Analysis of Fluids Seminars

Princeton-Tokyo Fluid Mechanics Workshop

Speaker: 
Various, see below
Date: 
Nov 7 2017 - 9:30am
Event type: 
Analysis of Fluids and Related Topics
Room: 
Jadwin Hall, Room 407 (Nov 7 & 8) Fine 110 (Nov 9)
Abstract: 

A 3-day workshop bringing together researchers from Princeton and Tokyo working in the field of mathematical fluid dynamics. New recent results and possible future directions will be discussed.

Workshop Organizers:  Peter Constantin, Yoshikazu Giga, Vlad Vicol

Tuesday, November 7th  

Jadwin Hall, Room 407

9:30am – Arrival

10:00am – Welcome and introduction

10:15 – 11:15am – Tristan Buckmaster

11:15 – 11:30am – Coffee Break

11:30 – 12:30pm – Prof. Tsuyoshi Yoneda

12:30 – 2:00pm – LUNCH

2:00 – 2:20pm – Kengo Nakai

2:25 – 2:45pm – Federico Pasqualoto

2:50 – 3:10pm – Ken Furukawa

3:30pm – Math Afternoon Tea – Common Room 

Wednesday, November 8th

Jadwin Hall, Room 407

9:30am – Arrival

10:00am – Welcome and introduction

10:15 – 11:15am – Prof. Takahito Kashiwabara

11:15 – 11:30am – Coffee Break

11:30 – 12:30pm – Mihaela Ignatova

12:30 – 2:20pm – LUNCH

2:25 – 2:45pm – Tatsu-Hiko Miura

2:45 – 3:05pm –Joonhyun La

3:10 – 3:30pm – Huanyuan Li

3:30pm – Math Afternoon Tea – Common Room 

Thursday, November 9th

Fine Hall, Room 110

9:30am – Arrival

10:00am – Intro

10:15am – 11:15am – Huy Nguyen

11:15am – 11:30am – Coffee Break

11:30am – 12:30pm – Theodore Drivas

12:30pm End of Program

Analysis of Fluids and Related Topics: Solutions after blowup in ODEs and PDEs: spontaneous stochasticity

Speaker: 
Alexei Mailybaev, Nacional de Matemática Pura e Aplicada (IMPA)
Date: 
Oct 19 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

We discuss the extension of solutions beyond a finite blowup time, i.e., the time at which the system ceases to be Lipschitz continuous. For larger times solutions are defined first by using a (physically motivated) regularization of equations and then taking the limit of a vanishing regularization parameter. We report on several generic situations when such a limit leads to stochastic solutions defining uniquely a probability to choose one or the other (non-unique) path. Moreover, such solutions appear to be independent of the details of regularization procedure, thus, following from the properties of original ideal system alone. In this talk, we provide some rigorous results for systems of ODEs with singularities by using the methods of dynamical system theory applied to renormalized equations. Then, we demonstrate numerical results confirming that such scenarios are realized in infinite dimensional models of hydrodynamic turbulence. Part of the results is a joint work with T.D.Drivas.

Analysis of Fluids and Related Topics: The optimal design of wall-bounded heat transport

Speaker: 
Ian Tobasco, University of Michigan
Date: 
Oct 12 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

Flowing a fluid is a familiar and efficient way to cool: fans cool electronics, water cools nuclear reactors, and the atmosphere cools the Earth. In this talk, we discuss a class of problems from fluid dynamics concerning the design of incompressible wall-bounded flows achieving optimal rates of heat transport for a given flow intensity budget. Guided by a perhaps unexpected connection between this optimal design problem and various “energy-driven pattern formation” problems from materials science, we construct flows achieving nearly optimal rates of heat transport in their scaling with respect to the intensity budget. The resulting flows share striking similarities with self-similar elastic wrinkling patterns, such as can be seen in the shape of a hanging drape or nearby the edge of a torn plastic sheet. They also remind of (carefully designed versions) of the complex multi-scale patterns seen in turbulent fluids. Nevertheless, we prove that in certain cases natural buoyancy-driven convection is not capable of achieving optimal rates of cooling. This is joint work with Charlie Doering.

Analysis of Fluids and Related Topics: Well-posedness for Stochastic Continuity Equations with Rough Coefficients.

