| Date: | September 7, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Yannis Angelopoulos, CALTECH |
| Title: | Late-time tails for linear waves on subextremal black hole spacetimes |
| Abstract: | I will present some recent work (that has been done jointly with Stefanos Aretakis and Dejan Gajic) on the precise asymptotics of linear waves on the exterior (up to and including the event horizon) of subextremal black holes. Particular examples of such spacetimes are the full subextremal families of Reissner-Nordstrom and Kerr black hole spacetimes. I will also discuss the special case of linear waves localized in angular frequency, and I will talk about some potential applications. |
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| Date: | September 14, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Jacek Jendrej, LAGA, Université Sorbonne Paris Nord |
| Title: | Soliton resolution for equivariant wave maps |
| Abstract: | I will present a joint work with Andrew Lawrie from MIT on equivariant wave maps with values in a sphere. We prove that every solution of finite energy decomposes, as time passes, into a superposition of harmonic maps (solitons) and radiation. The proof builds on prior works of Côte, and Jia and Kenig, who proved that such a decomposition holds for a sequence of times. We combine their ideas with modulation analysis of multi-solitons.
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| Date: | September 21, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Jonas Lührmann, Texas A&M |
| Title: | Asymptotic stability of kinks in 1D nonlinear scalar field theories |
| Abstract: | Nonlinear scalar field theories on the line such as the phi^4 model or the sine-Gordon model feature soliton solutions called kinks. They are expected to form the building blocks of the long-time dynamics for these models. In this talk I will survey recent progress on the asymptotic stability problem for kinks and I will present a recent result (joint work with W. Schlag) on the asymptotic stability of the sine-Gordon kink under odd perturbations. |
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| Date: | September 28, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Tarek Elgindi, Duke University |
| Title: | Remarks on the long-time behavior of 2d Euler flows |
| Abstract: | We will discuss the issue of infinite-time filamentation of vorticity in 2d Euler flows. We establish a trichotomy that applies to all sufficiently regular steady states of the 2d Euler equation. Namely, it is either that a given steady state is unstable pointwise, or that it possesses a time-periodic flow map (i.e. it is isochronal), or that generic solutions starting nearby exhibit filamentation and grow at least linearly in C^1 as time grows to infinity. In a large class of important cases, we show that (at least) filamentation must occur. This is joint work with Theodore Drivas. |
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| Date: | October 5, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Hatem Zaag, CNRS & Université Sorbonne Paris Nord |
| Title: | Profile of a touch-down solution to a nonlocal MEMS model |
| Abstract: | We consider a non local PDE of parabolic type modelling a MicroElectrMechanical System (MEMS). A MEMS is an electronic device available in every day life (microphone, etc.). It consists in an elastic membrane hanging above a rigid plate connected to an electric source and a capacitor. If ever the membrane reaches the rigid plate, we say that we have a "touch-down", which damages the device and has henceforth to be avoided. In this talk, we show that touch-down may occur, for a special class of initial data, under some conditions on parameters. We also describe the "shape" (in fact the profile) of the solution near the touch-down time. |
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| Date: | October 12, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Kevin Zumbrun, Indiana University |
| Title: | Stability of roll wave solutions in inclined shallow-water flow |
| Abstract: | We review recent developments in stability of periodic roll-wave solutions of the Saint Venant equations for inclined shallow-water flow. Such waves are well-known instances of hydrodynamic instability, playing an important role in hydraulic engineering, for example, flow in a channel or dam spillway. Until recently, the analysis of their stability has been mainly by formal analysis in the weakly unstable or ``near-onset`` regime. However, hydraulic engineering applications are mainly in the strongly unstable regime far from onset. We discuss here a unified framework developed together with Blake Barker, Mat Johnson, Pascal Noble, Miguel Rodrigues, and Zhao Yang for the study of roll wave stability across all parameter regimes, by a combination of rigorous analysis and numerical computation. The culmination of our analysis is a complete stability diagram, of which the low-frequency stability boundary is, remarkably, given explicitly as the solution of a a cubic equation in the parameters of the solution space. |
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| Date: | October 19, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Yuanyuan Feng, Pennsylvania State University |
| Title: | Dissipation enhancing flows and applications |
| Abstract: | In this talk, we would first introduce the dissipation enhancing flows. We would focus on the dissipation time of mixing flows, shear flows and the planar helical flows. Then we will apply these flows to Kuramoto Sivashinsky equations in 2d or 3d to get global existence of the solution. |
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| Date: | October 26, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | William Rundell, Texas A&M University |
| Title: | Inverse problems for fractional and other nonlocal operators |
