral invariant of the associated locally symmetric space. While the theory of Anosov subgroups has provided a successful generalization of convex cocompactness to higher-rank Lie groups, a corresponding spectral theory of resonances has been lacking.

In this talk, I will propose a definition of resonances for Anosov subgroups, building on microlocal methods for open hyperbolic dynamics introduced by Dyatlov and Guillarmou. Furthermore, I will explain how these analytic tools establish the meromorphic continuation of multivariate Poincaré series.

Abstract: Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure global well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In the first part of the talk, we will discuss the connection of Gibbs measures with the classical Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation. In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of Hamiltonian PDEs, including nonlinear Schrödinger equations with defocusing interactions. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari.

In the second part of the talk, we study (local) Gibbs measures for focusing nonlinear Schrödinger equations. These measures have to be localized by a truncation in the mass in one dimension and in the Wick-ordered (renormalized) mass in dimensions two and three. We show that local Gibbs measures correspond to suitably localized KMS states.

This is joint work with Zied Ammari and Andrew Rout.