Speaker: 
Sam Punshon-Smith, University of Maryland
Date: 
Oct 5 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

According to the theory of Diperna/ Lions, the continuity equation associated to a Sobolev (or BV) vector field with bounded divergence has a unique weak solution in L^p. Under the addition of white in time stochastic perturbations to the characteristics of the continuity equation, it is known that uniqueness can be obtained under a relaxation of the regularity conditions and the requirement of bounded divergence.  In this talk, we will consider the general stochastic continuity equation associated to an Itô diffusion with irregular drift and diffusion coefficients and discuss conditions under which the equation has a unique solution. Using the renormalization approach of DiPerna/Lions we will present a proof of uniqueness of solutions to the stochastic transport with additive noise and a drift in L^q_t L^p_x, satisfying the subcritical Ladyzhenskaya–Prodi–Serrin criterion 2/q + d/p < 1.
 

Analysis of Fluids and Related Topics: Some recent results on wave turbulence and quantum kinetics theories

Speaker: 
Minh Binh Tran, University of Wisconsin
Date: 
Sep 28 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

Wave turbulence is a branch of science studying the out-of-equilibrium statistical mechanics of random nonlinear waves of all kinds and scales.

Despite the fact that wave fields in nature are enormous diverse; to describe the processes of random wave interactions, there is a common mathematical concept; the wave kinetic equations.

After the production of the first Bose-Einstein Condensates (BECs), there has been an explosion of physics research on the kinetic theory associated to BECs and their thermal clouds.

In this talk, we will summarize our recent progress on this...

Analysis of Fluids and Related Topics: Global well-posedness for the 2D Muskat problem

Speaker: 
Stephen Cameron, University of Chicago
Date: 
Sep 21 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

The Muskat problem was originally introduced by Muskat in order to model the interface between water and oil in tar sands. In general, it describes the interface between two incompressible, immiscible fluids of different constant densities in a porous media. In this talk I will prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the initial data is monotonic or has slope strictly less than 1. The curvature of these solutions solutions decays to 0 as t goes to infinity, and they are unique when the initial data is C1,ϵ. We do this by constructing a modulus of continuity generated by the equation, just as Kiselev, Naverov, and Volberg did in their proof of the global well-posedness for the quasi-geostraphic equation.

Analysis of Fluids and Related Topics: Vortex sheets in domains with boundary

Speaker: 
Helena Nussenzveig Lopes IM-UFRJ
Date: 
May 11 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

In this talk we consider the vanishing viscosity problem for incompressible fluid flow in a smooth, bounded domain. We are particularly concerned with the shear layer occurring near the boundary, a phenomenon which we explore in a few examples of flows with symmetry. We use this discussion as motivation for a framework for weak solutions of the inviscid equations which allow for exchange between circulation around the boundary and vorticity in the bulk of the fluid.

Analysis of Fluids and Related Topics: Traveling-standing water waves and their stability

Speaker: 
Jon Wilkening , UC Berkeley
Date: 
May 4 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

We describe a computational framework for computing hybrid traveling-standing waves that return to a spatial translation of their initial conditions at a later time. We introduce two parameters to describe these waves, and explore bifurcations from pure traveling or pure standing waves to these more general solutions of the free-surface Euler equations.  Next, we combine Floquet theory in time and Bloch theory in space to study the stability of traveling-standing waves to harmonic and subharmonic perturbations. For the latter, we have developed new boundary integral methods for the spatially quasi-periodic Dirichlet-Neumann operator. While much is known about the spectral stability of pure traveling waves, this is the first study of general subharmonic perturbations of pure standing waves.  Our unified approach for traveling-standing waves simplifies the eigenvalue problem that arises in the pure traveling case as well.   We conclude with a discussion of general quasi-periodic solutions of the free-surface Euler equations and present preliminary calculations of some simple cases.

Analysis of Fluids and Related Topics: Investigating stability and finding new steady vortex flows through numerical bifurcation approaches

Speaker: 
Paolo Luzzatto-Fegiz , UC Santa Barbara
Date: 
Apr 27 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

