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.
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.
Title: On the solution of some nonautonomous evolution equations
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.
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.
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.
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.
Title: Formulations of Plateau problem and the existence of minimizers
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.
Title: Random data Cauchy problems for nonlinear Schrodinger 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.
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.
Title: On Bomberie-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 wasinvited to lecture at the 2014 International Congress of Mathematicians.
Title: Dielectric Boundary Force in Biomolecular Solvation
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.
Title: Electric-Field-Induced Instabilities in Liquid-Crystal Films
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.
Title: Parallelizable Block Iterative Methods for Stochastic Processes
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 naïve 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).
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 varios diffusion processes. We will employ a general framework of energetic variational approaches, consising 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.
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.
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'etat from the Universite 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 the 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. He is also the founder of the Banff International Research Station and its Scientific Director. In 2011, he became the Scientific Director of the MPrime network of Centres of Excellence.
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.
Title: Models for ''mixtures'', multifluid flows.
Abstract: We will discuss various issues on mathematical modeling of 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.
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.
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.
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.
Title: Classical solutions for a nonlinear Fokker-Planck equation arising in Computational Neuroscience
Abstract: Wwe analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describng the collective behavior of an ensemble of neurons. These equations were obtained as a diusive approximation of the mean-field limit of a stochastic dierential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coeffiients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation wth nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networs we give local well-posedness of classical solutions together with a blow up criterium.
Title: Optimal transport and regularity of c-convex potentials
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 introduction to the topic of optimal transport and give an impressionistic description of Caffarelli's regularity theory for the Monge Ampere. We will see when and how such a theory can be pushed to general costs (such as those in a Riemannian manifold). The new observatio is that in general regularity arises not so much from affine invariance (as it is exploited in the Euclidean setting), but rather from two opposite inequalities for the Mahler volume of c-convex sets (a kind of generalized Blaschke-Santal inequalities). Based on joint work with Jun Kitagawa.
Title: A multiscale method coupling network and continuum models in porous media
Abstract: In this talk, we present a numerical multiscale method for coupling a conservation law for mass at the continuum scale with a discrete network model that describes the pore scale flow in a porous medium. We developed single-phase flow algorithms and extended the methods to two-phase flow, for the situations in which the saturation. Our coupling method for the pressure equation uses local simulations on small sampled network domains at the pore scale to evaluate the continuum equation and thus solve for the pressure in the domain. For local simulation, it often requires a suitable initialization. We introduce a choice of initialization from a optimization problem, which is often used in image processing. We present numerical results for single-phase flows with nonlinear flux-pressure dependence, as well as two-phase flow.
Title: A PDE approach to fractional diffusion: a priori and a posteriori error analyses
Abstract: We study solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the Dirichlet to Neumann map for a degenerate/singular elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of weighted Sobolev spaces. Motivated by the rapid decay of the solution of this problem, we propose a truncation that is suitable for numerical approximation. We discretize this truncation using first degree tensor product finite elements. We derive a priori error estimates in weighted Sobolev spaces. The estimates exhibit optimal regularity but suboptimal order for quasi-uniform meshes. For anisotropic meshes, instead, they are quasi-optimal in both order and regularity. A posteriori error estimation techniques are discussed and analyzed. To illustrate the method's performance we present numerical experiments.
Title: Optimal regularity estimates for non linear continuity equations
Abstract: We prove compactness and hence existence for solutions to a class of non linear transport equations. The corresponding models combine the features of linear transport equations and scalar conservation laws. We introduce a new method which gives quantitative compactness estimates compatible with both frameworks.
Title: Keller-Segel, fast-diffusion, and functional inequalities
Abstract: We will show how the critical mass classical Keller-Segel system and the critical displacement convex fast-diffusion equation in two dimensions are related. On one hand, the critical fast diffusion entropy functional helps to show global existence around equilibrium states of the critical mass Keller-Seel system. On the other hand, the critical fast diffusion flow allows to show functional inequalities such as the Logarithmic HLS inequality in simple terms who is essential in the behavior of the subcritical mass Keller-Sgel system. HLS inequalities can also be recovered in several dimensions using this procedure. It is crucial the relation to the GNS inequalities obtained by DelPino and Dolbeault. This talk corresponds to two works in collaboration with E. Carlen and A. Blanchet, and with E. Carlen and M. Loss.
