Colloquia

Special Colloquium

Wojciech Ozanski, Florida State University

Friday, January 12th, 3:30 – 4:30 pm, KAP 414

Hydrodynamic instabilities and PDEs in fluid mechanics

Abstract: We will discuss the relation between hydrodynamic instabilities and well-posedness questions in the most important PDEs arising in incompressible fluid mechanics: the Euler equations and the Navier-Stokes equations. We will introduce some exciting recent developments which enable us to rigorously construct unstable perturbations of some families of steady $2$D and $3$D flows.

We will first describe the concept of an instability and demonstrate some new techniques for constructing instabilities in a simple example of a 2D shear flow. We will then consider steady solutions to the 3D incompressible Euler equations in the form of vortex columns, that is vector fields of the form $u=V(r)e_{\theta } + W(r) e_z$, where $r$ denotes the distance to the axis of rotation and $e_\theta$ and $e_z$ denote the standard cylindrical unit vectors, for a family of profiles $V,W$. We will demonstrate the first construction of infinitely many, genuinely three-dimensional modes of instabilities of some vortex columns, which take the form of `ring modes’, localized around $r=r_0$, for some $r_0>0$. We will describe the relevance of such instabilities and related open problems in the context of experimental and numerical phenomena observed in incompressible fluids.

_______________________________________________________________________________________________

Eva Miranda, Universitat Politecnica de Catalunya

Monday, January 29th, 3:30 – 4:30 pm, KAP 414

Towards a “Fluid Computer”

Abstract: Is hydrodynamics capable of performing computations? (Moore, 1991). Can a mechanical system (including a fluid flow) simulate a universal Turing machine? (Tao, 2016). Using a mirror between contact geometry and fluid dynamics unveiled by Etnyre and Ghrist 24 years ago, we find affirmative answers to Moore’s and Tao’s questions. Our methodology integrates techniques from Alan Turing with modern geometry, specifically contact geometry, leading to the conceptual design of a “Fluid Computer” in three dimensions. This construction notably reveals the existence of undecidable fluid paths.
Tao’s initial question emerged from a research program aimed at addressing the Navier–Stokes existence and smoothness problem. Could such a Fluid computer be used to address this Millennium prize problem? We will end the talk with some speculative ideas of a new Fluid computer construction à la Feynman (the “hybrid computer”).
This talk is based on joint work with Robert Cardona, Daniel Peralta-Salas, and Francisco Presas. The new hybrid computer model is based on work in progress with Ángel Gonzalez Prieto and Daniel Peralta-Salas.

_______________________________________________________________________________________________

 Xuecheng Wang, Tsinghua University

NOTE: This talk has been reschedule from Monday, Feb. 5th to Wednesday, Feb. 7th

Wednesday, February 7th, 3:30 – 4:30 pm, KAP 414

Title: Global regularity of the 3D relativistic Vlasov-Maxwell system

Abstract: The 3D relativistic Vlasov-Maxwell system is one of the fundamental models in the plasma physics, which describes the dynamics of electron and ions under the electromagnetic field created by particles themselves. In this talk, we discuss the existence of global solution and the propagation of regularity of the 3D relativistic Vlasov-Maxwell system for both the small initial data case and the large initial data case.

_______________________________________________________________________________________________

Matthew Schrecker, University of Bath

NOTE: This talk has been reschedule from Monday, Feb. 12th to Wednesday, Feb. 14th

Wednesday, February 14th, 3:30 – 4:30 pm, KAP 414

Title: Gravitational Landau Damping

Abstract: In the 1960s, Lynden-Bell, studying the dynamics of galaxies around steady states of the gravitational Vlasov-Poisson equation, described a phenomenon he called “violent relaxation,” a convergence to equilibrium through phase mixing analogous in some respects to Landau damping in plasma physics. In this talk, I will discuss recent work on this gravitational Landau damping for the linearised Vlasov-Poisson equation and, in particular, the critical role of regularity of the steady states in distinguishing damping from oscillatory behaviour in the perturbations. This is based on joint work with Mahir Hadzic, Gerhard Rein, and Christopher Straub.

_______________________________________________________________________________________________

Zaher Hani, University of Michigan

Monday, February 26th, 3:30 – 4:30 pm, KAP 414

Title: Hilbert’s sixth problem for nonlinear waves

Abstract: Hilbert’s sixth problem asks for a mathematically rigorous justification of the macroscopic laws of statistical physics from the microscopic laws of dynamics. The classical setting of this problem is the justification of Boltzmann’s kinetic equation from Newtonian particle dynamics. This justification has been proven for short times, starting with the work of Lanford in 1975, but its long time justification remains one of the biggest open problems in kinetic theory.

If classical colliding particles are replaced with interacting waves, one formally obtains what is known as “wave kinetic theory”, which is sometimes also called “wave turbulence theory”. This theory of statistical physics for waves has been developed, starting in the late 1920s, for wave systems that arise in various scientific disciplines like many-particle quantum physics, oceanography, climate science, etc. The central mathematical problem there is also the justification of a kinetic equation, known as the wave kinetic equation, starting from the Hamiltonian PDE that governs the corresponding microscopic system. In this talk, we shall describe the state of the art of this problem, leading to a most recent joint work with Yu Deng (USC), in which we give the first instance of a long time justification of a nonlinear (particle or wave) collisional kinetic limit.

