## Applied Mathematics Archives

**Fall 2017**

**Speaker:**Bertram Düring (Department of Mathematics University of Sussex)

**Date and Time:** Thursday, December 7, 11am-12pm

**Place:** Elliott School (1957 E St. #214)

**Title:** A Lagrangian Scheme for the solution of nonlinear diffusion equations**Abstract:** Nonlinear diffusion equations whose dynamics are driven by internal energies and given external potentials, e.g., the porous medium equation and the fast diffusion equation, have received a lot of interest in mathematical research and practical applications alike in recent years. Many of them have been interpreted as gradient flows with respect to some metric structure. When it comes to solving partial differential equations of gradient flow type numerically, it is natural to ask for appropriate schemes that preserve the equations' special structure at the discrete level.

In this talk we present a Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in multiple space dimensions. The scheme applies to a large class of nonlinear diffusion equations. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. We discuss the consistency of the scheme in two space dimensions and present numerical experiments for the porous medium equation.

**Joint Analysis/Applied Math seminar**

Title: Green's functions and the existence of the gauge for local and

non-local time-independent Schrodinger equations

Abstract: We discuss a series of papers with Igor Verbitsky (and in one case, with Fedor Nazarov), dealing with time-independent Schrodinger operators L = - Laplacian -q, where the potential q is non-negative. The "gauge," as it is known in the probability literature, is the solution of Lu=0 on a domain, with u=1 on the boundary of the domain. We obtain estimates for the Green's function of L, and conditions for the existence of the gauge, under very general conditions on the potential q. The conditions, which describe how rapidly q can blow up near the boundary, are close to being necessary. We also discuss related results when the Laplacian is replaced

by the fractional Laplacian. The conclusions are based on a general estimate for the kernel of a Neumann series (I-T)^{-1} associated with an integral operator T of norm less than 1.

**Speaker:** Xiang Xu, Ph.D., Old Dominion University**Date and Time:** Friday, November 17, 11am-12pm**Place:** Rome 771**Title:** Eigenvalue preservation for the Beris-Edwards system modeling nematic liquid crystals**Abstract: **The Beris-Edward equations are a hydrodynamic system modeling nematic liquid crystals in the setting of Q-tensor order parameter. Mathematically speaking it is the incompressible Navier-Stokes equations coupled with a Q-tensor equation of parabolic type.

**Speaker:** Bertram Düring (Department of Mathematics University of Sussex)**Date and Time:** Thursday, December 7, 5-6pm**Place:** Rome 771**Title:** A Lagrangian Scheme for the solution of nonlinear diffusion equations**Abstract:** Nonlinear diffusion equations whose dynamics are driven by internal energies and given external potentials, e.g., the porous medium equation and the fast diffusion equation, have received a lot of interest in mathematical research and practical applications alike in recent years. Many of them have been interpreted as gradient flows with respect to some metric structure. When it comes to solving partial differential equations of gradient flow type numerically, it is natural to ask for appropriate schemes that preserve the equations' special structure at the discrete level.

In this talk we present a Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in multiple space dimensions. The scheme applies to a large class of nonlinear diffusion equations. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. We discuss the consistency of the scheme in two space dimensions and present numerical experiments for the porous medium equation.

**Speaker:** Jacek Jendrej (postdoc at Univ of Chicago)**Date and time:** Wednesday, September 13, 11am-12noon **Place:** Rome 202**Title:** Two-bubble dynamics for the equivariant wave maps equation

**Abstract: ** I will consider the energy-critical wave maps equation with values in the sphere in the equivariant case, that is for symmetric initial data. It is known that if the initial data has small energy, then the corresponding solution scatters. Moreover, the initial data of any scattering solution has topological degree 0. I try to answer the following question: what are the non-scattering solutions of topological degree 0 and the least possible energy? Such "threshold" solutions would have to decompose asymptotically into a superposition of two ground states at different scales, with no radiation.

