PDE-Applied Math Archives for Academic Year 2015

Nonlinear Schrödinger equations: The interplay of modeling, numerics and physics

When: Fri, August 14, 2015 - 11:00am
Where: Math 1311
Speaker: Norbert Mauser (Wolfang Pauli Institute and Univ. of Vienna) -

Fractional thin film equations and hydraulic fractures

When: Thu, September 17, 2015 - 3:30pm
Where: Math 3206
Speaker: Antoine Mellet (UMD) -

Transonic problems on multidimensional conservation laws

When: Thu, September 24, 2015 - 3:30pm
Where: Math 3206
Speaker: Eun Heui Kim (California State University Long Beach) -

Spectral representation of generalized Laguerre semigroups and hypocoercitivity

When: Thu, October 1, 2015 - 3:30pm
Where: Math 3206
Speaker: Pierre Patie (Cornell University) -
Abstract: The first aim of this to talk is to present an original methodology for developing the spectral representation of a class of non-self-adjoint (NSA) invariant semigroups. This class is defined in terms of self-similar semigroups on the positive real line and we name it the class of generalized Laguerre semigroups. Our approach is based on an in-depth analysis of an intertwinning relationship that we establish between this class and the classical Laguerre semigroup which is self-adjoint. We proceed by discussing substantial difficulties that one may face when studying the spectral representation of NSA operators.
Finally, we also show that our approach enables us to get precise information regarding the speed of convergence towards stationarity. In particular, we observe in some cases the hypocoercivity phenomena which, in our context, can be interpreted in terms of the spectral norms.

Energy scaling laws for compressed thin elastic sheets

When: Thu, November 5, 2015 - 3:30pm
Where: Math 3206
Speaker: Ian Tobasco (Courant Institute) -
Abstract: A long-standing open problem in elasticity is to identify the minimum energy scaling law of a crumpled sheet of paper as thickness tends to zero. Though much is known about scaling laws for thin sheets in tensile settings, the compressive regime is mostly unexplored. I will discuss the analysis of two examples: an axially compressed thin elastic cylinder, and an indented cone. My focus in this talk will be the dependence of the minimum energy on the thickness and loading in the Foppl-von Karman model. I will prove upper and lower bounds for these scalings. The material for this talk is drawn from two papers in preparation; the work on indented cones is in collaboration with H. Olbermann and S. Conti.

Anomalous diffusion for some kinetic equations

When: Thu, November 19, 2015 - 3:30pm
Where: Math 3206
Speaker: Marjolaine Puel (University of Nice Sophia-Antipolis) -
Abstract: Kinetic equations involve a large number of variables, time, space and velocity and one important part of the study of those equation consists in giving an approximation of their solution for large time and large observation length. For example, when we model collisions via the linear Boltzmann equation, it is well known that when the equilibria are given by Gaussian distributions, we can approximate the solution by the product of an equilibrium that gives the dependence with respect to velocity multiplied by a density depending on time and position that satisfies a diffusion equation. But different models like inelastic collisions lead to heavy tails equilibria for which depending on the power of the tail, we get different situations. When the diffusion coefficient is no more defined, in the case of linear Boltzmann, the density satisfies a fractional diffusion equation. The same kind of problem arises when the interaction between particles are modeled via the Fokker Planck operator with an additional difficulty. I will present a probabilistic method to study the critical case where we obtain still a diffusion but with an anomalous scaling and present the problems arising for the subcritical exponents.

The Optimal Transportation Problem: applications and numerical methods

When: Thu, February 4, 2016 - 3:30pm
Where: Math 3206
Speaker: Prof. Adam Oberman (McGill University (joint Numerical Analysis/PDE seminar)) - http://www.adamoberman.net/
Abstract: The Optimal Transportation problem has been the subject of a great deal of attention by theoreticians in last couple of decades. The Wasserstein (or Earth Mover) distance allows for the metrization of the space of probability measures. However computation of these distances (and the associated maps) has been intractable, except for very small problems.

Current applications of Optimal Transportation include: Freeform Illumination Optics for shaping light or laser beams, Shape Interpolation (in computer graphics), Machine learning (comparing histograms), discretization of nonlinear PDEs (using the gradient flow in the Wasserstein metric), parameter estimation in geophysics, matching problems in mathematical economics, and Density Functional Theory in physical chemistry.

Recent advances have allowed for more efficient computation of solutions of the Monge-Kantorovich problem of optimal transportation. In the special, but important case of quadratic costs, the map can be obtained from the solution of the elliptic Monge-Ampere partial differential equation with nonstandard boundary conditions. For more general costs, the Kantorovich plan can be approximated by a finite dimensional linear program. In this talk we will compare the cost and quality of the solutions obtained by two different methods.

I will also discuss some nonlinear PDE problems (curvature flows, 2-Hessian equation) which can be solved using similar techniques to those applied to the Monge-Ampere PDE.

