The HADES seminar on Tuesday, October 10th will be at 3:30pm in Room 740.

Speaker: Ely Sandine

Abstract: I will discuss implosion for the equations describing a gas which is compressible, isothermal and self-gravitating. Under the hypotheses of radial symmetry and self-similarity, the equations reduce to a system of ODEs which has been extensively studied by the astrophysics community using numerical methods. One such solution, discovered by Larson and Penston in 1969, was recently rigorously proved to exist by Guo, Hadžić and Jang. In this talk, I will discuss rigorous existence of a subset of the discrete family of solutions found numerically by Hunter in 1977.

The HADES seminar on Tuesday, October 3rd will be at 3:30pm in Room 740.

Speaker: Anuj Kumar

Abstract: In this talk, we are concerned with DiPerna–Lions’ theory for the transport equation. In the first part of the talk, I will discuss a few results regarding the nonuniqueness of trajectories of the associated ODE. Alberti ’12 asked the following question: are there continuous Sobolev vector fields with bounded divergence such that the set of initial conditions for which the trajectories are not unique is of full measure? We construct an explicit example of divergence-free H\”older continuous Sobolev vector field for which trajectories are not unique on a set of full measure, which then answers the question of Alberti. The construction is based on building an appropriate Cantor set and a “blob flow” vector field to translate cubes in space. The vector field constructed also implies nonuniqueness in the class of measure solutions. The second part to talk is a more recent work jointly with E. Bruè and M. Colombo. We construct nonunique solutions of the continuity equation in the class L^\infty in time and L^r in space. We prove nonuniqueness in the range of exponents beyond what is available using the method of convex integration and sharply match with the range of uniqueness of solutions from Bruè, Colombo, De Lellis’ 21.

The HADES seminar on Tuesday, September 26th will be at 3:30pm in Room 740.

Speaker: Krutika Tawri

Abstract: In this talk, we will present a constructive approach to investigate
the existence of martingale solutions to a benchmark fluid-structure interaction
problem that involves an incompressible, viscous fluid interacting with a linearly
elastic membrane subjected to a multiplicative stochastic force. The fluid flow
is described by the Navier-Stokes equations while the elastodynamics of the
thin structure is modeled by the Koiter shell equations. We will discuss the
challenges arising due to the random motion of the time-dependent fluid domain
and present our recent findings. This is joint work with Sunčica Čanić.

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.

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.

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.

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.