Apr 12, 2021
Monday
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07:50 AM - 08:00 AM
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Welcome
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- Location
- SLMath: Online/Virtual
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08:00 AM - 08:50 AM
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Gravity capillary wave equations on the circle, normal forms and long time existence: a review
Jean Marc Delort (Université de Paris XIII (Paris-Nord))
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
Since the beginning of the program, results on long time existence for solutions to gravity/capillary wave equations with small data have been described, in particular in the series of lectures of Daniel Tataru during the Introductory workshop. In this talk, we aim at giving a review of a set of results concerning the possibility of going past the "cubic life span" for solutions to the gravity-capillary system with small data in the periodic setting.
- Supplements
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09:00 AM - 09:50 AM
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The KdV hierarchy at H^{-1} regularity
Herbert Koch (Rheinische Friedrich-Wilhelms-Universität Bonn)
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
This is a report on joint work with Friedrich Klaus and Baoping Liu. We prove wellposedness for all equations of the KdV hierarchy in H^{-1}. The proof follows the outline of Killip and Visan for KdV: We prove tightness, and convergence of the difference flow simultaneously for all equations of the hierarchy.
In contrast to Killip and Visan we reduce the problem via a modified Miura map to the Gardner hierarchy.
- Supplements
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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On the implosion of a viscous compressible fluid
Pierre Raphael (Université Nice Sophia-Antipolis)
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- Location
- SLMath: Online/Virtual
- Video
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I will describe an implosion mechanism for three dimensional viscous compressible fluids, and will make connections with highly oscillatory blow up mechanisms for energy super critical defocusing non linear wave models.
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Apr 13, 2021
Tuesday
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08:00 AM - 08:50 AM
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Arnold's variational principle and its application to the stability of viscous planar vortices
Thierry Gallay (Université Grenoble Alpes (Université de Grenoble I - Joseph Fourier))
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- Location
- SLMath: Online/Virtual
- Video
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We revisit the variational approach to nonlinear stability of planar flows, which was developed by V. I. Arnold around 1965. In particular, we study the coercivity properties of the quadratic form that describes the second variation of the energy at a radially symmetric vortex with strictly decreasing vorticity profile. We also show that this quadratic form can be used to obtain a new proof of nonlinear stability for the Lamb-Oseen vortices, which are self-similar solutions of the two-dimensional Navier-Stokes equations. This is all joint work with Vladimir Sverak.
- Supplements
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09:00 AM - 09:50 AM
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Heterogeneities in fluid mechanics
Didier Bresch (Université de Savoie (Chambéry); Centre National de la Recherche Scientifique (CNRS))
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- Location
- SLMath: Online/Virtual
- Video
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Pressure law which depends on the density and on time and space variables $t$ and $x$ for compressible flows or anelastic constraint on the velocity field in the incompressible setting for the rate of flow are examples of heterogeneities that may complicate mathematical analysis in the weak regularity context. I will explain different mathematical difficulties occuring in such situations and new developments recently obtained in collaboration with P.-E. Jabin and F. Wang.
- Supplements
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Incompressible limit for the free surface Navier-Stokes system
Frederic Rousset (Université Paris-Saclay)
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- Location
- SLMath: Online/Virtual
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We consider the Navier-Stokes system for compressible fluids in a low Mach number regime. The aim is to get the existence of a strong solution on an interval of time independent of the Mach number. The main issue, which also occurs when studying the system in a fixed domain with a slip boundary condition, is the conjonction of the time oscillations and a boundary layer in space. Joint work with Changzhen Sun and N. Masmoudi.
- Supplements
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12:10 PM - 01:10 PM
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Virtual "Reception"
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- Location
- SLMath: Online/Virtual
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Apr 15, 2021
Thursday
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08:00 AM - 08:50 AM
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Dynamics of Newtonian stars
Juhi Jang (University of Southern California)
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- Location
- SLMath: Online/Virtual
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- Abstract
A classical model to describe the dynamics of Newtonian stars is the gravitational Euler-Poisson system. The Euler-Poisson system admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In this talk, I will discuss some recent progress on those star solutions with focus on expansion and collapse. The talk is based on joint works with Yan Guo and Mahir Hadzic.