In 1875, Lord Kelvin stated an energy-based argument for determining equilibrium and stability in vortex flows. The possibility of implementing Kelvin’s argument, in the form of a simple bifurcation approach, had been the subject of debate. In this talk, we build on results from dynamical systems theory to show that, by constructing solution families through isovortical rearrangements, one obtains a bifurcation diagram that contains stability information. To detect bifurcations to new equilibrium families, we propose calculating vortices that have been made “imperfect” through the introduction of asymmetries in the vorticity field. The resulting approach can in principle be used to determine the number of positive-energy (likely unstable) modes for each solution belonging to a family of steady vortices. However, to compute full bifurcation diagrams, we must numerically solve the Euler equations in an unbounded domain, without prescribing any symmetry. These requirements pose challenges in achieving convergence (due to degeneracies in the Euler equations) and accuracy (as small-scale features develop). We introduce a numerical method that overcomes these limitations. We apply the overall numerical bifurcation methodology to several families of flow equilibria, including elliptical vortices, opposite-signed vortex pairs (of both rotating and translating type), single and double vortex rows, as well as gravity waves. For all flows, we discover new families of equilibria, which exhibit lower symmetry, and obtain stability properties in agreement with detailed linear stability analyses. We conclude by highlighting general solution features that still require explanation.

Analysis of Fluids and Related Topics: Culmination of the inverse cascade - mean flow and fluctuations

Speaker: 
Anna Frishman , Princeton PCTS
Date: 
Apr 13 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

An inverse cascade, energy transfer to progressively larger scales, is a salient feature of two-dimensional turbulence. If the cascade reaches the system scale, it creates a coherent flow expected to have the largest available scale and conform with the symmetries of the domain. In a square doubly periodic domain, the mean flow is expected to take the form of a vortex dipole. The velocity profile of a corresponding single vortex was recently obtained analytically and subsequently confirmed numerically. I will describe the next step in the derivation: using the mean velocity profile to predict features of the turbulent fluctuations.  I will also address the mean flow in a doubly periodic (non-square) rectangle. For a rectangle, the mean flow with zero total momentum was believed to be unidirectional, with two jets along the short side. I will describe how direct numerical simulations reveal that neither the box symmetry is respected nor the largest scale is realized: the flow is never purely unidirectional since the inverse cascade produces coherent vortices, whose number and relative motion are determined by the aspect ratio. This spontaneous symmetry breaking is closely related to the hierarchy of averaging times. Long-time averaging restores translational invariance due to vortex wandering along one direction, and gives jets whose profile, however, can be deduced neither from the largest-available-scale argument, nor from the often employed maximum-entropy principle or quasi-linear approximation.

Analysis of Fluids and Related Topics: Determining the Global Dynamics of the 2D Navier-Stokes Equations by 1D ODE

Speaker: 
Edriss Titi, Texas A&M and Weizmann Institute
Date: 
Apr 7 2017 - 2:00pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 224
Abstract: 

One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters -- finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I  will show how to explore this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipative systems  to design finite-dimensional feedback control for stabilizing their solutions. Notably, it is observed that this very same approach can be implemented for designing data assimilation algorithms of weather prediction based on discrete measurements. In addition, I will also show that the long-time dynamics of the Navier-Stokes equations can be imbedded in an infinite-dimensional dynamical system that is induced by  an ordinary differential equations, named {\it determining form}, which is governed by a globally Lipschitz vector field. Remarkably, as a result of this machinery  I will eventually show that the global dynamics of the Navier-Stokes equations is be determining by only one parameter that is governed by an ODE.  The Navier-Stokes equations  are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs, in particular to various dissipative  reaction-diffusion systems and geophysical models.

Analysis of Fluids and Related Topics: Incompressible flow around small obstacles *Please note this is an additional seminar (5:30)

Speaker: 
Milton Lopes , Instituto de Matemática, UFRJ
Date: 
Apr 6 2017 - 5:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

 In this talk we examine several results concerning the asymptotic behavior of incompressible flow outside small obstacles, including a single obstacle in two and three space dimensions, several small obstacles, viscous and inviscid flows and moving obstacles.

Analysis of Fluids and Related Topics: Stochastic homogenization for reaction-diffusion equations

Speaker: 
Andrej Zlatos, UC San Diego
Date: 
Apr 6 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

We study spreading of reactions in random media and prove that homogenization takes place under suitable hypotheses.  That is, the medium becomes effectively homogeneous in the large-scale limit of the dynamics of solutions to the PDE.  Hypotheses that guarantee this include fairly general stationary ergodic KPP reactions, as well as homogeneous ignition reactions in up to three dimensions perturbed by radially symmetric impurities distributed according to a Poisson point process.  In contrast to the original (second-order) reaction-diffusion equations, the limiting "homogenized" PDE for this model are (first-order) Hamilton-Jacobi equations, and the limiting solutions are discontinuous functions that solve these in a weak sense.  A key ingredient is a novel relationship between spreading speeds and front speeds for these models (as well as a proof of existence of these speeds), which can be thought of as the inverse of a well-known formula in the case of periodic media, but we are able to establish it even for more general stationary ergodic media.