**Spring 2017**

**Title: **Scattering below the ground state for the radial focusing NLS

**Speaker: ** Jason Murphy, UCLA

**Date and Time: April 2****8, 2017**, Friday,1:00 pm-2:00pm

**Place:** Rome 771

**Abstract: **We consider scattering below the ground state for the radial cubic focusing NLS in three dimensions. Holmer and Roudenko originally proved this via concentration compactness and a localized virial estimate. We present a simplified proof that avoids the use of concentration compactness, relying instead on the radial Sobolev embedding and a virial/Morawetz estimate. This is joint work with Benjamin Dodson.

**Title: **Singularity Formation in Nonlinear Derivative Schrödinger Equations

**Speaker: ** Gideon Simpson, Drexel University

**Date and Time: April 19****, 2017**, Wednesday,2:30 pm-3:30pm

**Place:** Rome 771

**Abstract: **Direct numerical simulation of an $L^2$ supercritical variant of the derivative nonlinear Schrödinger equation suggests that there is a finite time singularity. Subsequent exploration with the dynamic rescaling method provided more detail about the blowup and a recent refined asymptotic analysis of the blowup solution gives predictions of the blowup rates. Due to the mixed hyperbolic-dispersive nature of the equation, these methods have limited the proximity to the blowup time. Using a locally adaptive meshing method, we are able to overcome these difficulties.

**Title: **Existence analysis of a single-phase flow mixture with van der Waals pressure

**Speaker: ** Nicola Zamponi

**Date and Time: ****March 8, 2017**, Thursday,3:00 pm-4:00pm

**Place:** Rome 771

**Abstract: **The transport of single-phase fluid mixtures in porous media is described by cross-diffusion equations for the chemical concentrations. The equations are obtained in a thermodynamic consistent way from mass balance, Darcy's law, and the van der Waals equation of state for mixtures. Including diffusive fluxes, the global-in-time existence of weak solutions in a bounded domain with equilibrium boundary conditions is proved, using the boundedness-by-entropy method. Based on the free energy inequality, the large-time convergence of the solution to the constant equilibrium concentration is shown. For the two-species model and specific diffusion matrices, an integral inequality is proved, which reveals a maximum and minimum principle for the ratio of the concentrations. Without diffusive fluxes, the two-dimensional pressure is shown to converge exponentially fast to a constant. Numerical examples in one space dimension illustrate this convergence.

**Title: **Droplet phase in a nonlocal isoperimetric problem under confinement

**Speaker: ** Prof. Ihsan Topaloglu, Virginia Commonwealth University

**Date and Time: ****February 23 2017**, Thursday, 5:15 pm-6:15pm

**Place:** Rome 352

**Abstract: ** In this talk I will consider the small volume-fraction asymptotic limit of a nonlocal isoperimetric functional with a confinement term. This functional is derived as the sharp interface limit of a variational model for self-assembly of diblock copolymers under confinement by inclusion of nanoparticles in the system. Looking at confinement densities which are spatially variable and attain a nondegenerate maximum, I will present a two-stage asymptotic analysis in the sense of Γ-convergence wherein a separation of length scales is captured due to the competition between the nonlocal repulsive and confining attractive effects in the energy. The results will also relate to existence and non-existence of minimizers of a recently well-studied nonlocal isoperimetric functional which appears in the liquid drop model. This is a joint work with S. Alama, L. Bronsard, and R. Choksi.