- Supplements
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09:00 AM - 09:50 AM
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Two dimensional gravity water waves at low regularity
Albert Ai (University of Wisconsin-Madison)
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- Location
- SLMath: Online/Virtual
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In this talk, we will consider the low regularity well-posedness problem for the two dimensional gravity water waves. This quasilinear dispersive system admits an interesting structure which we exploit to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier energy estimates of Hunter-Ifrim-Tataru. These results allow us to significantly lower the regularity threshold for local well-posedness, even without using dispersive properties. Combined with nonlinear vector field Sobolev inequalities, an idea first introduced by the last two authors in the context of the Benjamin-Ono equations, these improvements extend to global solutions for small and localized data. This is joint work with Mihaela Ifrim and Daniel Tataru.
- Supplements
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Riemann's non-differentiable function and the binormal curvature flow
Luis Vega Gonzalezs (Universidad del País Vasco/Euskal Herriko Unibertsitatea)
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- Location
- SLMath: Online/Virtual
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In a joint work with Valeria Banica, we make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious non- linear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids. If time permits, I will also mention some recent results on intermittency in the context of the linear Schrödinger equation.
- Supplements
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Apr 16, 2021
Friday
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08:00 AM - 08:50 AM
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On the Nernst-Planck-Navier-Stokes System
Peter Constantin (Princeton University)
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- Location
- SLMath: Online/Virtual
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09:00 AM - 09:50 AM
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Analytic Solutions For The Water-Waves System
Nicolas Burq (Université Paris-Saclay)
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- Location
- SLMath: Online/Virtual
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In this talk I will present some results on the Cauchy problem for the gravity water-wave equations, in a domain with flat bottom and in arbitrary space dimension. I will show that if the data are of size $\eps$ in a space of analytic functions which have a holomorphic extension in a strip of size $\sigma$, then the solution exists up to a time of size $C/\eps$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $\sigma - C'\eps t$. This question comes from motivations from control theory which force us to consider analytic solutions. I will actually start the talk with these motivations.
This is joint work with T. Alazard and C. Zuily.
- Supplements
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Vacuum States in Hydrodynamic Models
Roberto Camassa (University of North Carolina)
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- SLMath: Online/Virtual
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Apr 19, 2021
Monday
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08:00 AM - 08:50 AM
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The quartic integrability and long time existence of steep water waves in 2d
Sijue Wu (University of Michigan)
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- Location
- SLMath: Online/Virtual
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It is known since the work of Dyachenko \& Zakharov in 1994 that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non-trivial resonant manifold. In this talk I will present a recent result that proves this partial integrability from a different angle. We construct a sequence of energy functionals $\mathfrak E_j(t)$, directly in the physical space, which are explicit in the Riemann mapping variable and involve material derivatives of order $j$ of the solutions for the 2d water wave equation, so that $\frac d{dt} \mathfrak E_j(t)$ is quintic or higher order. We show that if some scaling invariant norm, and a norm involving one spacial derivative above the scaling of the initial data are of size no more than $\varepsilon$, then the lifespan of the solution for the 2d water wave equation is at least of order $O(\varepsilon^{-3})$, and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size $\varepsilon$, then the lifespan of the solution is at least of order $O(\varepsilon^{-5/2})$. Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.
- Supplements
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09:00 AM - 09:50 AM
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Three-dimensional steady water waves with vorticity
Erik Wahlén (Lund University)
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- Location
- SLMath: Online/Virtual
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In recent years there has been a lot of progress on two-dimensional steady water waves with vorticity and three-dimensional waves with zero vorticity (irrotational flow). The three-dimensional case with vorticity remains largely unexplored. Part of the reason is that the problem (generally) doesn't reduce to an elliptic free boundary problem. In my talk I will discuss recent progress on this topic. I will mainly focus on an ongoing work where we consider doubly periodic, symmetric gravity-capillary waves with small vorticity, by treating the problem as elliptic-hyperbolic and using ideas from the theory of magnetohydrostatic equilibria.
This is joint work with D. S. Seth and K. Varholm.