Why gradient flows of some energies good for defect equilibria are not good for dynamics, and an improvement

When: Thu, February 11, 2016 - 3:30pm
Where: Math 3206
Speaker: Prof. Amit Acharya, Joint CSCAMM/Applied Math Seminar (Civil and Environmental Engineering, Carnegie Mellon University) -
Abstract: Line defects appear in the microscopic structure of crystalline materials (e.g. metals) as well as liquid crystals, the latter an intermediate phase of matter between liquids and solids. Mathematically, their study is challenging since they correspond to topological singularities that result in blow-up of total energies of finite bodies when utilizing most commonly used classical models of energy density; as a consequence, formulating nonlinear dynamical models (especially pde) for the representation and motion of such defects is a challenge as well. I will discuss the development and implications of a single pde model intended to describe equilibrium states and dynamics of these defects. The model alleviates the nasty singularities mentioned above and it will also be shown that incorporating a conservation law for the topological charge of line defects allows for the correct prediction of some important features of defect dynamics that would not be possible just with the knowledge of an energy function.

The flow near the ground state for semilinear evolution equations

When: Tue, February 16, 2016 - 11:00am
Where: Math 3206
Speaker: P. Raphael (University of Nice) -
Abstract: I will consider the question of the study of the flow near the ground state solitary wave for semilinear heat or Schrodinger type equations. I will illustrate on some recent examples both the problem of construction of non trivial dynamics, in particular singularity formation, and the one of the complete classification of the flow. The approach will allow us to distinguish the construction problem, and in particular the one of minimal elements (with one or possibly more blow up bubbles), and the more involved stability problem. Applications will be given in particular in the mass and energy critical settings.

Unsteady flow barriers at scales from geophysics to microfluidics: identification, control, and optimizing transport

When: Thu, February 25, 2016 - 3:30pm
Where: Math 3206
Speaker: Sanjeeva Balasuriya (University of Adelaide) -
Abstract: Unsteady flows typically possess blobs of particles moving coherently, in addition to regions in which extensive mixing occurs. The boundaries of each of these structures might be considered to be "unsteady flow barriers." Exactly defining what these are is, however, problematic. Such flow barriers may be thought to demarcate geophysical features such as the Antarctic Circumpolar Vortex (ozone hole), or the interface between two fluids that one desires to mix together for DNA synthesis in a microfluidic device. This talk will examine some recent results on the control of unsteady flow barriers, and how one might attempt to optimize mixing across an unsteady flow barrier. Additionally, some ongoing work on how these ideas can be used to correct errors in oceanic velocity data obtained from satellite observations will be briefly discussed.

Nonlocal reaction-diffusion equations and evolutionary ecology

When: Thu, March 24, 2016 - 3:30pm
Where: Math 3206
Speaker: Olga Turanova (UCLA) -
Abstract: We consider a reaction-diffusion equation with a nonlocal reaction term that arises as a model in evolutionary ecology. We study asymptotic behavior and global bounds for solutions of this PDE.

The H-principle in Fluid Mechanics

When: Thu, March 31, 2016 - 3:30pm
Where: Math 3206
Speaker: Douglis lecture: Laszlo Szekelyhidi (University of Leipzig) -
Abstract: It is known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence.

In joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we interpret the wild behaviour of weak solutions of Euler as an instance of Gromov’s h-principle. In this lecture I will explain this connection and outline recent progress concerning Onsager’s conjecture.

A Lagrangian Fluctuation-Dissipation Relation for Scalar Turbulence

When: Fri, April 1, 2016 - 2:00pm
Where: 1313.0
Speaker: Theo Drivas (Johns Hopkins) -
Abstract: A common approach to calculate the solution of a scalar advection-diffusion
equation is by a Feynman-Kac representation which averages over stochastic Lagrangian
trajectories going backward in time to the initial conditions and boundary data. The trajectories
are obtained by solving SDE's with the advecting velocity as drift and a backward Itō term representing the scalar diffusivity. In this framework, we present an exact formula for scalar dissipation in terms
of the variance of the scalar values acquired along each random trajectory. As an important
application, we study the connection between anomalous scalar dissipation in turbulent flows for
large Reynolds and Péclet numbers and the spontaneous stochasticity of the Lagrangian particle
trajectories. The latter property corresponds to the Lagrangian trajectories remaining random
in the limit Re,Pe→∞, when the backward Itō term formally vanishes but the advecting velocity
field becomes non-Lipschitz. For flows on domains without boundaries (e.g. tori, spheres) and
for wall-bounded flows with no-flux Neumann conditions for the scalar, we prove that spontaneous
stochasticity is necessary and sufficient for anomalous scalar dissipation. The fluctuation-dissipation
relation provides a Lagrangian representation of scalar dissipation also in turbulent flows where
present experiments suggest that dissipation is tending to zero as Re,Pe→∞. We discuss an
illustrative example of Rayleigh-Bénard convection with imposed heat-flux at the top and bottom
plates. Our formula here shows that the scalar dissipation is given by the variance of the local
time densities of the stochastic particles at the heated boundaries. The ``ultimate regime'' of
turbulent convection predicted by Kraichnan-Spiegel occurs when the near-wall particle densities
are mixed to their asymptotic uniform values in a large-scale turnover time. The current observations
of vanishing scalar dissipation require that fluid particles be trapped at the wall and remain unmixed
for many, many large-scale turnover times. This talk presents joint work with Gregory Eyink.