**Fall 2016**

**Title: **Long time behavior of solutions to the 2D Keller-Segel equation with degenerate diffusion

**Speaker: ** **YAO YAO (Georgia Tech) **

http://people.math.gatech.edu/

**Date and Time: October 11**, 2016, Tuesday, 3:45pm

**Place:** Rome 204

**Abstract: ** The Keller-Segel equation is a nonlocal PDE modeling the collective motion of cells attracted by a self-emitted chemical substance. When this equation is set up in 2D with a degenerate diffusion term, it is known that solutions exist globally in time, but their long-time behavior remains unclear. In a joint work with J.Carrillo, S.Hittmeir and B.Volzone, we prove that all stationary solutions must be radially symmetric up to a translation, and use this to show convergence towards the stationary solution as the time goes to infinity. I will also discuss another joint work with K.Craig and I.Kim, where we let the power of degenerate diffusion go to infinity in the 2D Keller-Segel equation, so it becomes an aggregation equation with a constraint on the maximum density. We will show that if the initial data is a characteristic function, the solution will converge to the characteristic function of a disk as the time goes to infinity with certain convergence rate.

**Title: ** Nonlocal Models and Peridynamics

**Speaker: ****Speaker: ** **Michael Parks (****Sandia National Laboratories****)**

http://www.sandia.gov/~

**Date and Time: October 18**, 2016, Tuesday, 3:45pm

**Place:** Rome 204

**Abstract: **Nonlocal models are receiving increasing interest from scientific and engineering communities due to their ability to describe physical processes which are not well represented by PDE-based models. In particular, nonlocal models are useful in that they can resolve phenomena at multiple length scales, making them suitable models for multiscale processes. I will survey peridynamics, a nonlocal extension of classical continuum mechanics, as a representative nonlocal model. I will show several motivating examples and discuss the impact of nonlocality upon the computational structure of the model, reviewing discretization techniques, conditioning results, and solution methods. I’ll highlight recent results from the numerical analysis of peridynamics, including some non-intuitive results caused by the interplay of multiple length scales.

**Title: **Exponential tails for the non-cutoff Boltzmann equation

**Speaker: ****Maja Taskovich (UPenn)**

https://www.math.upenn.edu/~ta

**Date and Time: November 8**, 2016, Tuesday, 3:45pm

**Place:** Rome 204

**Abstract:** The Boltzmann equation models the evolution of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (so called Grad's cutoff), with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties, which motivates further analysis of the equation in this setting.

We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime, by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments, which can be understood as a generalization of exponential moments. An interesting aspect of the result is that the singularity rate of the angular kernel affects the order of tails that can be propagated in time. This is based on joint works with Alonso, Gamba, Pavlovic and with Gamba, Pavlovic.

**Title: **Concentration Compactness for Critical Radial Wave Maps

**Speaker: ** Jonas Lührmann (Johns Hopkins University)

http://www.math.jhu.edu/~

**Date and Time: December 6**, 2016, Tuesday, 3:45pm

**Place:** Rome 204

**Abstract: **The wave maps equation is the natural generalization of the linear wave equation for scalar-valued fields to fields that take values in a Riemannian manifold. In this talk we consider radially symmetric, energy critical wave maps from (1+2)-dimensional Minkowski space into the unit sphere and prove global regularity and scattering for essentially arbitrary smooth data of finite energy. In addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. This extends and complements the beautiful classical work of Christodoulou-Tahvildar-Zadeh and Struwe as well as of Nahas on radial wave maps in the case of the unit sphere as the target.

The proof proceeds along the concentration compactness/rigidity method of Kenig-Merle. I will explain the severe difficulties of implementing this strategy for energy critical wave maps due to strong low-high interactions in the wave maps nonlinearity and how these difficulties can be overcome by introducing a "twisted" profile decomposition. This is joint work with Elisabetta Chiodaroli and Joachim Krieger.

Speaker: Yongyong Cai, Purdue University Mathematics Department

Date and Time: Friday April 22nd at 2-3

Place: Rome 771

states and the dynamics, will be discussed. In the second part, a nonlinear Schroedinger equation with wave operator (NLSW) will be discussed. The NLSW is NLSE perturbed by the wave operator with strength described by a dimensionless parameter, which causes high oscillation in time and brings significant difficulties in designing and analyzing numerical methods with uniform accuracy. We will propose a uniformly accurate method for NLSW and apply it to Kleign-Gordon equation in the nonrelativistic limit regime.