- Supplements
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Wave propagation on vorticity discontinuities and the Burgers-Hilbert equation
John Hunter (University of California, Davis)
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- Location
- SLMath: Online/Virtual
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The two-dimensional incompressible Euler equations have solutions for vorticity discontinuities (planar analogs of vortex patches) that separate two half-spaces with constant but distinct vorticities. Vorticity discontinuities are linearly stable but support unidirectional waves that propagate along the discontinuity and decay exponentially into the interior of the fluid. The linearized frequency of these waves is independent of their wave numbers, which leads to interesting and unusual weakly nonlinear, nondispersive dynamics for discontinuities with small slope. In particular, we formulate contour dynamics equations for the motion of vorticity discontinuities before they filament, and prove that a quadratically nonlinear Burgers-Hilbert equation provides an asymptotic description of the motion of small-slope vorticity discontinuities on cubically nonlinear time-scales. This is joint work with Joseph Biello, Ryan Moreno-Vasquez, Jingyang Shu, and Qingtian Zhang.
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Apr 20, 2021
Tuesday
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08:00 AM - 08:50 AM
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Long time behavior in a flow-structure interaction
Irena Lasiecka (University of Memphis)
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- Location
- SLMath: Online/Virtual
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Flow-structure interactions are ubiquitous in nature. Problems such as attenuation of turbulence or flutter in an oscillating structure [Tacoma bridge] are prime examples of relevant applications. Mathematically, the models are represented by a 3 D Euler Equation coupled to a nonlinear dynamic elasticity on a 2 D manifold. Strong boundary-type coupling at the interface between the two media is at the center of the analysis. This provides for a rich mathematical structure, opening the door to several unresolved problems in the area of nonlinear PDE's, dynamical systems and related harmonic analysis and geometry. This talk aims at providing a brief overview of recent developments in the area along with a presentation of some recent advances addressing the issues of control and long time behavior [partial structural attractors] subject to mixed boundary conditions arising in modeling of the interface between the two environments.
- Supplements
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09:00 AM - 09:50 AM
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Separation and Circulation in the Stationary Prandtl Equation
Anne-Laure Dalibard (Sorbonne Université)
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- Location
- SLMath: Online/Virtual
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In this talk, I will describe some recent results about the separation phenomenon within the stationary Prandtl equation.
The Prandtl equation describes the behavior of a fluid with small viscosity close to the boundary. In the presence of an adverse pressure gradient, it has been observed experimentally that the boundary layer may detach itself from the boundary. This separation has been described in a formal way by Goldstein in 1958.
With Nader Masmoudi, we gave a rigorous proof of the validity of the Goldstein singularity, and a precise, quantitative description of the behaviour of the solution in the vicinity of the separation point.
I will also tackle some issues related to the well-posedness of the Prandtl system in the presence of a recirculation bubble, i.e. downstream of the separation point. This topic is a work in progress with Frédéric Marbach and Jean Rax.
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Stable shock wave formation for the compressible Euler equations
Tristan Buckmaster (Princeton University)
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- Location
- SLMath: Online/Virtual
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I will talk about recent work with Steve Shkoller, and Vlad Vicol, regarding shock wave formation for the compressible Euler equations.
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11:50 AM - 01:00 PM
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Virtual "Reception"
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- Location
- SLMath: Online/Virtual
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Apr 22, 2021
Thursday
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08:00 AM - 08:50 AM
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Global existence for the 2D Kuramoto-Sivashinsky equation
Anna Mazzucato (Pennsylvania State University)
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- Location
- SLMath: Online/Virtual
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I will present recent results concerning global existence for the Kuramoto-Sivashinsky equation (KSE) in 2 space dimensions with and without growing modes. The KSE is a model of long-wave instability in dissipative systems. This is joint work with David Ambrose (Drexel).
- Supplements
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09:00 AM - 09:50 AM
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Handling congestion in fluid equations
Charlotte Perrin (Centre National de la Recherche Scientifique (CNRS))
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- Location
- SLMath: Online/Virtual
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This talk addresses the mathematical analysis of fluid models including a maximum packing constraint. These equations arise naturally for instance in the modeling of mixtures like suspensions or in the modeling of collective motion. I will present recent existence and stability results on two classes of PDEs systems corresponding to two modeling approaches. The first one, called the "soft approach", is based on compressible equations complemented with constitutive laws that are singular close to the maximal packing constraint. The second one, the "hard approach", is based on a free boundary problem between a congested domain with incompressible dynamics and a free domain with compressible dynamics.