Title: Computing Transition Paths in Stochastic Systems
Abstract: Many processes in nature are modeled using stochastic differential equations. Examples include thermally-activated small-scale processes in physics and chemistry such as conformational changes in molecules, chemical reactions, magnetization switches and nucleation events. Other examples come from such fields as evolutionary biology and stochastically modeled computer networks. I will consider two stochastic systems: a system without detailed balance evolving according to a non-gradient stochastic differential, and a stochastic network associated with the Lennard-Jones-38 cluster. In both cases the noise is small so that the direct simulation is inefficient for studying transitions between metastable states. In the first system, I will introduce a method for computing the quasipotential, a function that determines the transition rates and transition paths in the case of small noise. I will present the results of analysis of the second system using three different approaches: the discrete transition path theory approach, the zero-temperature approach, and a heuristic approach.
Title: G-equations in the modeling of turbulent flame speeds
Abstract: Predicting turbulent flame speed ($s_T$) is a fundamental problem in turbulent combustion theory. Several simplified models have been proposed to study $s_T$. The G-equation (A Hamilton-Jacobi level set equation) is a very popular model in turbulent combustion. Two important projects are (1) establish the theoretical existence of $s_T$ and (2) determine the dependence of turbulent flame speeds on the turbulence intensity (think of the relation between the spreading velocity of wild fire and strength of the wind). In this talk, I will present some theoretical results under the G-equation model. These are joint works with Jack Xin.
Title: An extremal problem in quasiconformal mappings from a PDE point of view.
Abstract: I will talk about a classical L^\infty variational problem where one asks to minimize the essential supremum of the dilation of a homemorphism in a given class of competitors. I will show that, given sufficient regularity, the problem can be related to a non-linear system of PDE reminiscent of the scalar infinity-Laplacian.
Title: A high-order method for simulating turbulent flow on moving and deforming unstructured grids
Abstract: High-order numerical schemes which are based on locally discontinuous polynomial approximations on unstructured meshes are particularly attractive for nonlinear convection-dominated problems in complex geometry. This talk focuses on spectral difference (SD) method for unsteady viscous flow problems on moving and deforming grids. The SD method deals with the strong form of the compressible Navier-Stokes equations. It uses local interpolation of conservative variables and their fluxes within a universal computational cell. At the interfaces of cells, an approximate Riemann solver is employed. The SD method was designed to achieve superior computational efficiency by avoiding volume and surface quadratures altogether compared to the Discontinuous Galerkin (DG) method. Meanwhile, the SD method is able to obtain optimal spatial order accuracy. We verify the SD method using a cylindrical compressible Taylor-Couette flow with exact solution. Spatial accuracies, such as 3rd-order, 4th-order, and 5th-order, are successfully verified. Subsequently, the method is further validated by the simulation of subsonic viscous flows past a plunging airfoil. The SD algorithms are now applied for high-order accurate simulations of a flapping wing behind an oscillating cylinder. Interesting transitional flow behavior is successfully predicted by our massively parallel 3D computation on moving and deforming grid.
Short Bio: Dr. Chunlei Liang is an assistant professor in the department of Mechanical and Aerospace Engineering at George Washington University. He obtained his PhD from University of London in 2005 advised by Dr. George Papadakis (currently Imperial College London) and Prof. Michael Yianneskis (currently King's College London). Prior to joining GWU, he conducted three years postdoctoral research at Stanford University under the directions of Professor Antony Jameson and Professor Parviz Moin. Dr Liang is a senior member of American Institute of Aeronautics and Astronautics, and a editorial board member of Computers and Fluids, an Elsevier journal. His research interests are developing numerical solvers for both incompressible and compressible flows and modeling real-world challenging fluid flow problems through massively parallel computing. Dr Liang directs the Computational Aero and Hydrodynamics Laboratory (CAHL) in GWU. His research group website is http://www.seas.gwu.edu/~chliang.
Title: Regularity and decay estimates of dissipative equations
Abstract: We establish Gevrey class regularity of solutions to dissipative equations. The main tools are the Kato-Ponce inequality for Gevrey estimates in Sobolev spaces and the Gevrey estimates in Besov spaces using the paraproduct decomposition. As an application, we obtain temporal decay of solutions for a large class of equations including the Navier-Stokes equations, the subcritical quasi-geostrophic equations.
