PDE-Applied Math Archives for Academic Year 2013

Optimal transport and regularity of c-convex potentials

When: Thu, September 12, 2013 - 3:30pm
Where: Math 3206
Speaker: Nestor Guillen (UCLA) - http://www.math.ucla.edu/~nestor/
Abstract: The question of regularity in optimal transport and related equations
of Monge Ampere type has seen a lot of activity in the past few
decades. Starting from the usual quadratic cost in R^n and now ranging
arbitrary costs in Riemannian manifolds (and the related reflector
antenna problems)

In this talk, we will give an impressionistic description of
Caffarelli's regularity theory for the Monge Ampere equation in
Euclidean space, which strongly uses the affine invariance of the
equation. We will see when and how such a theory can be pushed to
general costs,. The new observation is that in general regularity
arises not so much from affine invariance, but rather from two
opposite inequalities for the Mahler volume of c-convex sets (a kind
of generalized Blaschke-Santaló inequalities). The validity of such
inequalities are closely tied to the fourth order Ma-Trudinger-Wang
tensor of the cost but they do not require the C^4 regularity of the
cost. Based on joint work with Jun Kitagawa.

Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation

When: Thu, October 10, 2013 - 3:30pm
Where: Math 3206
Speaker: Xuwen Chen (Brown University) -
Abstract: We consider the dynamics of N bosons in one dimension. We derive rigorously the one dimensional focusing cubic NLS with a quadratic trap as the N->\infty limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system.

On the relativistic Vlasov-Maxwell System from 2D to 3D

When: Thu, October 24, 2013 - 3:30pm
Where: Math 3206
Speaker: Robert Strain (U. Penn) -
Abstract: We consider the relativistic Vlasov-Maxwell (VM) system with initial data of unrestricted size. In the 3D case, since the work of Glassey-Strauss in 1986, we know that as long as the 3D momentum support remains bounded then solutions can be continued and they will remain regular. We prove that as long as there exists a plane upon which the momentum support remains bounded then solutions can be continued and they will remain regular. In the 2D and the 2.5D cases, Glassey-Schaeffer proved in a series of works (1997-1998) that for regular initial data with compact momentum support VM has global in time solutions. They further prove that the electromagnetic fields are bounded in space with possibly double exponential growth in time. We prove in the 2D and the 2.5D cases that if regular initial data has suitable bounded moments, then VM has global in time solutions with fields that grow at most polynomially in time. These are joint works with Jonathan Luk.

Coagulation dynamics of random shocks and phase boundaries -(Joint PDE/Numerical Analysis)

When: Thu, November 7, 2013 - 2:30pm
Where: Math 3206 (Unusual Time)
Speaker: Robert Pego (Carnegie Mellon University) -
Abstract: Smoluchowski's coagulation equation is a simple kinetic model for clustering, and its dynamics are still poorly understood except for a few cases `solvable' by Laplace transform. However, among these cases are some directly connected with classic problems in PDE. Two relate to random solutions of the inviscid Burgers equation and models of the coarsening of patterns of phase boundaries for the Allen-Cahn equation. I plan to review existing results on scaling dynamics for these particular models of ballistic aggregation and annihilation, indicate current progress and describe some of the numerous open problems in this subject.

Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti

When: Thu, November 14, 2013 - 3:30pm
Where: Math 3206
Speaker: Natasa Pavlovic (The University of Texas at Austin) - http://www.ma.utexas.edu/users/natasa/
Abstract: The derivation of nonlinear dispersive PDE, such as the nonlinear
Schrodinger (NLS) or nonlinear Hartree (NLH) equations, from many body
quantum dynamics is central topic in mathematical physics, which has been
approached by many authors in a variety of ways. In particular, one way to
derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an
infinite system of coupled linear non-homogeneous PDE. The most involved
part in such a derivation of NLS consists in establishing uniqueness of
solutions to the GP. That was achieved in seminal papers of
Erdos-Schlein-Yau. Recently, with T. Chen, C. Hainzl and R. Seiringer
we obtained a new, simpler proof of the unconditional uniqueness of
solutions to the cubic Gross-Pitaevskii hierarchy in R^3. One
of the main tools in our analysis is the quantum de Finetti theorem.

In the talk, we will present a brief review of the derivation of NLS
via the GP, describing the context in which the new uniqueness result
appears, and will then focus on the uniqueness result itself.

