Analysis of Fluids Seminars

Analysis of Fluids and Related Topics: Lattice Hydrodynamics

Speaker: 
Dennis Sullivan, Stony Brook University
Date: 
Feb 22 2018 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

We construct a particular lattice model of 3D incompressible fluid motion with viscosity parameter. The construction follows the momentum derivation of  the continuum model switching to combinatorial topology just before taking the calculus limit. The lattice consists of two interpenetrating face centered cubic lattices. [the structure of NaCl]. The lattice of sites organizes a chain complex L of four vector spaces built from overlapping uniform cubes, faces, edges and sites giving a multilayered covering of periodic three space. There are two nilpotent operators on L, a duality  involution, each of odd degree, and a combinatorial Laplacian on L. The result of the momentum derivation is an ODE on one degree of L which is a combinatorial version of the continuum model. The goal of work in progress is to use the model both to derive theory and to compute at a given scale.

Analysis of Fluids and Related Topics: Rogue waves and large deviations in nonlinear Schroedinger models

Speaker: 
Eric Vanden-Eijnden, Courant Institute of Mathematical Sciences, New York University
Date: 
Feb 8 2018 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

The appearance of rogue waves in deep sea is investigated using the modified nonlinear Schroedinger (MNLS) equation with random initial conditions that are assumed to be Gaussian distributed, with a spectrum approximating the JONSWAP spectrum obtained from observations of the North Sea. It is shown that by supplementing the incomplete information contained in the JONSWAP spectrum with the MNLS dynamics one can reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. This is achieved by identifying ocean states that are precursors to rogue waves, which also permits their early detection. Our findings indicate that rogue waves in MNLS obey a large deviation principle—i.e., they are dominated by single realizations—which we calculate by solving an optimization problem.  This method generalizes to estimate the probability of extreme events in other deterministic dynamical systems with random initial data and/or parameters, by using prior information about the nature of their statistics.

Analysis of Fluids and Related Topics: New Integrals of Motion and Singularities in 2D Fluid Dynamics with Free Surface

Speaker: 
Sergey Dyachenko, University of Illinois at Urbana-Champaign.
Date: 
Dec 14 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

We study the problem of 2D incompressible fluid dynamics with free surface, we assume
the the fluid is ideal and the flow is potential. Following the conformal mapping
technique we reformulate the problem to surface variables and demonstrate the existence
of previously undiscovered constants of motion associated with singularities in the
analytic continuation of conformal map and complex potential. In numerical simulations
we recover the analytic structure of the surface shape and observe simple poles and
branch point singularities of the square-root type. We use the Alpert-Greengard-Hagstrom
method to recover the location, type and magnitude of the singularities. We show how
the approach of square-root type singularities may be responsible for the breaking of
waves in the ocean, following the nonlinear stage of modulational instability.

Analysis of Fluids and Related Topics: Winding of Brownian trajectories and heat kernels on covering spaces

Speaker: 
Gautam Iyer, Carnegie Mellon University
Date: 
Dec 7 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

We study the long time behaviour of the heat kernel on Abelian covers of compact Riemannian manifolds. For manifolds without boundary work of Lott and Kotani-Sunada establishes precise long time asymptotics. Extending these results to manifolds with boundary reduces to a "cute" eigenvalue minimization problem, which we resolve for a Dirichlet and Neumann boundary conditions. We will show how these results can be applied to studying the "winding" of Brownian trajectories in Riemannian manifolds.

Analysis of Fluids and Related Topics: SINGULARITY FORMATION IN THE CONTOUR DYNAMICS FOR 2D EULER EQUATION ON THE PLANE, SPEAKER: SERGUEI DENISSOV

Speaker: 
Serguei Denissov, University of Wisconsin
Date: 
Nov 30 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

We will study 2d Euler dynamics of centrally symmetric pair of patches on the plane. In the presence of exterior regular velocity, we will show that these patches can merge so fast that the distance between them allows double-exponential upper bound which is known to be sharp. The formation of the 90 degree corners on the interface and the self-similarity analysis of this process will be discussed. For a model equation, we will prove existence of the curve of smooth stationary solutions that originates at singular stationary steady state.

Analysis of Fluids and Related Topics: Well-posedness problems for the magneto-hydrodynamics models

Speaker: 
Mimi Dai, University of Illinois at Chicago
Date: 
Nov 16 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

We will talk about some recent results on the well-posedness problems in Sobolev spaces for the magneto-hydrodynamics with and without Hall effect, i.e., the Hall MHD and classical MHD models. One of the purposes of the work is to search the optimal Sobolev space of well-posedness for the two models. Another purpose is to understand the nonlinear Hall term $\nabla\times((\nabla\times b)\times b)$ in the Hall MHD, which appears more singular than $u\cdot\nabla u$ in the NSE, but with special geometry.

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  *CLICK ON NAME LINKS BELOW TO VIEW PRESENTATION SLIDES

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 Pasqualotto

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: Self-similar structure of caustics and shock formation

Speaker: 
Jens Eggers, University of Bristol
Date: 
Oct 26 2017 - 4:30pm
Event type: 
Analysis of Fluids and Related Topics
Room: 
Fine Hall 322
Abstract: 

Caustics are places where the light intensity diverges, and where the wave front has a singularity. We use a self-similar description to derive the detailed spatial structure of a cusp singularity, from where caustic lines originate. We use this insight to study shock formation in the dKP equation, as well as shocks in classical compressible Euler dynamics. The spatial structure of these shocks is that of a caustic, and is described by the same similarity equation.

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.