Title: Global solutions of a nonlinear beam equation with exponential nonlinearity in four space dimensions
Abstract: Fourth order nonlinear PDEs have been widely investigated in recent years. In this talk, we will discuss the Cauchy problem of a fourth order nonlinear wave equation with an exponential nonlinear term, and we prove the global well-posedness of the equation in the energy space H^2. In dimension d = 4, this kind of nonlinearity is a natural critical nonlinearity with respect to the energy space. Our results are mainly based on some improved Strichartz estimates.
Title: Effective Boundary Conditions on a Thermally Insulated Body by Anisotropic Coatings
Abstract: Of concern is the scenario of protecting a thermally conducting body f rom overheating by an anisotropically conducting coating, thin compared to the s cale of the body. We assume that either the whole thermal tensor of the coating is small, or it is small in the directions normal to the boundary of the body ( the case of what we call "optimally aligned coating"). We study the asymptotic b ehavior of the solution to the heat equation as the thickness of the coating shr inks. It turns out that the effective (limiting) condition on the boundary of th e body can be the standard ones (Dirichlet, Neumann and Robin) or something that are non-local involving the Dirichlet-to-Neumann mapping or the Hilbert transfo rmation on the circle, depending on the scaling relationship between the thermal tensor of the coating and its thickness. In this fashion, we not only discove r some new boundary conditions, give new physical interpretations of the known b oundary conditions, but also identify scaling laws that ensure the well-insulate dness of the conducting body. I will also present a result on the lifespan of th e effective Neumann boundary condition, obtained in my student's Ph.D thesis.
Title: Topological quantum computation
Abstract: Quantum computing models have the potential to perform tasks such as\ factoring integers and simulating quantum physics exponentially faster than an\ y known classical algorithms, thus revolutionizing information science. But th\ e construction of a large-scale quantum computer is still in its infancy due to\ the decoherence of quantumness. One promising way to defeat decoherence is vi\ a topology. I will give an introduction to this approach to building a large-s\ cale quantum computer, as pursued at Microsoft Station Q (http://stationq.ucsb.\ edu/), and discuss the mathematical and scientific challenges.
Title: On the optimal control of singular PDE's: incomplete data problems and/or ill posed problems
Abstract: We present a method by J.-L. Lions well adapted to the control of PDE's of incomplete. We give a simple example in the elliptic case. Then we show that the method can be used to the control of the ill-posed backward heat equation.
Title: Bose-Einstein condensation and the Kompaneets equation
Abstract: Solutions of the Kompaneets equation can develop a Bose-Einstein condensate in finite-time. We wish to extend the solution beyond the blow-up time. We establish global solutions for a model of the Kompaneets equation that exhibits the same phenomenon.
Title: Fully nonlinear partial differential equations and their numerical solutions
Abstract: In the past thirty years tremendous progresses have been made on the development of the viscosity solution theory for fully nonlinear 2nd order PDEs. However, in contrast with the success of the PDE theory, until very recently there has been essentially no progress on how to reliably compute these viscosity solutions. This lack of progress is due to the facts that (a) viscosity solutions often are only conditionally unique; (b) the notion of viscosity solutions is non-variational and non-constructive, hence, it is extremely difficult (if it is possible) to mimic at the discrete level. In this talk, I shall first review some recent advances (and attempts) in numerical methods for fully nonlinear 2nd order PDEs, in particular, the Monge-Ampere type equations. I shall then focus on discussing a newly developed methodology (called the vanishing moment method) and the induced new notion of weak solutions (called moment solutions) as well as the relationship between viscosity solutions and moment solutions. Recent developments in Galerkin type mnumerical methods such as finite element methods, spectral methods, mixed methods for fully nonlinear 2nd order PDEs based on the vanishing moment methodology will be reviewed. Finally, I shall present some numerical results for the Monge-Ampere equation and the prescribed Gauss curvature equation, and also discuss a few applications such as the semigeostrophic flow, the Monge-Kantorovich optimal mass transport, the Monge-Ampere gravitational model, and the affine maximal surface problem which all give rise interesting (and difficult) fully nonlinear PDE problems.
