Georgia Institute of Technology College of Sciences

In this talk, we will discuss dispersive and local decay estimates for a class of non-self-adjoint Schrödinger operators that naturally arise from the linearization of nonlinear Schrödinger equations around a solitary wave. We review the spectral properties of these linearized operators, and discuss how threshold resonances may appear in their spectrum. In the presence of threshold resonances, it will be shown that the slow local decay rate can be pinned down to a finite rank operator corresponding to the threshold resonances. We will also discuss examples of non-self-adjoint operators that arise from linearizing around solitons in other contexts. 

It is well-known that a spacetime which expands sufficiently fast can stabilize the fluid for relativistic/Einstein-fluid systems. One may wonder whether the expansion of the fluid, instead of the background spacetime geometry, is also able to achieve a similar stabilizing effect. As an attempt to address this question, we consider the free boundary relativistic Euler equations in Minkowski background M1+3 equipped with a physical vacuum boundary, which models the motion of relativistic gas. For the class of isentropic, barotropic, and polytropic gas, we construct an open class of initial data which launch future-global solutions. Such solutions are spherically symmetric, have small initial density, and expand asymptotically linearly in time. In particular, the asymptotic rate of expansion is allowed to be arbitrarily close to the speed of light. Therefore, our main result is far from a perturbation of existing results concerning the classical Euler counterparts. This is joint work with Marcelo Disconzi and Chenyun Luo.

The Reissner–Nordström spacetime models a spherically symmetric and time-independent charged black hole in general relativity. The Cauchy horizon in the interior of such a black hole is subject to an infinite blueshift instability. In 1989, Poisson and Israel discovered a nonlinear manifestation of this instability in the spherically symmetric setting called "mass inflation," where the Hawking mass becomes identically infinite at the Cauchy horizon. 

We complete the first proof of mass inflation for a wave-type matter model, namely the spherically symmetric Einstein–Maxwell–scalar field system. This result follows from a large-data decay result for the scalar field in the black hole exterior combined with works of Dafermos, Luk–Oh, and Luk–Oh–Shlapentokh-Rothman.

 In this talk we present a perturbative proof of the asymptotic stability of the solitary wave solutions for the 1D focusing cubic Schrödinger equation under small perturbations in weighted Sobolev spaces. The strategy of our proof is based on the space-time resonances approach based on the distorted Fourier transform and modulation techniques to capture the asymptotic behavior of the solution. A major difficulty throughout the nonlinear analysis is the slow local decay of the radiation term caused by the threshold resonances in the spectrum of the linearized operator around the solitary wave. The presence of favorable null structures in the quadratic terms mitigates this problem through the use of normal form transformations.