Category Archives: Fall 2023

Low regularity well-posedness for the surface quasi-geostrophic front equation

The HADES seminar on Tuesday, September 19th will be at 3:30pm in Room 740.

Speaker: Ovidiu-Neculai Avadanei

Abstract: We consider the well-posedness of the generalized surface quasi-geostrophic (gSQG) front equation. In the present paper, by making use of the null structure of the equation, we carry out a paradifferential normal form analysis in order to obtain balanced energy estimates, which allows us to prove the low regularity local well-posedness of the g-SQG front equation in the non-periodic case at a low level of regularity (in the SQG case, it is only one half of a derivative above scaling). In addition, we establish global well-posedness theory for small and localized rough initial data, as well as modified scattering, by using the testing by wave packet approach of Ifrim-Tataru.

This is joint work with Albert Ai.

Convergence of Lindblad Dynamics towards Fokker-Planck Equations beyond Ehrenfest time

The HADES seminar on Tuesday, September 12th will be at 3:30pm in Room 740.

Speaker: Zhen Huang

Abstract: The goal of this talk is to introduce the topic of semi-classical analysis of open quantum systems to the audience.
Semi-classical analysis of closed quantum systems is a very well-established topic (for example, see Zworski’s book). However, rigorous analytical studies of open quantum systems in the semi-classical regimes are rarely done so far. This is partly because open quantum dynamics often do not have properties as nice as Schrodinger equations. The lack of analytic results also hinders the design and analysis of numerical algorithms.
Quantum-classical correspondence in the Schrodinger equation is well known to hold for O(log(1/h)) time scale (h is the non-dimensionalized Planck constant). We will discuss a very recent work that addresses the quantum-classical correspondence for the simplest open quantum system model (which is still complicated), i.e. Lindblad dynamics. We present a rigorous proof that a classical description is valid for O(1/sqrt(h)) time, which is much longer than the Ehrenfest timescale. We will also discuss several open questions along this line, and possible generalizations to more complicated open quantum systems.

Illposedness for dispersive equations: Degenerate dispersion and Takeuchi-Mizohata condition

The HADES seminar on Tuesday, September 5th will be at 3:30pm in Room 740.

Speaker: Sung-Jin Oh

Abstract: In this talk, my aim is to provide a unified viewpoint on two illposedness mechanisms for dispersive equations, namely degenerate dispersion and (the failure of) the Takeuchi-Mizohata condition. For a linear dispersive equation, degenerate dispersion is a property of the principal term in the presence of degenerating coefficients, and the Takeuchi-Mizohata condition concerns the effect of the subprincipal term. First, I will demonstrate how these two effects manifest in the context of wave packet construction. Then, I will exhibit a simple energy and duality argument (similar to testing by wave packets of Ifrim-Tataru) that allows one to extend this illposedness phenomenon to a variety of quasilinear(!) degenerate dispersive PDEs, including singular generalized SQG, surface growth model, Rosenau-Hyman model, etc. This talk is mostly based on joint projects with In-Jee Jeong and Dongho Chae.

Three Things About Polynomials

The HADES seminar on Tuesday, August 29th will be at 3:30 pm in Room 740.

Speaker: Ruixiang Zhang

Abstract: I will talk about three interesting ingredients that go into the results on Hörmander type operators I presented at APDE seminar (joint with Shaoming Guo and Hong Wang). They are all related to algebraic or geometric properties of multivariate polynomials.