_______________________________________________________________________________________________

Pranava Jayanti, USC

Monday, March 18th, 3:30 – 4:30 pm, KAP 414

Title: Mass Transfer and Global Solutions in a Micro-Scale Model of Superfluidity

Abstract: We investigate a micro-scale model of superfluidity derived by Pitaevskii in 1959 to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. This system consists of the nonlinear Schr\”odinger 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 $\mathbb{T}^d$ $(d=2,3)$ 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 obtaining 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.

_______________________________________________________________________________________________

Frederi Viens, Rice University

Monday, March 25th, 3:30 – 4:30 pm, KAP 414

Title: Analysis on Wiener space for Yule’s correlation statistic: asymptotics and applicability to attribution testing in climate time series

Abstract: The empirical correlation statistic for two time series of length $n$ is known as Pearson’s correlation. As $n$ tends to infinity, it is known to converge to the true correlation coefficient for the pair of models from whence the series came, as long as these models satisfy some mild stationarity and weak memory conditions. This certainly works well in practice when the series are actually i.i.d. measurements, and applied scientists the world over are aware of this. Things start to go sideways when the memory among datapoints in each sequence is much stronger (the normality of asymptotic fluctuations get destroyed), and even worse, when the stationarity assumption is significantly violated. Famously, convergence to the true correlation will fail dramatically in the case of random walks, as was noticed in 1926 by G. Udny Yule using empirical calculations (done by hand!). The empirical correlation, then known as “Yule’s nonsense correlation” statistic, is asymptotically diffuse, over the entire interval (−1,1). Many decades later, one still runs into vexing instances of applied scientists who draw incorrect attribution conclusions based on invalid inference about correlations of time series, including, arguably, in climate science, in ignorance of Yule’s original observation.

We will describe the mathematical question of understanding the asymptotics of Yule’s correlation for random walks, including an explicit expression for its variance when the random walks are Gaussian, and a surprisingly rapid rate of convergence to a limiting diffuse “nonsense correlation” object. These results, based on Wiener chaos calculus, appeared in a paper with Philip Ernst and Dongzhou Huang, in Stochastic Processes and their Applications, in April 2023. We will mention work in progress and a conjectured framework for an exotic conditional central limit theorem with interesting practical implications. For instance, for modeling questions in paleoclimatology, this framework would allow a more principled methodology for climate change attribution, with better robustness properties with respect to model misspecification.

The same framework could also be used to develop a metric for assessing the consistency, broadly construed, of pairs of so-called climate ensembles. Time permitting, we will discuss informally our reporting, in a new paleo-climatology preprint with J. Emile-Geay, G. Hakim, F. Zhu, and D. Amrhein, of some challenging mathematical questions which could explain why these and other robust tools are severely needed in this field of application.

_______________________________________________________________________________________________

Joonhyun La, Princeton University and KIAS

Monday, April 1st, 3:30 – 4:30 pm, KAP 414

Title: Local well-posedness and smoothing of MMT kinetic wave equation

Abstract: In this talk, I 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).

_______________________________________________________________________________________________

Ioan Bejenaru, UC San Diego

Monday, April 15th, 3:30 – 4:30 pm, KAP 414

Title: Dynamics of equivariant Schrödinger Maps near solitons

Abstract: We review the current literature of the near soliton dynamics for Schrödinger Maps and its parabolic cousin Harmonic Map Heat Flow; we will discuss stability, blow-up and the more complex scenario of “eternal oscillations”. We will introduce some new results in the context of Schoedinger Maps.

_______________________________________________________________________________________________

Jacob Bedrossian, UCLA

Monday, April 22nd, 3:30 – 4:30 pm, KAP 414

Title: Nonlinear dynamics in stochastic systems

Abstract: In this overview talk we discuss several results regarding the dynamics of stochastic systems arising in or motivated by fluid mechanics. First, we discuss proving “Lagrangian chaos” in stochastic fluid mechanics, that is, demonstrating a positive Lyapunov exponent for the motion of a particle in the velocity field arising from the stochastic Navier-Stokes equations. We describe how this chaos can be used to deduce qualitatively optimal almost-sure exponential mixing of passive scalars, which can be used to provide a rigorous derivation of the power spectrum of passive scalar turbulence in certain regimes. Next, we describe more recently developed methods for obtaining strictly positive lower bounds and some quantitative estimates on the top Lyapunov exponent of weakly-damped stochastic differential equations, such as Lorenz-96 model or Galerkin truncations of the 2d Navier-Stokes equations (called “Eulerian chaos” in fluid mechanics). We discuss upcoming work which combines many of the above ideas to study “symmetry breaking” in Lorenz-96 with degenerate forcing, giving an example of non-uniqueness of stationary measures (while providing a unique “physically correct” measure). Related ideas regarding nonlinear energy transfer in degenerately damped systems will also be discussed if time permits. All of the work except for the last (joint with Kyle Liss) is joint with Alex Blumenthal and Sam Punshon-Smith.

_______________________________________________________________________________________________