Speaker: Simone Mazzini Bruschi, University of Brasilia - Brazil

Date and Time: Thursday March 31st at 4-5

Place: Rome 771

Speaker: Professor Lili Ju, University of South Carolina

Date and Time: March 10, 4-5pm

Place: Rome 771

September 9 (Wednesday) 4:00PM - 5:00PM, Monroe 267

Speaker: Yannick Sire, Johns Hopkins University

Title: Bounds on eigenvalues on riemannian manifolds"

Abstract: I will describe several recent results with N. Nadirashvili where we construct extremal metrics for eigenvalues on riemannian surfaces. This involves the study of a Schrodinger operator. As an application, one gets isoperimetric inequalities on the 2-sphere for the third eigenvalue of the Laplace Beltrami operator.

September 10 (Thursday) 5:00 PM - 6:00 PM, Monroe 350.

Speaker: Yihong Du, University of New England, Australia

Title: Reaction-diffusion equations and spreading of species

Abstract: In this talk, I will firstly give a brief review of some

pioneering works (of Fisher, Kolmogorov-Petrovsky-Piscunov, and Skellam)

on traveling waves and constant spreading speed. Then I will look at the

theory of Aronson-Weinberger that models the spreading by suitable Cauchy

problems. Finally I will describe some recent theory obtained with my

collaborators based on nonlinear free boundary problems, and compare it with

results arising from the Cauchy problem along the lines of Aronson-Weinberger.

November 11 (Wednesday) 2:00PM - 3:00PM, Monroe 267

Speaker: Ming Yan, Michigan State University

Title: ARock: Asynchronous Parallel Coordinate Updates

Abstract:We propose ARock, an asynchronous parallel algorithmic framework for finding a fixed point to a nonexpansive operator. In the framework, a set of agents (machines, processors, or cores) updates a sequence of randomly selected coordinates of the unknown variable in a parallel asynchronous fashion. As special cases of ARock, novel algorithms in linear algebra, convex optimization, machine learning, distributed and decentralized optimization are introduced. We show that if the nonexpansive operator has a fixed point, then with probability one the sequence of points generated by ARock converges to a fixed point. Very encouraging numerical performance of ARock is observed on solving linear equations, sparse logistic regression, and other large-scale problems in recent data sciences. This is joint work with Zhimin Peng, Yangyang Xu, and Wotao Yin.

Spring 2015

February 18 (Wednesday) 4:00 PM - 5:00 PM, Monroe 267.

Speaker: Aynur Bulut

University of Michigan

Title: Random data Cauchy problems for nonlinear Schr\”odinger and wave equations

Abstract: We will discuss recent progress on probabilistic local and global well-posedness results for the Nonlinear Schrödinger and Nonlinear Wave equations. In these problems one considers randomly chosen initial data, distributed as a Gaussian process, and with low regularity properties (supercritical with respect to the scaling of the nonlinearity). In particular, our data belong almost surely to the ill-posed regime for these problems, and probabilistic considerations are therefore essential. Tools used in the approach include sharp a priori bounds for the nonlinear evolutions and associated linearizations, algebraic structure arising from the Hamiltonian nature of the problems, and careful analysis of frequency interactions.

February 20 (Fridauy) 2:30 PM - 3:30 PM, Monroe 250.

Speaker: Francesco Maggi,

University of Texas, Austin

Title: Formulations of Plateau problem and the existence of minimizers<p>

Abstract: We review various mathematical formulations of the Plateau-Lagrange problem and present a measure-theoretic approach to compactness of surfaces leading to various existence results.

April 8 (Wednesday) 4:00 PM - 5:00 PM, Monroe 267

Speaker: James Colliander, University of Toronto

Title: Big frequency cascades in the nonlinear Schrodinger evolution

Abstract: I will outline a construction of an exotic solution of the nonlinear

Schrodinger equation that exhibits a big frequency cascade. Recent advances

related to this construction and some open questions will be surveyed.