PDE/ODE models of motility in active biosystems

When: Thu, November 21, 2013 - 3:30pm
Where: Math 3206
Speaker: Leonid Berlyand (Pennsylvania State University) -
Abstract: In the first part of the talk we present a review of our work on PDE/ODE models of swimming bacteria. First, a stochastic PDE model is introduced for a dilute suspension of self-propelled bacteria. Using this model, an explicit asymptotic formula for the effective viscosity (EV) is obtained that explains the mechanisms of the drastic reduction of EV. Next, we introduce a model for semi-dilute suspensions with pairwise interactions and excluded volume constraints. We compute EV analytically (based on a kinetic theory approach) and numerically. We then analyze the onset of collective motion in bacterial suspensions by developing a PDE/ODE model that captures the phase transition in the bacterial suspension { an appearance of correlations and large scale structures due to interbacterial interactions. Collisions of the bacteria, ignored in most of the previous works, play an important role in this study.
Collaborators: Ryan, Haines (all PSU students); Aronson, Sokolov, Karpeev (all Argonne).
In the second part of the talk we consider a system of two parabolic PDEs arising in modeling of motility of eukaryotic cells on substrates. The two key properties of this system are (i) the presence of gradients in the coupling terms (gradient coupling) and (ii) the mass (volume) preservation constraints. We derive the equation for the motion of the cell boundary, which is the mean curvature motion perturbed by a novel nonlinear term and prove that the sharp interface property of initial conditions is preserved in time. The numerical solution of this equation shows that an asymmetry in the cell boundary persists leading to a net motion as observed in recent experiments.
Collaborators: Rybalko (Inst. Low Temp. Physics and Engineering) and Potomkin (PSU)

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

When: Wed, December 4, 2013 - 2:00pm
Where: Math 3206
Speaker: Jacob Bedrossian (Courant Institute, NYU) -
Abstract: We prove asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically mixed to small scales by an almost linear evolution and in general enstrophy is lost in the weak limit. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. Joint work with Nader Masmoudi.

Free boundaries in random domains, from an invariance principle to the homogenization of a free boundary problem

When: Thu, December 12, 2013 - 2:00pm
Where: Math 3206
Speaker: Nestor Guillen (UCLA) - http://www.math.ucla.edu/~nestor/
Abstract: Free boundary problems are models where an unknown physical field is coupled with an unknown submanifold of the underlying physical space (the "free boundary"), i.e. the temperature around a melting crystal, which interacts naturally with its solid/liquid interface. The analysis of such problems combines ideas from geometric measure theory and harmonic analysis. This talk deals with these ideas in a random setting, specifically, the analysis of a (one-phase) Hele-Shaw type model set in a domain with many (random) microscopic obstacles. The main result is that the free boundaries converge (in the macroscopic limit) to the solution of an effective, deterministic problem. This is made possible by new pointwise estimates for linear elliptic equations in perforated domains which are used to control the geometry of the interface. Many of the ideas and tools have a parallel in statistical mechanics, in fact, at the linear level, the proof leads to an invariance principle and estimates for the transition probabilities of reflected Brownian motion on perforated domains.

Fast algorithms for electronic structure analysis

When: Mon, December 16, 2013 - 11:00am
Where: Math 3206
Speaker: Lin Lin (Lawrence Berkeley National Laboratory) -
Abstract: Kohn-Sham density functional theory (KSDFT) is the most widely used electronic structure theory for molecules and condensed matter systems. For a system with N electrons, the standard method for solving KSDFT requires solving N eigenvectors for an O(N) * O(N) Kohn-Sham Hamiltonian matrix. The computational cost for such procedure is expensive and scales as O(N^3). We have developed pole expansion plus selected inversion (PEXSI) method, in which KSDFT is solved by evaluating the selected elements of the inverse of a series of sparse symmetric matrices, and the overall algorithm scales at most O(N^2) for all materials including insulators, semiconductors and metals. The PEXSI method can be used with orthogonal or nonorthogonal basis set, and the physical quantities including electron density, energy, atomic force, density of states, and local density of states are calculated accurately without using the eigenvalues and eigenvectors. The recently developed massively parallel PEXSI method has been implemented in SIESTA, one of the most popular electronic structure software using atomic orbital basis set. The resulting method can allow accurate treatment of electronic structure in a unprecedented scale. We demonstrate the application of the method for solving graphene-like structures with more than 20,000 atoms, and the method can be efficiently parallelized 10,000 - 100,000 processors on Department of Energy (DOE) high performance machines.