Title: Number of central configurations in celestial mechanics
Abstract: The motion of celestial body is described by a system of second order differential equations and it is called n-body problem. A central configuration plays the essential role in understanding the global structure of solutions of the n-body problem. A central configuration is an arrangement of the initial positions of masses that leads to special families of solutions of the n-body problem. There are different understandings of equivalence of central configurations in collinear n-body problem and we call them permutation equivalence angeometric equivalence when we count the number of central configurations. In the permutation equivalence, Euler found three collinear central configurations and Moulton generalized to n!/2 central configurations for any given mass $m$ in the collinear n-body problem under permutation equivalence. In particular, the number of central configurations becomes from 12 under permutation equivalence to 1 under geometric equivalence for four equal masses. The main result in this paper is the discovery of the explicit parametric expressions of the union H4 of the singular surfaces in the mass space (four distinct positive masses) which decrease the number of collinear central configurations under geometric equivalence.
Speaker: Carlos Castillo-Chavez, Arizona State University.
Title: Complexity and Epidemics: Influenza Epidemics in Mexico
Abstract: Disease dynamics are connected to biological, environmental and social processes that take place over multiple time scales and over various levels of social and biological organization. In today's world, epidemic outbreaks become instant potential health and/or economic global threats with increasing segments of the population playing active roles on the transmission patterns of infectio us diseases like influenza. Despite the myriad of complexities associated with d isease dynamics, macroscopic epidemic patterns emerge but finding ways of making effective use of this knowledge remains. I will address some of these challeng es in a historical context starting with the work of physicians-theoreticians li ke Bernoulli, Ross, Kermack and McKendrick. The lecture will be tied in to the e pidemiology of influenza with examples from the 2009 H1N1 pandemic that originat ed in Mexico.
Title: Deterministic Compressed Sensing for Images with Chirps and Reed-Muller Sequences
Abstract: A deterministic approach to compressed sensing via discrete frequency- modulated chirps (from harmonic analysis) or Reed-Muller sequences (from the cod ing theory) was proposed recently by R. Calderbank et al., however, any efficient reconstruction of higher dimensi onal signals was not possible. In particular, it was not applicable to a general image reconstruction or to a more sparse 2d signals such as medical images. We develop an effective reconstruction algorithm which incorporates several new ele ments to improve computational complexity and reconstruction fidelity in this ap plication regime and discuss the results in various imaging settings.
Title: An overview of Compressed Sensing
Abstract: In the few years since the foundational ideas of compressed sensing (C S) were set forth by Donoho, and Candes and Tao, the methodology had inspired a substantial body of research seeking to exploit sparsity in various classes of s ignals to enable efficient measurement approaches. I will give an overview on compressed sensing, including theory, a survey of cur rent methods, and applications. I will also review some basics of MRI and applic ation of CS to MRI as time permits.
Title: Steady states and asymptotic behavior of mean-field models for diblock copolymer melts.
Abstract: We study the mean-field models for diblock copolymer melts, describing the evolution of distributions of particle radii obtained by taking the small volume fraction limit of the free boundary problem where micro phase separation results in an ensemble of small balls of one component. In the dilute case, we identify all the steady states and show the convergence of solutions.
Title: Derivation of mean-field models for diblock copolymer melts in two dimensional case
Abstract: We study the free boundary problem describing the micro phase separation of diblock copolymer melts in two dimension in the regime that one component has small volume fraction such that micro phase separation results in an ensemble of small disks of one component. On some time scale, the evolution is dominated by coarsening and subsequent stabilization of the radii of the circles. Starting from the free boundary problem restricted to circles we rigorously derive the mean-field equations in this time regime. Our analysis is based on passing to the homogenization limit in the variational framework of a gradient flow.
Title: Axial Symmetry of Some Entire Solutions of Nonlinear Elliptic Equations
Abstract: In this talk, I will present some recent results on the axial symmetry of certain entire solutions which are anisotropic. The type of solutions includes stationary solutions for nonlinear Schrodinger equation, saddle solutions and traveling wave solutions for Allen-Cahn equations. Open problems will be discussed.