- Supplements
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Long time existence for the Euler-Coriolis system
Benoit Pausader (Brown University)
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- Location
- SLMath: Online/Virtual
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We consider the 3d incompressible Euler equation with a Coriolis force and show that small axially symmetric solutions lead to solutions which exist for a long time. This is a joint work with Yan Guo and Klaus Widmayer.
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Apr 23, 2021
Friday
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08:00 AM - 08:50 AM
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Locally dissipative solutions of the Euler equations
Camillo De Lellis (Institute for Advanced Study)
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- Location
- SLMath: Online/Virtual
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The Onsager conjecture, recently solved by Phil Isett, states that, below a certain threshold regularity, Hoelder continuous solutions of the Euler equations might dissipate the kinetic energy. The original work of Onsager was motivated by the phenomenon of anomalous dissipation and a rigorous mathematical justification of the latter should show that the energy dissipation in the Navier-Stokes equations is, in a suitable statistical sense, independent of the viscosity. In particular it makes much more sense to look for solutions of the Euler equations which, besides dissipating the {\em total} kinetic energy, satisfy as well a suitable form of local energy inequality. Such solutions were first shown to exist by Laszlo Szekelyhidi Jr. and myself. In this talk I will review the methods used so far to approach their existence and the most recent results by Isett and by Hyunju Kwon and myself.
- Supplements
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09:00 AM - 09:50 AM
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Global regularity and long time behavior of solutions of electroconvection models
Mihaela Ignatova (Temple University)
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- Location
- SLMath: Online/Virtual
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We consider models of electroconvection in bounded domains or with periodic boundary conditions. The models involve nonlocal operators. We describe results on global regularity and long time asymptotic behavior of solutions.
- Supplements
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Recent Progress in the Study of the Prandtl System and the Zero Viscosity Limit
Nader Masmoudi (New York University, Courant Institute)
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- Location
- SLMath: Online/Virtual
- Video
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TBA
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Apr 26, 2021
Monday
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08:00 AM - 08:50 AM
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Remarks on 2D Euler stationary states
Tarek Elgindi (Duke University)
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- SLMath: Online/Virtual
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We will discuss stability properties of stationary solutions to the 2D Euler equation.
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09:00 AM - 09:50 AM
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Vortex filament solutions of the 3D Navier-Stokes equations
Jacob Bedrossian (University of Maryland)
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- Location
- SLMath: Online/Virtual
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- Abstract
We consider solutions of the Navier-Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of 3d Navier-Stokes, as well as solutions which are locally approximately self-similar. This is joint work with Pierre Germain and Benjamin Harrop-Griffiths.
- Supplements
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Mode Two Solitary Waves in Stratified Flows
Paul Milewski (University of Bath)
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
There is a substantial literature on horizontally propagating waves in stratified flows. The vast majority of this work, particularly when it concerns nonlinear structures and solitary waves, focuses on “mode one”, that is, the "fast" waves of the system whereby all the pycnoclines (density jumps) are deflected with the same polarity. The simplest model for mode one waves is the two-layer flow of a lighter fluid above a heavier one bounded above and below by rigid boundaries. In that case the mode one wave is the only wave in the system. Mode two waves require (at least) one additional layer in order for the two interfaces to deflect with opposite polarity (mode two), or the same polarity (mode one). We shall consider the three-layer problem in this talk, considering KdV and MCC-like models, and the full Euler equations. We shall describe the problem and consider the question: do mode two solitary waves exist in the Euler equations?
- Supplements
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Apr 27, 2021
Tuesday
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08:00 AM - 08:50 AM
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Shock formation for compressible Euler
Vlad Vicol (New York University, Courant Institute)
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- Location
- SLMath: Online/Virtual
- Video
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I will discuss recent joint work with T Buckmaster and B Shkoller concerning the formation of finite time point-shocks in the full compressible Euler system.