April 10 (Friday) 2:00 PM - 3:00 PM, Monroe 352.

Speaker: Juhi Jang, University of California, Riverside

Title: On the kinetic Fokker-Planck equation with absorbing barrier

Abstract: We discuss the well-posedness theory of classical solutions to the kinetic

Fokker-Planck equation in bounded domains with absorbing boundary conditions. We show

that the solutions are smooth up the boundary away from the singular set and they are

Holder continuous up to the singular set. This is joint work with H.J. Hwang, J. Jung,

and J.L. Velazquez.

April 22 (Wednesday) 4:00 PM - 5:00 PM, Monroe 267.

Speaker: Jose Vega-Guzman, Howard University

Title: On the solution of some nonautonomous evolution equations<p>

Abstract: Solution methods for certain linear and nonlinear evolution equations will be presented. Emphasis will be placed mainly on the analytical

treatment of nonautonomous differential equations, which are challenging to solve despite the existent numerical and symbolic computational software programs available. Ideas from the transformation theory are adopted allowing one to solve the problems under consideration from a non-traditional perspective. The Cauchy initial value problem is considered for a class of nonautonomous and inhomogeneous linear diffusion-type equation on the entire real line.

Explicit transformations are used to reduce the equations under study to their corresponding standard forms emphasizing on natural relations with certain Riccati(and/or Ermakov)-type systems. This results will serve as a base to create a strong commutative relation, and to construct explicitly the minimum-uncertainty squeezed states for quantum harmonic oscillators. For the last, it is shown that the product of the variances

attains the required minimum value only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied.

February 25 (Wednesday) 4:00 PM - 5:00 PM, Monroe 267.

Speaker: Alina Chertock,

North Carolina State University

Title: Interaction Dynamics of Singular Wave Fronts Computed by Particle Methods

Abstract: Some of the most impressive singular wave fronts seen in Nature are the transbasin oceanic internal waves, which may be observed from a space shuttle, as they propagate and interact with each other. The characteristic feature of these strongly nonlinear waves is that they reconnect whenever any two of them collide transversely. The dynamics of these internal wave fronts is governed by the so-called EPDiff equation, which, in particular, coincides with the dispersionless case of the Camassa-Holm (CH) equation for shallow water in one- and two-dimensions.

In this talk, I will present a particle method for the numerical simulation and investigation of solitary wave structures of the EPDiff equation in one and two dimensions. I will also discuss the extension of the presented particle method to a family of strongly nonlinear equations that yield traveling wave solutions and can be used to model a variety of fluid dynamics.

I will also provide global existence and uniqueness results for this family of fluid transport equations by establishing convergence results for the particle method. The lattes is accomplished by using the concept of space-time bounded variation and the associated compactness properties.

Finally, I will present numerical examples that demonstrate the performance of the particle methods in both one and two dimensions. The numerical results illustrate that the particle method has superior features and represent huge computational savings when the initial data of interest lies on a submanifold. The method can also be effectively implemented in straightforward fashion in a parallel computing environment for arbitrary initial data.

January 27 (Tuesday) 4:00 PM - 5:00 PM, Monroe 267.

Speaker: Juncheng Wei,

University of British Columbia

Title: On Bombieri-De Giorgi-Giusti Minimal Graph and Its Applications

Abstract: I will first discuss the refined estimates of Bomberie-De Giorgi-Giusti minimal graph. Then I will give several imprtant applications including the counter-example to De Giorgi's conjecture, Serrin's overdetermined problem, and translating graph of mean curvature flow.

Bio: Prof. Juncheng Wei at University of British Columbia is a Canada Research Chair in Partial Differential Equations. He is a prolific researcher who has written over 280 articles since 1995. His paper coauthored with M. Del Pino and M. Kowalczyk (Annals of Mathematics 174 (2011)) resolved the De Giorgi's Conjecture in dimensions greater than 8. For this and other significant works, he was invited to lecture at the 2014 International Congress of Mathematicians.