Title: Soft Matter approaches to Membranes and Matrices
Abstract: From membranes to matrices, biology is rife with structures and patterns that motivate mimickry with a goal toward clarifying determinants of function. In order to clarify mechanisms and effects of segregation or flexibility in bio-function, two distinct types of polymeric systems will be described. Block copolymers are described as self-assembling but electrostatically segregating systems(1). Fibrous protein matrices in tissues are imitated in their elasticity with crosslinked hydrogels, demonstrating the potent influence of tissue elasticity E on biological processes such as cell differentiation (2-4)and on structures such as nuclei - which are viscoelastic sacks filled chromatin fibers (5).
References: (1) D.A. Christian, A. Tian, W.G. Ellenbroek, I. Levental, P.A. Janmey, A.J. Liu, T. Baumgart, D.E. Discher. Spotted vesicles, striped micelles, and Janus assemblies induced by ligand binding. Nature Materials 8: 843-849 (2009). (2) A.E.X. Brown, R. Litvinov, D.E. Discher, P. Purohit, J. Weisel. Multiscale mechanics of fibrin polymer: Gel stretching with protein unfolding and loss of water. Science 325: 741-744 (2009). (3) A. Engler, S. Sen, H.L. Sweeney, and D.E. Discher. Matrix elasticity directs stem cell lineage specification. Cell 126: 677-689 (2006). (4) A. Zemel, F.Rehfeldt, A.E.X. Brown, D.E. Discher, and S.A. Safran. Optimal matrix rigidity in the self-polarization of stem cells. Nature Physics (to appear). (5) J.D. Pajerowski, K.N. Dahl, F.L. Zhong, P.J. Sammak, and D.E. Discher. Physical plasticity of the nucleus in stem cell differentiation. PNAS 104: 15619-15624 (2007).
Title: Turing patterns and standing waves of FitzHugh-Nagumo type systems
Abstract: There are many interesting patterns observed in activator-inhibitor systems. A well-known model is the FitzHugh-Nagumo system. In conjunction with calculus of variations, there is a close relation between the stability of a steady state and its relative Morse index. We give a sufficient condition in diffusivity for the existence of standing wavefronts joining with Turing patterns.
Title: Epitaxial growth: A two-scale perspective
Abstract: Epitaxial growth provides a paradigm of an intimate connection of two scales: The nanoscale, where line defects are evident and discrete schemes are used; and the macroscale where PDEs for thermodynamic variables are invoked. In this talk, I will address recent progress and challenges in understanding crystal surface motion in light of descriptions of these scales. Deterministic and stochastic aspects will be discussed.
Title: Simple PDE model of spot replication in any dimension
Abstract: We propose a simple PDE model which exhibits self-replication of spot solutions in any dimension. This model is analysed in one and higher dimensions. In one dimension, we rigorously demonstrate that the conditions proposed by Nishiura and Ueyama for self-replication are satisfied. In dimension three, two different types of replication mechanisms are analysed. The first type is due to radially symmetric instability, whereby a spot bifurcates into a ring. The second type is non-radial instability, which causes a spot to deform into a peanut-like shape, and eventually split into two spots. Both types of replication are observed in our model, depending on parameter choice. Numerical simulations are shown confirming our analytical results. This is a joint work with Chiun-Chuan Chen.
Title: Compressible Navier-Stokes equations on domains with corners
Abstract: In this talk I will talk about existence and regularity result for the compressible Navier-Stokes equations on bounded domains with corners. The solution is constructed by the decomposition into singular and regular parts near non-convex vertices, based on the corner singularity expansion, and the regular part is shown to be twice differentiable. So the flow variables do not have enough regularities at the non-convex vertices and their derivatives blow up, in particular the corner singularities are propagated into the region by the transport character of the continuity equation and generate certain interior layers and discontinuity jumps. Such structure may possibly have applications in controlling cavity flow oscillations in a compressible flow, for instance, resolving the resonant amplitude tones near corners, etc.
Title: Some recent investigations on motion by mean curvature in inhomogeneous medium
Abstract: We will present some results for the interfacial propagation in inhomogeneous medium. The prototype equation is given by motion by mean curvature. The key feature is the interaction between the mean curvature of the interface and the underlying spatial inhomogeneity. We will describe the transition between the pinning and de-pinning of the interface and the existence of pulsating waves. Some recent investigations on the pinning threshold, front propagations between patterns, contact line dynamics and random walks in random medium will also be mentioned.