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09:00 AM - 09:50 AM
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Unbounded growth of the energy density associated to the Schrödinger map and the binormal flow
Valeria Banica (Sorbonne University, Laboratoire Jacques-Louis Lions)
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- Location
- SLMath: Online/Virtual
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In this talk I shall consider the binormal flow equation, which is a model for the dynamics of vortex filaments in Euler equations. Geometrically it is a flow of curves in three dimensions, explicitly connected to the 1-D Schrödinger map with values on the 2-D sphere, and to the 1-D cubic Schrödinger equation. Although these equations are completely integrable we show the existence of an unbounded growth of the energy density. The density is given by the amplitude of the high frequencies of the derivative of the tangent vectors of the curves, thus giving information of the oscillation at small scales. In the setting of vortex filaments the variation of the tangent vectors is related to the one of the direction of the vorticity, that according to the Constantin-Fefferman-Majda criterion plays a relevant role in the possible development of singularities for Euler equations. This is a joint work with Luis Vega.
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Motion of several slender rigid filaments in a Stokes flow
Franck Sueur (Université de Bordeaux)
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- Location
- SLMath: Online/Virtual
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In this talk I would present a joint work with Richard H\"ofer and Christophe Prange regarding the dynamics of several slender rigid bodies moving in a flow driven by the three-dimensional steady Stokes system in presence of a smooth background flow. More precisely we consider the limit where the radii of these slender rigid bodies tend to zero with a common rate $\epsilon$, while their volumetric mass density is held fixed, so that the positions occupied by the bodies shrink into separated massless curves. While for each positive $\epsilon$, the bodies’ dynamics are given by the Newton equations and correspond to some coupled second order ODEs for the positions of the bodies, we prove that the limit equations are decoupled first order ODE whose coefficients only depend on the limit curves and on the background flow. These coefficients appear through appropriate renormalized Stokes' resistance tensors associated with each limit curve, and through renormalized Fax\’en-type force and torque associated with the limit curves and the background flow.
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12:10 PM - 01:10 PM
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Virtual "Reception"
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- Location
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Apr 29, 2021
Thursday
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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Modeling inviscid water waves
Emmanuel Dormy (École Normale Supérieure)
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- Location
- SLMath: Online/Virtual
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We consider numerical strategies to handle two-dimensional water waves in a fully non-linear regime. The free-surface is discretized via lagrangian tracers and the numerical strategy is constructed carefully to include desingularizations, but no artificial regularizations. We approach the formation of singularities in the wave breaking problem and also model solitary waves and the effect of an abruptly changing bottom.
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Apr 30, 2021
Friday
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08:00 AM - 08:50 AM
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Universality and possible blowup in fluid equations
Terence Tao (University of California, Los Angeles)
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- Location
- SLMath: Online/Virtual
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We discuss some possible (and still speculative) routes to establishing finite time blowup in fluid equations (and other PDE), focusing in particular on methods based on establishing universality properties for such equations.
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09:00 AM - 09:50 AM
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Vortex filament solutions for the Navier-Stokes equations
Pierre Germain (New York University, Courant Institute)
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- Location
- SLMath: Online/Virtual
- Video
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I will present a construction of solutions of the Navier-Stokes equations for data whose vorticity are concentrated on 1D curves (as measures). This corresponds to large locally self-similar data, for which the usual perturbative approach to local well-posedness does not apply, and for which a number of interesting questions arise. These data are also of fundamental importance from a physical perspective, since vortex filaments are expected to play a crucial role in 3D flows. This is joint work with J. Bedrossian and B. Harrop-Griffiths.
- Supplements
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10:00 AM - 11:00 AM
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Gathertown Break
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- Location
- SLMath: Online/Virtual
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11:00 AM - 11:50 AM
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On spectra of certain linearized operators
Vladimir Sverak (University of Minnesota Twin Cities)
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- Location
- SLMath: Online/Virtual
- Video
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We discuss some calculations of spectra and their consequences, focussing mostly on recent joint work with Hao Jia concerning long-distance behavior of the steady-state solutions of the 3d Navier-Stokes equations.
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