PDE-Applied Math Archives for Academic Year 2017


Characteristic fast marching method for approximating the viscosity solution of generalized eikonal equation

When: Thu, September 15, 2016 - 3:30pm
Where: MATH0407
Speaker: Daisy Dahiya (UMD) -
Abstract: The propagation of a wavefront in an inhomogeneous moving medium is governed by generalized eikonal equation. If the medium of propagation is at rest then the governing equation is the eikonal equation. Fast marching method is a computationally efficient numerical method for approximating the viscosity solution of eikonal equation. But the method fails in a moving medium. In this work we present a generalization of fast marching method for approximating the viscosity solution of generalized eikonal equation.

On a Boltzmann mean field model for knowledge growth

When: Thu, September 22, 2016 - 3:30pm
Where: MATH 0407
Speaker: Alexander Lorz (Universite Pierre et Marie Curie - Paris 6) - http://www.alexanderlorz.com/
Abstract: We analyze a Boltzmann type mean field game model for knowledge
growth, which was proposed by Lucas and Moll. We discuss the underlying
mathematical model, which consists of a coupled system of a Boltzmann type
equation for the agent density and a Hamilton-Jacobi-Bellman equation for the
optimal strategy. We study the analytic features of each equation separately
and show local in time existence and uniqueness for the fully coupled system.
Furthermore we focus on the existence of special solutions,
which are related to exponential growth in time - so called balanced growth path
solutions.
This is joint work with Martin Burger and Marie-Therese Wolfram

On the surface signature of internal waves in the ocean

When: Thu, October 6, 2016 - 3:30pm
Where: MATH 0407
Speaker: Philippe Guyenne (University of Delaware) - http://www.math.udel.edu/~guyenne/
Abstract: Based on a Hamiltonian formulation of a two-layer ocean, we consider the situation in which internal waves are treated in the long-wave regime while surface waves are described in the modulation regime. We derive an asymptotic model for surface-internal wave interactions, in which the nonlinear internal waves evolve according to a KdV equation while the smaller-amplitude surface waves propagate at a resonant group velocity and their envelope is described by a linear Schrodinger equation.
In the case of an internal soliton of depression, for small depth and density ratios of the two layers, the Schrodinger equation is shown to be in the semi-classical regime in analogy with quantum mechanics, and thus admits localized bound states. This leads to the phenomenon of trapped surface modes, which propagate as the signature of the internal wave, and thus it is proposed as a possible explanation for bands of surface roughness above internal waves in the ocean.
Some numerical simulations taking oceanic parameters into account are also performed to illustrate this phenomenon. This is joint work with Walter Craig and Catherine Sulem.


Deterministic and stochastic aspects of fluid mixing

When: Thu, October 13, 2016 - 3:30pm
Where: MATH0407
Speaker: Michele Coti Zelati (Department of Mathematics - UMD) - http://www.math.umd.edu/~micotize/
Abstract: The process of mixing of a scalar quantity into a homogenous fluid is a familiar physical phenomenon that we experience daily. In applied mathematics, it is also relevant to the theory of hydrodynamic stability at high Reynolds numbers - a theory that dates back to the 1830's and yet only recently developed in a rigorous mathematical setting. In this context, mixing acts to enhance, in certain
senses, the dissipative forces. Moreover, there is also a transfer of information from large length-scales to small length-scales vaguely analogous to, but much simpler than, that which occurs in turbulence. In this talk, we focus on the study of the implications of these fundamental processes in linear settings, with particular emphasis on the long-time dynamics of deterministic systems (in terms of sharp decay estimates) and their stochastic perturbations (in terms of invariant measures).

A Non-local Variational Problem Arising from Studies of Nonlinear Charge Screening in Graphene Monolayers

When: Thu, October 20, 2016 - 3:30pm
Where: MATH 0407
Speaker: Cyrill Muratov (New Jersey Institute of Technology) - https://web.njit.edu/~muratov/
Abstract: This talk is concerned with energy minimizers in an orbital-free density functional theory that models the response of massless fermions in a graphene monolayer to an out-of-plane external charge. The considered energy functional generalizes the Thomas-Fermi energy for the charge carriers in graphene layers by incorporating a von-Weizsaecker-like term that penalizes gradients of the charge density. Contrary to the conventional theory, however, the presence of the Dirac cone in the energy spectrum implies that this term should involve a fractional Sobolev norm of the square root of the charge density. We formulate a variational setting in which the proposed energy functional admits minimizers in the presence of an out-of-plane point charge. The associated Euler-Lagrange equation for the charge density is also obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. In addition, a bifurcation from zero to non-zero response at a finite threshold value of the external charge is proved. This is joint work with J. Lu (Duke University) and V. Moroz (Swansea University).