Antoine Mellet, University of Maryland, College Park.
Title: Fronts propagation in heterogeneous medium
Abstract: Fronts propagation in homogeneous medium (modeling, for instance, invasions in population dynamics or flame propagation in combustion theory) is classically described by the so-called Traveling Waves solutions. In the past ten years or so, significant progress have been made in generalizing this notion of solutions to more general frameworks. I will review some of the results obtained in the context of flame propagation. One of the main issue that will be discussed is the determination of the effective speed of propagation for such fronts.
Mike Coleman, George Washington University.
Title: An Isotropic Model for the Configuration and Accuracy of Deployable Elastic Antenna Reflectors
Abstract: Deployable aperture antenna reflectors are constructed by seaming together two elastic canopies and then pressurizing the enclosure with an inflation gas. The desired shape of an antenna reflector is a paraboloid and therefore the same shape is sought for the canopies described. While the canopies are designed to reach a nearly parabolic shape upon inflation, the elastic nature of the material allows them to deform when subject to external forces. We investigate how the external support structure, the internal pressure and gravitational loading forces affect the geometric accuracy of the reflector. The equilibrium configuration of the reflector is found by minimizing the total energy of a corresponding discretized system. Included in the energy computation are: the gravitational potential of the reflector material, the hydrostatic pressure of the inflation gas, the film strain energy of the reflector membrane and the strain energy in the supporting tendons. We model the membrane assuming a linear stress-strain constitutive relation for the material and approximate the surface by a piecewise isotropic constant-strain faceted surface. The reflector's geometry can be assessed using an RMS computation and an analysis of the electric and magnetic fields that are reflected to the far field zone of the antenna. Results demonstrate the manner in which the various external forces affect the accuracy of the reflector surface.
Sarah Day, College of William and Mary.
Title: Computer-assisted proofs for dynamical systems
Abstract: With recent advances in computing power, numerical studies of nonlinear dynamical systems have become increasing more popular. However, errors inherent to such studies may obscure the dynamics or, in the very least, raise doubts about the existence of numerically observed structures. Furthermore, unstable behavior, an intrinsic element of complicated systems, may be difficult to track even with very careful numerical work. I will discuss topological techniques which allow for the rigorous detection of dynamical structures of various stability types. Towards the end of the talk, I will focus on more recent work on expanding these techniques that led to the computation of a rigorous lower bound on the topological entropy (one measurement of complexity) for the (chaotic) Henon map.
Stan Alama, McMaster University.
Title: Thin film limits in the Ginzburg-Landau model of superconductivity
Abstract: We consider the full three-dimensional Ginzburg-Landau functional, which models the physical state of a superconductor in a given applied magnetic field. Assuming the superconductor occupies a thin domain of varying thickness, we first study variational convergence to a simplified two-dimensional model as the thickness parameter tends to zero. This limit is done within the frame work of de Giorgi's Gamma convergence. We identify three interesting asymptotic regimes, depending on the strength of the applied magnetic field in terms of the film thickness. In the most interesting ("critical") regime, we discover two curious new phenomena involving the component of the magnetic field parallel to the limiting plane. We then study the limiting energies in each regime in the London limit to determine the location of vortices. We show that, near the lower critical field, vortices may concentrate on points or on curves in the domain, depending on the choice of applied field and film profile. This represents two separate collaborations, one with Bronsard and Galvao-Sousa, and the other with Bronsard and Millot.
Michael Ward, University of British Columbia.
