The HADES seminar on Tuesday, May 7th, will be at 3:30pm in Room 748.

Speaker: Pranava Jayanti

Abstract: We introduce the physics of superfluidity, including two mathematical models. We begin with a micro-scale description of the interacting dynamics between the superfluid and normal fluid phases of Helium-4 at length scales much smaller than the inter-vortex spacing. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of solutions in $\mathbb{T}^n$ (for n=2,3) for a power-type nonlinearity, beginning from small initial data. Depending upon the strength of the nonlinear self-interactions, we obtain solutions that are global or almost-global in time. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while deriving time-independent a priori estimates. We compare two different approaches: purely energy based, versus a combination of energy estimates and maximal regularity. The results are from recent collaborations with Juhi Jang and Igor Kukavica.
We will also briefly discuss some results pertaining to a macro-scale model known as the HVBK equations, some of which is joint work with Konstantina Trivisa.

The HADES seminar on Tuesday, April 30th, will be at 3:30pm in Room 939.

Speaker: Ning Tang

Abstract: The study of low-regularity inextendibility criteria for Lorentzian manifolds is motivated by the strong cosmic censorship conjecture in general relativity. In this expository talk I will present a result by Jan Sbierski on the C^0 inextendibility of the maximal analytic Schwarzschild spacetime. I will start with a proof for the continuous inextendibility of Minkowski spacetime, followed by a comparison between this and the continuous inextendibility of Schwarzschild exterior. Then I will sketch the proof of continuous inextendibility of Schwarzschild interior.

The HADES seminar on Tuesday, April 23rd, will be at 3:30pm in Room 939.

Speaker: Andrew Scharf

Abstract: Multilayered poroelastic structures are found in many biological tissues, such as cartilage and the cornea, and find use in the design of bioartificial organs and other bioengineering applications. Motivated by these applications, we analyze the interaction of a free fluid flow modeled by the time-dependent Stokes equation and a multilayered poroelastic structure consisting of a thick Biot layer and a thin, linear, poroelastic membrane separating it from the Stokes flow. The resulting equations are linearly coupled across the thin structure domain through physical coupling conditions such as the Beavers-Joseph-Saffman condition. I will discuss previous work in which weak solutions were shown to exist by constructing approximate solutions using Rothe’s method. While a number of partitioned numerical schemes have been developed for the interaction of Stokes flow with a thick Biot structure, the existence of an additional thin poroelastic plate in the model presents new challenges related to finite element analysis on multiscale domains. As an important step toward an efficient numerical scheme for this model, we develop a novel, fully discrete partitioned method for the multilayered poroelastic structure problem based on the fixed strain Biot splitting method. This work is carried out jointly with Sunčica Čanić and Jeffrey Kuan at the University of California, Berkeley and Martina Bukač at the University of Notre Dame.

The HADES seminar on Tuesday, March 12th, will be at 2:00pm in Room 748. (NOTE THE UNUSUAL SPACE AND TIME)

Speaker: Federico Pasqualotto

Abstract: In this talk, I will review recent results concerning the singularity formation problem for 3d incompressible fluids. In particular, I will focus on the role of angular regularity and explain why higher angular regularity makes blow-up constructions harder. I will finally outline recent work in collaboration with Tarek Elgindi for the 3d Euler equations on R^3, in which we construct the first singularity scenario entirely smooth in the angular variable.

The HADES seminar on Tuesday, January 30th, will be at 3:30pm in Room 939 (not in 740 this semester!).

Speaker: Joonhyun La

Abstract: In this talk, we will prove local well-posedness of kinetic wave equation arising from MMT equation, which is introduced by Majda, Mclaughlin, and Tabak and is one of the standard toy models to study wave turbulence. Surprisingly, our result reveals a regularization effect of the collision operator, which resembles the situation of non-cutoff Boltzmann. This talk is based on a joint work with Pierre Germain (Imperial College London) and Katherine Zhiyuan Zhang (Northeastern).