Stochastic Homogenization of Reaction-Diffusion Equations in Stationary-Ergodic and Isotropic Media

When: Thu, April 7, 2016 - 3:30pm
Where: 3206.0
Speaker: Jessica Lin (University of Wisconsin, Madison) -
Abstract: We consider reaction-diffusion equations with combustion
nonlinearity in stationary-ergodic and isotropic environments in
dimensions $d≤3$. We prove the existence of asymptotic, deterministic
speeds of propagation for solutions with both spark-like and
front-like initial data in random heterogeneous media. This leads to a
general stochastic homogenization result which shows that on average,
the large-scale large-time behavior is governed by a deterministic
Hamilton-Jacobi equation modeling front propagation. Applications
include predicting the evolution of forest fires in random isotropic
environments. This talk is based on joint work with Andrej Zlatos.

A Structure Preserving Discretization For Liquid Crystals With External Fields (joint Numerical Analysis and PDE Seminar)

When: Thu, April 14, 2016 - 3:30pm
Where: Math 3206
Speaker: Prof. Shawn Walker (Department of Mathematics and center for computation and technology, Louisiana State University)) - https://www.math.lsu.edu/~walker/
Abstract: We present a finite element method (FEM) for computing equilibrium configurations of liquid crystals with variable degree of orientation. The model consists of a Frank-like energy with an additional "s" parameter that allows for line defects with finite energy, but leads to a degenerate elliptic equation for the director field. Our FEM uses a special discrete form of the energy that does not require regularization, and allows us to obtain a stable (gradient flow) scheme for computing minimizers of the energy. We also include external fields to model the so-called "Freedricksz Transition". Simulations in 2-D and 3-D are presented to illustrate the method.

On ill-posedness of Euler system with non-local terms

When: Thu, April 21, 2016 - 3:30pm
Where: Math 3206
Speaker: Agnieszka Swierczewska-Gwiazda (Institute of Applied Mathematics and Mechanics, University of Warsaw (joint KI-Net PDE-Applied Math seminar)) -
Abstract: The talk will concern the issue of existence of weak solutions to the Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity appearing in collective behavior patterns.

We consider several modifications of the Euler system of fluid dynamics including its pressureless variant driven by non-local interaction repulsive-attractive and alignment forces in the space dimension $N=2,3$. These models arise in the study of self-organisation in collective behavior modeling of animals and crowds.

We adapt the method of convex integration, adapted to the incompressible Euler system by De Lellis and Szekelyhidi, to show the existence of infinitely many global-in-time weak solutions for any bounded initial data. Then we consider the class of dissipative solutions satisfying, in addition, the associated global energy balance (inequality).

The discussed result is in a certain sense negative result concerning stability of particular solutions. It turns out that the solutions must be sought in a stronger class than that of weak and/or dissipative solutions. We essentially show that there are infinitely many weak solutions for any initial data and that there is a vast class of velocity fields that gives rise to infinitely many admissible (dissipative) weak solutions. We may therefore infer that the class of weak solutions is not convenient for analysing certain qualitative properties such as stability and formation of the flock patterns. However, we also show that the strong solutions are robust in a larger class of all admissible (dissipative) weak solutions leading to the possibility of establishing certain stability results of flock solutions. We establish a weak-strong uniqueness principle for the pressure driven Euler system with non-local interaction terms as well as for the pressureless system with Newtonian interaction.

The talk is based on the following result: J. A. Carrillo, E. Feireisl, P. Gwiazda, and A. \'Swierczewska-Gwiazda. Weak solutions for Euler systems with non-local interactions, arXiv:1512.03116

Quantum Dots and Dislocations: Dynamics of Materials Defects

When: Wed, May 4, 2016 - 3:15pm
Where: Math 3206
Speaker: Aziz Lecture: Prof. Irene Fonseca (Department of Mathematical Science, Carnegie Mellon University) - http://www.math.cmu.edu/math/faculty/Fonseca
Abstract: The formation and assembly patterns of quantum dots have a significant impact on the optoelectronic properties of semiconductors. We will address short time existence for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained three-dimensional films. Further, the nucleation of misfit dislocations will be analyzed. This is joint work with Nicola Fusco, Giovanni Leoni and Massimiliano Morini.

Homogenization of Integral Energies Under Periodically Oscillating Differential Constraints

When: Thu, May 5, 2016 - 3:30pm
Where: Math 3206
Speaker: Prof. Irene Fonseca (Department of Mathematical Science, Carnegie Mellon University) - http://www.math.cmu.edu/math/faculty/Fonseca
Abstract: A homogenization result for a family of integral energies is presented, where the fields are subjected to periodic first order oscillating differential constraints in divergence form. We will give an example that illustrates that, in general, when the operators differential operators have non constant coefficients then the homogenized functional maybe be nonlocal, even when the energy density is convex. This is joint work with Elisa Davoli, and is based on the theory of A-quasiconvexity with variable coefficients and on two-scale convergence techniques.