Zika Virus Dynamics: the Implications of the Sexual Transmission Pathway

When: Thu, October 27, 2016 - 3:30pm
Where: MATH 0407
Speaker: Fola B. Agusto (Dept of Ecology and Evolutionary Biology - University of Kansas) - https://sites.google.com/site/agustofb/


Effective models for Ginzburg-Landau vortices

When: Wed, November 2, 2016 - 3:15pm
Where: CSIC 4122
Speaker: Sylvia Serfaty (Douglis Lecture) (NYU) - http://www.math.nyu.edu/~serfaty/
Abstract: Ginzburg-Landau type equations are models for superconductivity,
superfluidity, Bose-Einstein
condensation. A crucial feature is the presence of quantized vortices, which
are topological zeroes of the complex-valued solutions. This talk will
review some results
on the derivation of effective models to describe the statics and dynamics
of these vortices,
with particular attention to the situation where the number of vortices
blows up with the
parameters of the problem. In particular we will present new results on
the derivation of mean field limits
for the dynamics of many vortices starting from the parabolic
Ginzburg-Landau equation or
the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.

Nonlinear Flow in Microfluidic Devices

When: Thu, November 10, 2016 - 3:30pm
Where: MATH0407
Speaker: Kaitlyn Hood (MIT) - http://www.kaitlynhood.com
Abstract: Typically, microfluidic devices are modeled by linear PDEs because the length scales and velocity scales are small so that the Reynolds number is close to zero. However, in some medical devices, large flow velocities are used to access nonlinear inertial effects. In this case, the flow is described by the Navier-Stokes equations where the Reynolds number is moderately large, on the order of 10 to 100. I will discuss a mathematical model using numerical methods combined with singularity solutions via perturbation methods. This model reduces computational complexity and produces a scaling law that can be used to design microfluidic devices.


On the Musket problem

When: Thu, November 17, 2016 - 3:30pm
Where: Kirwan Hall 0407
Speaker: Robert Strain (University of Pennsylvania) - https://www.math.upenn.edu/~strain/
Abstract: The Muskat problem models the dynamics of an interface between two incompressible immiscible fluids with different characteristics, in porous media. The phenomena have been described using the experimental Darcy’s law. Saffman and Taylor (1958) related this problem with the evolution of an interface in a Hele-Shaw cell since both physical scenarios can be modeled analogously. In this talk we will discuss existence results, singularity results, and long time decay behavior of the Muskat problem in 2D and in 3D.

PDE models of biological growth

When: Thu, December 8, 2016 - 3:30pm
Where: CSIC 4122
Speaker: Alberto Bressan (Penn State University) - https://www.math.psu.edu/bressan/
Abstract: Living tissues, such as stems, leaves and flowers in plants and bones in animals, grow into a great variety of shapes. In some cases, Nature has found ways to control this growth with remarkable accuracy.

In this talk I shall discuss some free boundary problems modeling controlled growth, namely

(I) Growth of 1-dimensional curves in R^3 (plant stems), where stabilization
in the vertical direction is achieved by a feedback response to gravity.

(II) Growth of 2 or 3-dimensional domains, controlled by the concentration of a morphogen, coupled with the minimization of an elastic deformation energy.

Some recent existence, uniqueness, and stability results will be presented, together with numerical simulations. Further research directions will be discussed.

Turbulent weak solutions of the Euler equations

When: Tue, January 17, 2017 - 2:00pm
Where: Kirwan Hall 1308
Speaker: Vlad Vicol (Princeton University) -
Abstract: Motivated by Kolmogorov's theory of hydrodynamic turbulence, we consider dissipative weak solutions to the 3D incompressible Euler equations. We show that there exist infinitely many weak solutions of the 3D Euler equations, which are continuous in time, lie in a Sobolev space H^s with respect to space, and they do not conserve the kinetic energy. Here the smoothness parameter s is at the Onsager critical value 1/3, consistent with Kolmogorov's -4/5 law for the third-order structure functions. We shall also discuss bounds for the second order structure functions, which deviate from the classical Kolmogorov 1941 theory. This talk is based on joint work with T. Buckmaster and N. Masmoudi.

Effective dynamics of nonlinear Schrodinger equations on large domains

When: Thu, March 2, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Zaher Hani (Georgia Tech) - http://people.math.gatech.edu/~zhani6/
Abstract: While the long-time behavior of small amplitude solutions to nonlinear dispersive and wave equations on Euclidean spaces (R^n) is relatively well-understood, the situation is marked different on bounded domains. Due to the absence of dispersive decay, a very rich set of dynamics can be witnessed even starting from arbitrary small initial data. In particular, the dynamics in this setting is characterized by out-of-equilibrium behavior, in the sense that solutions typically do not exhibit long-time stability near equilibrium configurations. This is even true for equations posed on very large domains (e.g. water waves on the ocean) where the equation exhibits very different behaviors at various time-scales.

