Author Archives: esandine

Analysis of micro- and macro-scale models of superfluidity

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 C^0 inextendibility of the maximal analytic Schwarzschild spacetime

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.

A loosely coupled splitting scheme for a fluid – multilayered poroelastic structure interaction problem

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.

Singularity formation in 3d incompressible fluids: the role of angular regularity

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.

Local well-posedness and smoothing of MMT kinetic wave equation

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).

Mode stability for Kerr(-de Sitter) black holes

The HADES seminar on Tuesday, November 28th, will be at 3:30pm in Room 740.

Speaker: Rita Teixeira da Costa

Abstract: The Teukolsky master equations are a family of PDEs describing the linear behavior of perturbations of the Kerr black hole family, of which the wave equation is a particular case. As a first essential step towards stability, Whiting showed in 1989 that the Teukolsky equation on subextremal Kerr admits no exponentially growing modes. In this talk, we review Whiting’s classical proof and a recent adaptation thereof to the extremal Kerr case. We also present a new approach to mode stability, based on uncovering hidden spectral symmetries in the Teukolsky equations. Part of this talk is based on joint work with Marc Casals (CBPF/UCD).

This talks complements yesterday’s Analysis & PDE seminar, but will be self-contained.

Methods for sharp well-posedness for completely integrable PDE

The HADES seminar on Tuesday, November 14th, will be at 3:30pm in Room 740.

Speaker: Thierry Laurens

Abstract:We will describe some of the methods used to prove sharp well-posedness for the Benjamin–Ono equation in the class of H^s spaces, namely, the method of commuting flows. Since its introduction by Killip and Visan in 2019, this groundbreaking approach to completely integrable systems has been adapted to a wide variety of models in order to prove sharp well-posedness results that were previously inaccessible. In this talk, we will describe some of the overarching principles of the method of commuting flows, with a focus on how these ideas were implemented in the case of the Benjamin–Ono equation. This is based on joint work with Rowan Killip and Monica Visan.

Construction of nonunique solutions of the transport and continuity equation for Sobolev vector fields in DiPerna–Lions’ theory

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.

On a nonlinearly coupled stochastic fluid-structure interaction model.

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.

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.