Title: Diffusion on and Inside a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points
Abstract: A common biophysical feature associated with cellular signal transduct ion is that a diffusing surface-bound molecule must arrive at a localized signal ing region on the cell membrane before a signaling cascade can be initiated. The question then arises of how quickly such signaling molecules can arrive at the localized signaling regions. To model this problem, one must calculate the mean first passage time for a diffusing particle confined to the surface of a sphere in the presence of N partially absorbing hemispherical traps of asymptotically s mall radii. The rate at which the small diffusing molecule becomes captured by o ne of the traps is determined by asymptotically calculating the principal eigenv alue for the Laplace-Beltrami operator on the sphere with small localized traps. A related problem, referred to as the narrow escape problem, is to determine th e mean first passage time for a Brownian particle located initially inside a sph ere that has an almost entirely reflecting boundary except for the presence of N non-overlapping absorbing windows or traps of small measure where the Brownian particle can ultimately escape or be trapped. For each of these two problems, as ymptotic results for the mean first passage time are given in the limit of small trap radii, and a history of results for some related singular perturbation pro blems is given. Our analysis relies largely on the method of matched asymptotic expansions, together with detailed properties of the Neumann Green's function fo r the Laplacian either on or inside the sphere. The optimal spatial arrangments of N small traps that minimize the mean first passage time are shown to be close ly related to the determination of Fekete points related to the classic optimiza tion problem of determining the minimum energy configuration for N repelling Cou lomb charges on the unit sphere.
Qiang Du, Penn State University.
Title: Diffuse interface modeling of some interface problems involving elastic energy contributions
Abstract: We report some recent works on the diffuse interface modeling and simulation of some interface problems in materials science and biology. We present particular examples on the study of deformation of biomimetic vesicle membranes and homogeneous nucleation in anisotropic elastic solids. In both cases, elastic energy contributions are taken into account. We consider various theoretical and computational issues related to diffuse interface models and present some simulations results.
Jieun Lee, The George Washington University.
Title: Phase separation on a vesicle membrane and the corresponding isoperimetric problem, Part 2.
Abstract: In the second part of this series we take the approximate solution constructed last time and look for an exact solution nearby. One key step here is to analyze the linearized operator at the approximate solution. The linearized operator has an approximate kernel and the error of the approximate solution must be perpendicular to this kernel. This implies that the small patch must appear at a critical point of the Gauss curvature of the membrane surface. Further analysis reveals that this critical point must be a local maximum for the patch to be stable.
Cyrill Muratov, New Jersey Institute of Technology.
Title: A variational approach to front propagation in infinte cylinders.
Abstract: In their classical 1937 paper, Kolmogorov, Petrovsky and Piskunov proved that for a particular class of reaction-diffusion equations on a line the solution of the initial value problem with the initial data in the form of a unit step propagates at long times with constant velocity equal to that of a certain special traveling wave solution. This type of a propagation result has since been established for a number of general classes of reaction-diffusion-advection problems in cylinders. In this talk I will show that actually in the problems without advection or in the presence of transverse advection by a potential flow these results do not rely on the specifics of the problem. Instead, they are a consequence of the fact that the considered equation is a gradient flow in an exponentially weighted L^2 space generated by a certain functional, when the dynamics is considered in the reference frame moving with constant velocity along the cylinder axis. I will show that independently of the details of the problem only three propagation scenarios are possible in the above context: no propagation, a "pulled" front, or a "pushed" front. The choice of the scenario is completely characterized via a minimization problem.
Jieun Lee, The George Washington University.
Title: Phase separation on a vesicle membrane and the corresponding isoperimetric problem, Part 1.
Abstract: In a vesicle membrane made of two or more types of lipids, the segregated phase, where the two components separate to form phase domains, can be modeled by a Landau type free energy formulated on a surface. Here we study the situation where one component has a much smaller mass compared to the other and it forms a small, approximately round patch. We plan to show that a solution of this type to the Euler-Lagrange equation of the system can be constructed via a singular perturbation argument. In the first part of this project we construct an approximate solution and show that it nearly satisfies the Euler-Lagrange equation. We will give a very good estimate on the error of this approximate solution. In the next talk, using this error estimate we will show that an exact solution can be found near the approximate solution. Moreover the patch must appear at the point where the surface's Gauss curvature attains maximum. The talk will cover some basic concepts in differential geometry including length element, area element, first fundamental form, geodesic polar coordinates, and Gauss curvature.
Tao Luo, Georgetown University. Title: Stability of Rotating Star Solutions of the Compressible Euler-Poisson Equations with Applications to White Dwarf Stars.
Abstract: In this talk, I will talk about the stability of rotating star solutions for the compressible Euler-Poisson Equations. The rotating star solutions are axi-symmetric steady-state solutions of the compressible isentropic Euler-Poisson equations in 3 spatial dimensions, with prescribed angular momentum and total mass. The stability of those solutions is proved by using several conservative quantities, such as mass, energy and angular momentum, for the evolutionary Euler-Poisson equations. Those results apply to white dwarf stars. This is joint work with Joel Smoller.