In this talk, we shall consider the nonlinear Schr\"odinger equation posed on a large box of size $L$. We will analyze the various dynamics exhibited by this equation when $L$ is very large, and exhibit a new type of dynamics that appears at a particular large time scale (that we call the resonant time scale). The rigorous derivation of this dynamics relies heavily on tools from analytic number theory. This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah (all at Courant Institute, NYU).

A new Heintze-Karcher inequality with boundary; with an application to small droplets.

When: Thu, March 9, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Matias Delgadino (ICTP) -
Abstract: Motivated by an application of H-K inequality to characterize almost mean constant mean curvature surfaces, we develop a H-K inequality for surface with boundary. As an application we characterize certain critical points of the capillarity energy.

Onsager conjecture and beyond

When: Thu, March 16, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Tristan Buckmaster (New York University) - http://www.cims.nyu.edu/~tristanb/
Abstract: In this talk I will discuss new results related to Onsager's conjecture and non-uniqueness to fluid equations.

BV estimates in optimal transport and applications

When: Thu, March 30, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Alpar Meszaros (UCLA) -
Abstract: In this talk the main question that I will consider is the regularity of solutions of certain variational problems in optimal transport. In particular I will be interested in the Wasserstein projection of a measure with BV density on the set of measures with densities bounded by a given BV function f. I will show that the projected measure is of bounded variation as well with a precise estimate of its BV norm. Of particular interest is the case f = 1, corresponding to a projection onto a set of densities with an $L^\infty$ bound, where one can prove that the total variation decreases by the projection. This estimate and, in particular, its iterations have a natural application to some evolutionary PDEs as, for example, the ones describing a crowd motion. In fact, as an application of our results, one can obtain BV estimates for solutions of some non-linear parabolic PDEs by means of optimal transport techniques. The talk is based on a joint work with G. De Philippis (SISSA, Italy), F. Santambrogio (Orsay, France) and B. Velichkov (Grenoble, France).

Pursuing random targets identified at random times: optimality, robustness, and computational cost

When: Thu, April 6, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Alexander Vladimirsky (Cornell University) - http://www.math.cornell.edu/~vlad/
Abstract: The classical tools of optimal control theory yield the best strategy for going from where you are right now to where you want to be in the future. But what if your target is selected randomly and is only revealed at a random later time T? Should you just do nothing until this happens? Should you only optimize the expected total cost or can you also provide some guarantees about the worst-case scenario?

I will use simple 1- and 2-dimensional examples to show how "free boundaries" and discontinuities arise based on our answers to the above questions. These phenomena pose different computational challenges & influence our choice of discretization/solution strategy for PDEs encoding the optimality.

This talk is meant to be self-contained and will not assume any prior background in control theory, dynamic programming, or Hamilton-Jacobi PDEs.

A Proof of Onsager’s Conjecture for the Incompressible Euler Equations

When: Thu, April 13, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Phil Isett (University of Texas, Austin) -
Abstract: In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder.  I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time.  The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity.  A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture.  The proof of the full conjecture combines convex integration using the “Mikado flows” introduced by Daneri-Székelyhidi with a new “gluing approximation” technique.  The latter technique exploits a special structure in the linearization of the incompressible Euler equations.


Min-max formulas for nonlocal elliptic operators and applications

When: Thu, May 4, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Nestor Guillen (University of Massachusetts) - http://people.math.umass.edu/~nguillen/
Abstract: A mapping $F$ between spaces of real valued functions is said to have the "global comparison property" (GCP) if $u\leq v$ everywhere with $u=v$ at some point $x$ means that $F(u)\leq F(v)$ at this point $x$. A classical result of Courrège says that a continuous linear map from $C^2(\mathbb{R}^d)$ to $C^0(\mathbb{R}^d)$ has the GCP if and only if it is a sum of jump and drift-diffusion operators. In work with Russell Schwab, we characterize nonlinear maps having the GCP as those given by a min-max of linear operators having the GCP. This result provides representation formulas for the Dirichlet-to-Neumann map of nonlinear elliptic equations, and for the interface velocity for various free boundary problems, respective applications will be discussed along with a list of related questions which are open.

Computing optimal transport maps via optimization

When: Thu, May 11, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Yanir Rubinstein (UMCP) - http://www.math.umd.edu/~yanir
Abstract: In joint work with M. Lindsey we rephrase the optimal transportation problem with quadratic cost--via a Monge-Ampere equation--as an infinite-dimensional optimization problem, which is often a convex problem. This leads us to define a natural finite-dimensional discretization to the problem and ultimately develop a numerical scheme for which we prove a convergence result.