January 28 (Wednesday) 4:00 PM - 5:00 PM, Monroe 267

Speaker: Mark Alber

University of Notre Dame

Title: Combined Multi-scale Modeling and Experimental Study of Bacterial Swarming and Blood Clot Formation

Abstract: As with most phenomena in biology and medicine, insight into emergent organizational and tissue level properties can be gained by, and indeed require, combination of multi-scale computational modeling and experiments. This approach will be demonstrated in this talk using two examples.

Surface motility such as swarming is thought to precede biofilm formation during spread of infection. Population of swarming bacteria P. aeruginosa, major infection in hospitals, will be shown to efficiently propagate as high density waves that move symmetrically as rings within swarms towards the extending tendrils. Multi-scale model simulations suggested a cell-cell coordination mechanism of wave propagation which was recently shown in experiments to moderate swarming direction of individual bacteria to avoid antibiotics.

In the second half of the talk a novel three-dimensional multi-scale model will be described and used to simulate receptor-mediated adhesion of deformable platelets during blood clot formation at the site of vascular injury under different shear rates of blood flow. Newly established correlations between structural changes and mechanical responses of fibrin networks exposed to compressive loads will be also described. Fibrin plays an important role in blood clot formation, wound healing, tissue regeneration and is widely employed in surgery as a sealant and in tissue engineering as a scaffold.

Fall 2014

**Applied Math seminar**

**Speaker**: Bo Li, Department of Mathematics, UC San Diego**Date/time**: December 3, 2014 1:00pm-2:00pm**Title**: Dielectric Boundary Force in Biomolecular Solvation**Place:** Monroe 267

Abstract:

A dielectric boundary in a biomolecular system is a solute-solvent (e.g., protein-water) interface that defines the dielectric coefficient to be one value in the solute region and another in solvent. The inhomogeneous dielectric medium gives rise to an effective dielectric boundary force that is crucial to the biomolecular conformation and dynamics. This talk begins with a review of the Poisson-Boltzmann theory commonly used for electrostatic interactions in charged molecular systems and then focuses on the mathematical description of the dielectric boundary force. A precise definition and explicit formula of such force are presented. The motion of a cylindrical dielectric boundary driven by the competition between the surface tension, electrostatic interaction, and solvent viscous force is then studied. Implications of the mathematical findings to biomolecular conformational stabilities are finally discussed.

** Speaker: **Chuck Gartland

Department of Mathematical Sciences, Kent State University

**Title:**Electric-Field-Induced Instabilities in Liquid-Crystal Films

**Date/Time:**Wednesday, October 15, 2014, 3:30 PM - 4:30 PM

**Place:**Monroe 267.

Abstract:

The orientational properties of materials in the liquid-crystal phase (characterized by the "director field", a unit vector field, in the simplest macroscopic continuum models) are strongly influenced by applied electric fields, which provide a common switching control mechanism in liquid-crystal-based technologies. In turn, the liquid-crystal medium, by virtue of its anisotropic and generally inhomogeneous nature, also influences the local electric field; so the equilibrium director field and electric field must be computed in a coupled, self-consistent way. Equilibria correspond to stationary points of the "free energy" (an integral functional of the director field and the electrostatic potential field), which fails to be coercive, in typical applications, due to both the negative-definite nature of the director/electric-field coupling term and the pointwise unit-vector constraints on the director field. We will discuss characterizations of local stability of equilibrium fields in such settings and anomalous behavior that can result even in simple geometries, such as those of classic Fredericks transitions.