| Abstract: | Fractional derivatives and powers of operators have been a well-studied topic over the last decade - and for good reason. Not only is there interesting mathematics involved but a myriad of applications from diverse areas of the physical sciences have shown that such operators in fact play a crucial role in correct modelling. In this talk we shall look at several inverse problems involving both classical derivatives and some of their fractional counterparts. The obvious question, which will come with some answers, is whether these two paradigms give similar uniqueness results. Many classical inverse problems for PDE are characterised by the fact that their solution is often highly ill-posed. This leads to the further question: are the ill-conditioning levels the same for both classical and fractional models? If not, can fractional operators be used as regularising approximations thus gaining intrinsic value even for classical derivatives? |
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| Date: | November 2, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Vincent Martinez, City University of New York Hunter College |
| Title: | Convergence analysis for a parameter estimation algorithm for the 2D Navier-Stokes equations |
| Abstract: | In this talk, we will discuss the problem of the determination of unknown parameters in a dynamical system via observations on a subset of the systems’s state variables. In our setup, we consider a 2D incompressible viscous fluid whose kinematic viscosity is assumed to be unknown, but its filtered velocity field is observed continuously-in-time. The algorithm studied is one proposed by Carlson, Hudson, and Larios (2020) in which a feedback control paradigm for data assimilation of PDEs, originally introduced by Azouani, Olson, and Titi (2014), is leveraged to provide systematic updates to the value of the viscosity based on the collected observations. Although several computational tests had been carried out that corroborate the convergence of this algorithm to the true value of viscosity, a rigorous proof remained elusive. We discuss a direct proof of convergence in the regime of observing a sufficiently large resolution of the velocity field, under the assumption that a certain non-degeneracy condition holds. The two main ingredients of the proof are the availability of higher-order sensitivity-type bounds and identification of a Lyapunov-type structure for the time-derivative of the energy of the velocity error. |
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| Date: | November 9, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Hao Jia, University of Minnesota |
| Title: | Linear vortex symmetrization: the spectral density function approach and Gevrey regularity |
| Abstract: | The two-dimensional incompressible Euler equation is globally well-posed but the long time behavior is very difficult to understand due to the lack of global relaxation mechanism. Numerical simulations and physical experiments
show that coherent vortices often become a dominant feature in two-dimensional fluid dynamics for a long time. The mathematical analysis of vortices, especially in connection to the so-called `vortex symmetrization` problem, has attracted a lot of attention in recent years. In this talk, after a quick review of recent developments in the study of nonlinear asymptotic stability of shear flows and the symmetrization problem for (the special case of) point vortices, we turn to the general vortex symmetrization problem and report a recent result with A. Ionescu for the linearized flow. The linearized problem has been analyzed before by Bedrossian-Coti Zelati-Vicol who obtained control on the profile of the vorticity field in Sobolev spaces with limited regularity. Our main new discovery is that in the vortex problem, unlike the shear flow case, it is no longer possible to obtain smooth control uniformly in time on a single modulated profile for the vorticity field. Rather, there are two such profiles. To address this issue (for future nonlinear
applications), we propose instead to control a new object, the so-called `spectral density function`, which is naturally associated with the linearized flow and can be bounded, for the linearized flow at least, in the same Gevrey space as the initial data.
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| Date: | November 16, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Antoine Remond-Tiedrez, University of Wisconsin - Madison |
| Title: | Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundary |
| Abstract: | To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations. These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by latent heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings). In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation. |
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| Date: | November 23, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Aseel Farhat, Florida State University |
| Title: | Intermittency in turbulence and the 3D Navier-Stokes regularity problem |
| Abstract: | We describe several aspects of an analytic/geometric framework for the three-dimensional Navier-Stokes regularity problem, which is directly inspired by the morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of three-dimensional turbulence. Among these, we present our proof that the scaling gap in the 3D Navier-Stokes regularity problem can be reduced by an algebraic factor within an appropriate functional setting incorporating the intermittency of the spatial regions of high vorticity. Furthermore, a turbulent cascade model based on the suitably defined scale of sparseness of the super-level sets of the higher-order derivatives of the velocity field is examined. In particular, a certain universality property of the ratios of the scales of sparseness at nearby (differential) levels is discovered.
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| Date: | November 30, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Vedran Sohinger, University of Warwick |
| Title: | Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs |
| Abstract: | Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In this talk, we will discuss the connection of Gibbs measures with the Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation. In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of nonlinear Hamiltonian PDEs. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari. |
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