Analysis of Fluids and Related Topics: Brief survey of computer assisted proofs for partial differential equations

Speaker: 
Jacek Cyranka, Rutgers University
Date: 
Mar 30 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

I will present a brief survey of computer assisted methods of studying partial differential equations that I have worked on. The methods I am going to discuss allow for obtaining proofs of the existence of particular solutions of a certain class of PDEs in a prescribed range of parameters.  I will discuss opportunities  and limitations of the presented approach. In particular most of the presented results have not been obtained using known techniques of 'classical analysis'.  I will focus on two particular examples from my research, namely 1) a proof of the existence of globally attracting solutions for the 1d viscous Burgers equation (with non-autonomous forcing) https://arxiv.org/abs/1403.7170, and 2) recent proof of the heteroclinic connections in the 1d Ohta-Kawasaki (diblock copolymers) model https://arxiv.org/abs/1703.01022

Analysis of Fluids and Related Topics: Mixing in Compressible Hydrodynamics as Diffusivities Approach Zero

Speaker: 
Daniel Lecoanet , Princeton University
Date: 
Mar 16 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

The Kelvin-Helmholtz (KH) instability is a prototypical hydrodynamic mixing process driven by velocity shear.  I will present simulations of the KH instability in compressible hydrodynamics.  Compressibility introduces baroclinic instabilities which can further enhance mixing.  I compare simulations run at specific Reynolds numbers to "implicit large eddy simulations'' (ILES) in which numerical errors play the role of a sub-grid scale diffusivity parameterization.  Many of the simulations were run using Dedalus, an open-source spectral code which can solve nearly arbitrary PDEs.  I will then discuss extrapolating our simulations to the limit of Re->infinity, extrapolating the ILES to the limit of resolution approaching infinity, and whether or not these two limits are the same.

An algebraic reduction of the `scaling gap' in the Navier-Stokes regularity problem

Speaker: 
Analysis of Fluids and Related Topics: Zoran Grujic, University of Virginia
Date: 
Mar 9 2017 - 5:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

Please note special time (5:30).  It is shown--within a mathematical framework based on the suitably defined scale of sparseness of the super-level sets of the positive and negative parts of the vorticity components--that the ever-resisting `scaling gap' in the 3D Navier-Stokes regularity problem can be reduced by an algebraic factor; all preexisting improvements have been logarithmic in nature, regardless of the functional set up utilized. The mathematics was inspired by morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of turbulent flows. This is a joint work with A. Farhat and Z. Bradshaw. 

Recent results on 2D density patches for inhomogeneous Navier-Stokes

Speaker: 
Analysis of Fluids and Related Topics: Francisco Gancedo, Universidad de Sevilla
Date: 
Mar 9 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

This talk is about the dynamics of a patch given by two fluids of different constant densities, evolving by the inhomogeneous Navier-Stokes equations. The main question to address is whether the regularity of the boundary of the initial patch is preserved in time. Using classical Sobolev spaces for the velocity, we establish the propagation of C1+γW2,∞anC^{2+\gamma}regularitiewith0<\gamma<1$. The results are based on new cancelations found for time dependent singular integrals given by the linear nonhomogeneous heat kernel acting on quadratic terms. 

 

Analysis of Fluids and Related Topics: On a Slightly Compressible Water Wave

Speaker: 
Chenyun Luo, Johns Hopkins University
Date: 
Mar 2 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

In this talk, I would like to go over some recent results on a compressible water wave. We generalize the apriori energy estimates for the compressible Euler equations established in Lindblad-Luo to when the fluid domain is unbounded.

Analysis of Fluids and Related Topics: "Self-similar vortex spirals"

Speaker: 
Volker Elling , University of Michigan
Date: 
Feb 9 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
322 Fine Hall
Abstract: 

We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The regions of positive and negative vorticity form algebraic spirals. Connections to the problem of non-uniqueness for the Euler equations will be discussed.

TBA

Speaker: 
Edriss Titi, Texas A&M University
Date: 
May 7 2015 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine 322