** Speaker: **Gil Ariel, Bar Ilan University, Tel Aviv, Israel

**Title:**Parallelizable Block Iterative Methods for Stochastic Processes

**Date/Time:**Wednesday, October 1, 2014, 1:00 PM - 2:00 PM

**Place:**Monroe 267.

**Abstract**: In many applications involving large systems of stochastic differential equations, the states space can be partitioned into groups which are only weakly interacting. For example, molecular dynamics simulations of large molecules undergoing Langevin dynamics may be divided into smaller components, each at equilibrium. If the components are decoupled, then the equilibrium distribution of the entire system is a product of the marginals and can be computed in parallel. However, taking interactions into account, the entire state of the system must be considered as a whole and naÃ¯ve parallelization is not possible. We propose an iterative method along the lines of the wave-form relaxation approach for calculating all component marginals. The method allows some parallelization between conditionally independent components, depending on the minimal coloring of the graph describing their mutual interactions. Joint work with Ben Leimkuhler and Matthias Sachs (University of Edinburgh).

Spring 2014

### April 3, 2014 (Thursday) 4:00 PM - 5:00 PM, Monroe 267.

Speaker: Xiaofeng Ren, George Washington University.

Title: A double bubble assembly as a new phase of a ternary inhibitory system

Abstract: A ternary inhibitory system is a three component system characterized by two properties: growth and inhibition. A deviation from homogeneity has a strong positive feedback on its further increase. In the meantime a longer ranging confinement mechanism prevents unlimited spreading. Together they lead to a locally self-enhancing and self-organizing process. The model considered here is a planar nonlocal geometric problem derived from the triblock copolymer theory. An assembly of perturbed double bubbles is mathematically constructed as a stable critical point of the free energy functional. Triple junction, a phenomenon that the three components meet at a single point, is a key issue addressed in the construction. Coarsening, an undesirable scenario of excessive micro-domain growth, is prevented by a lower bound on the long range interaction term in the free energy. The proof involves several ideas: perturbation of double bubbles in a restricted class; use of internal variables to remove nonlinear constraints, local minimization in a restricted class formulated as a nonlinear problem on a Hilbert space; and reduction to finite dimensional minimization. This existence theorem predicts a new morphological phase of a double bubble assembly.

### April 24, 2014 (Thursday) 4:00 PM - 5:00 PM, Monroe 267.

Speaker: Chun Liu, Pennsylvania State University.

Title: Energetic Variational Approaches: Onsager's Maximum Dissipation Principle, General Diffusion, Optimal Transport and Stochastic Integrals

Abstract: In the talk, I will explore the underlying mechanism governing various diffusion processes. We will employ a general framework of energetic variational approaches, consisting of in particular, Onsager's Maximum Dissipation Principles, and their specific applications in application is biology and physiology. We will discuss the roles of different stochastic integrals (Ito's form, Stratonovich's form and other possible forms), and the procedure of optimal transport in the context of general framework of theories of linear responses.

March 24, 2014 (Monday) 4:00 PM - 5:00 PM, Monroe 267 S

peaker: Nassif Ghoussoub, University of British Columbia

Title: Decoupling DeGiorgi systems via multi-marginal mass transport

Abstract: We exhibit a surprising relationship between elliptic gradient systems of PDEs, multi-marginal Monge-Kantorovich optimal transport problem, and multivariable Hardy-Littlewood inequalities. We show that the notion of an "orientable" elliptic system, conjectured to imply that --at least in low dimensions-- solutions with certain monotonicity properties are essentially $1$-dimensional, is equivalent to the definition of a "compatible" cost function, known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems. Orientable nonlinearities and compatible cost functions turn out to be also related to "sub-modular" functions, which appear in rearrangement inequalities of Hardy-Littlewood type. We use this equivalence to establish a decoupling result for certain solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport. Bio: Nassif Ghoussoub obtained his Doctorat d'état in 1979 from the Université Pierre et Marie Curie in Paris, France. His present research interests are in non-linear analysis and partial differential equations. He is currently a Professor of Mathematics, a "Distinguished University Scholar", and an elected member of the Board of Governors of the University of British Columbia. He was the founding Director of PIMS (Pacific Institute for the Mathematical Sciences), a co-founder of the MITACS Network of Centres of Excellence (Mathematics of Information Technology and Complex Systems) and a member of its Board of Directors for the periods. He is also the founder of BIRS (Banff International Research Station) and its Scientific Director. In 2011, he became the Scientific Director of the MPrime network of Centres of Excellence.