Fokas' method for linear evolution equations: a spectral interpretation

When: Thu, March 6, 2014 - 3:30pm
Where: Math 3206
Speaker: David Smith (University of Cincinnati) -
Abstract: We study linear, constant-coefficient evolution PDE, of arbitrary spatial order, in 1 space and 1 time dimension, equipped with an initial condition and arbitrary linear boundary conditions. Fokas' method allows us to derive a transform-inverse transform pair, tailored to any such well-posed problem and which may be used to solve that problem. By analogy with the classical trigonometric Fourier transform methods for the heat equation, we provide a spectral interpretation of the Fokas transforms. In the process, we define a new species of spectral functional and show how they diagonalize the non-self-adjoint spatial differential operator.

Inviscid limits for the stochastic Navier-Stokes equations

When: Thu, March 13, 2014 - 3:30pm
Where: Math 3206
Speaker: Vlad Vicol (Princeton university) -
Abstract: We discuss recent results on the behavior in the infinite Reynolds number limit of invariant measures for the 2D stochastic Navier-Stokes equations. Invariant measures provide a canonical object which can be used to link the fluids equations to the heuristic statistical theories of turbulent flow. We prove that the limiting inviscid invariant measures are supported on bounded vorticity solutions of the 2D Euler equations. This is joint work with N. Glatt-Holtz and V. Sverak.

Van der Corput estimates for oscillatory Riemann Hilbert problems

When: Thu, March 27, 2014 - 3:30pm
Where: Math 3206
Speaker: Yen Do (Yale University) -
Abstract: I will describe some motivation and ideas in a recent joint work with Philip T. Gressman, where we consider an operator analogue of the classical problem of finding the optimal decay rate for
oscillatory integrals. This problem arises naturally in the analysis of some oscillatory Riemann-Hilbert problems.

From Boltzmann to Euler: Hilbert’s 6th problem revisited

When: Thu, April 3, 2014 - 3:30pm
Where: Math 3206
Speaker: Marshall Slemrod (University of Wisconsin) -
Abstract: This talk addresses the hydrodynamic limit of the Boltzmann equation, namely the compressible Euler equations of gas dynamics. An exact summation of the Chapman–Enskog expansion originally given by Gorban and Karlin is the key to the analysis. An appraisal of the role of viscosity and capillarity in the limiting process is then given where the analogy is drawn to the limit of the Korteweg–de Vries–Burgers equations as a small parameter tends to zero.

Regularity, blow up, and small scale creation in fluids

When: Thu, April 17, 2014 - 3:30pm
Where: Colloquium Room 3206
Speaker: Alexander Kiselev ( University of Wisconsin at Madison) -
Abstract: The Euler equation of fluid mechanics describes a flow of inviscid and incompressible fluid, and has been first written in 1755. The equation is both nonlinear and nonlocal, and its solutions often create small scales easily and tend to be unstable. I will review some of the background, and then discuss a recent sharp result on small scale creation in solutions of the 2D Euler equation. I will also indicate links to the long open question of finite time blow up for solutions of the 3D Euler equation.

Fractional Stokes-Fourier limit for kinetic equations

When: Thu, April 24, 2014 - 3:30pm
Where: Math 3206
Speaker: Sara Merino (Cambridge University) -
Abstract: Fractional diffusion limits have been derived for collisional kinetic models conserving only the total mass (0-th moment). Their derivation is due, mainly, to the presence of a heavily tailed equilibrium distribution function in the collisional operator (instead of a Maxwellian) and a particular rescaling in time. In the present work, we extend the previous results to a linear kinetic model conserving the first three moments. Our approach is based on the 'moments methods' introduced by Antoine Mellet. In the limit we obtain the Stokes-Fourier equation with fractional laplacian, under some conditions. This is a joint work with Sabine Hittmeir from RICAM (Johann Radon Institute for Computational and Applied Mathematics).

Nonlinear Schrodinger systems with non-trivial boundary conditions

When: Thu, May 15, 2014 - 3:30pm
Where: Math 3206
Speaker: Gino Biondini (State University of New York at Buffalo, Department of Mathematics) -
Abstract: Nonlinear Schrodinger (NLS) equations are of great interest from both an applicative and a theoretical point of view. Despite having been intensely investigated over the last forty years, these systems still offer a number of challenges. Some of these involve the study of boundary value problems (BVPs) and problems in which non-trivial boundary conditions (BCs) are posed. In this talk I discuss a number of recent results in this area. In particular: (i) BVPs for the NLS equation on the half line with linearizable BCs, and (ii) IVPs for both focusing and defocusing, scalar and vector, NLS equations with non-zero
BCs at infinity.