March 19, 2014 (Wednesday) 4:00 PM - 5:00 PM, Monroe 267

Speaker: Qiang Du, Penn State University

Title: Nonlocal calculus, nonlocal balance laws and asymptotically compatible discretizations

Abstract: Nonlocality is ubiquitous in nature. While partial differential equations (PDE) have been used as effective models of many physical processes, nonlocal models and nonlocal balanced laws are also attracting more and more attentions as possible alternatives to treat anomalous process and singular behavior. In this talk, we exploit the use of a recently developed nonlocal vector calculus to study a class of constrained value problems on bounded domains associated with some nonlocal balance laws. The nonlocal calculus of variations then offers striking analogies between nonlocal model and classical local PDE models as well as the notion of local and nonlocal fluxes. We discuss the consistency of

nonlocal models to local PDE limits as the horizon, which measures the range of nonlocal interactions, approaches zero. In addition, we present asymptotically compatible discretizations that provide convergent approximations in the nonlocal setting with a nonzero horizon and are also convergent asymptotically to the local limit as both the horizon and the mesh size are taking to zero. Such asymptotically compatible discretizations can be more robust for multiscale problems with varying length scales.

Applied Mathematics Seminar Thursday Feb. 20th.

The speaker in Thierry Goudon from INRIA, Sophia Antipolis Research Center (France)

http://www-sop.inria.fr/members/Thierry.Goudon/index.html

The seminar will be at 3pm in 1957 E Street room 308

Title:"Models for "mixtures" ; multifluid flows "

Abstract: We will discuss various issues on mathematical modelingof mixture flows. The equations are characterized by the role of density gradients and unusual constraints on the velocity field.Coming back to a coupled fluid-kinetic description of the flows, we derive a hierarchy of models that generalizes the Kazhikov-Smagulov system. We exhibit some stability properties of the system.

February 21, 2014 (Friday) 4:00 PM - 5:00 PM, Monroe 267.

Speaker: Mingfeng Zhao, University of Connecticut.

Title: A Liouville-type theorem for higher order elliptic systems

Abstract: By using a Rellich-Pohozaev identity and an adapted Souplet's idea about the measure and feedback arguments, we prove that there are no positive solutions to higher order Lane-Emden system provided some conditions. Our result is a higher order analogue of Souplet's result for Lane-Emden system. This is a joint work with Frank Arthur and Xiaodong Yan.

Fall 2013

October 18 (Friday) 1:00 PM - 2:00 PM, Govermnent 102.

Speaker: Russell Schwab, Michigan State University.

Title: Recent Topics in Integro-Differential Equations

Abstract: We will give a brief overview of some recent results on the analysis of elliptic integro- differential equations (which are the natural class of generators of Markov processes) from the perspective of nonlinear elliptic equations. We will discuss some regularity results and possibly some applications to Neumann homogenization.

November 8 (Friday) 1:00 PM - 2:00 PM, Government 102.

Speaker: Brittney Froese, University of Texas, Austin.

Title: Finite Difference Methods for Nonlinear Elliptic Equations with Application to Optimal Transport

Abstract: We describe the use of finite difference methods for solving nonlinear elliptic partial

differential equations (PDEs). We show that simple techniques, which work for linear equations, may fail for nonlinear equations. We describe a framework for developing convergent finite difference methods for nonlinear degenerate elliptic equations. Focusing specifically on optimal transport, a challenging problem that is important to both theoretical and applied mathematics, we construct robust numerical methods for the Monge-Ampere equation with transport boundary conditions. A range of computational examples demonstrate the effectiveness and efficiency of the method.