Chiun-Chuan Chen, National Taiwan University.
Title: Topological degree for a Liouville type equation with singular source
Abstract: We consider a Liouville type equation in two dimensional domains which arises from prescribing Gaussian curvature problem, the mean field limit of vortices in Euler flows, and limit cases of Chern-Simons models. The solutions can blow up when the total mass tends to some critical values. We will discuss how the delta functions in the source term affect the blowup behavior of the solutions and present a combinatorial formula of the Leray-Schauder degree for the problem.
Tobias Baumgart, University of Pennsylvania.
Title: Of domains and boundaries - phase behavior and mechanics of biomembrane models
Abstract: Lipid bilayer membranes of biological cells are likely to be non-random mixtures of membrane components. Lateral membrane domain formation is thought to be involved in essential functions of the membrane, including signaling, sorting, and trafficking. In order to elucidate the physical, mechanical, and physico-chemical basis and consequences of membrane heterogeneity, model systems have been developed. These typically consist of ternary lipid mixtures that under suitable conditions segregate into two fluid phases, a liquid ordered, and a liquid disordered phase. Of particular interest to us are the boundaries of domains in membranes with phase coexistence due to interfacial tension (line tension) at fluid phase boundaries. We find that this line tension couples to three dimensional membrane shape, modulating biologically relevant phenomena, including vesicle budding and fission. Line tension a function of membrane composition, using micropipette aspiration of giant vesicles, as well as capillary wave spectroscopy of thermal boundary fluctuations. We demonstrate that these two complementary techniques probe different line tension regimes. Furthermore, we are developing experimental methods to investigate the partitioning of both lipids and proteins among curvature gradients. We find that lipids are not detectably sorted among membrane with steep curvature difference, whereas peripherally membrane binding proteins are efficiently sorted. We discuss the biological relevance of our findings.
Susan Gillmor, George Washington University.
Title: Lipid Behavior, Cell Membranes and Dimpled Vesicles
Abstract: We begin with a thorough analysis of lipids and their role in cell membranes. The membrane structure, its fluidity, line tension and lipid phases are all part of cellular dynamic processes. As we dissect the membrane, we explore lipid models to delve into their behavior in a controlled manner. From our vesicle model, we find parameters for lipids alone to give us mimics of the red blood cell iconic biconcaved shape. From Sheetz and Singer classic study (PNAS 71, 4457-4461 (1974)), perturbations to the exterior lipid leaflet of the membrane promotes the cup-shaped stomatocytes while additives to the interior have the opposite effect. ADE (area-difference elasticity) theory on vesicle shape points to leaflet area asymmetry, bending elasticity and volume as determining variables. Our analysis isolates the interface energetics of why the shape is favored and points to transient pores enabling the vesicle to deflate and to assume a lower profile. Membrane fluidity and phase are key to the RBC transformations.
Maria Emelianenko, George Mason University.
Title: Mesoscale modeling of polycrystals: understanding stochastic events in microstructure evolution
Abstract: Preparing a texture suitable for a given purpose is a central problem in materials science, which presents many challenges for mathematical modeling, simulation, and analysis. In recent years we have witnessed a changing paradigm in the materials laboratory with the introduction of automated data acquisition technologies. This has permitted a more accurate characterization of materials properties and the collection of statistics on a vast scale, both of which pave the road to a better understanding of the way materials evolve in nature and to optimizing aspects of material behavior to better fit technological needs. In this talk, I will focus on the mesoscopic behavior of a model grain boundary system and on understanding the role of topological reconfigurations during evolution. We have explored several evolution equations based on pure probabilistic and stochastic descriptions and compared against the results provided by large-scale simulations and experiments. The advantages and limitations, numerical characteristics and possible extensions of these approaches to higher dimensions will be discussed.
Frank Baginski, George Washington University.
Title: The equilibrium shape of bubbles, balloons, and blood cells
Abstract: Soap bubbles, stratospheric balloons, and blood cells are examples of membranes whose equilibrium configuration is shaped by surface energy. The surface energy may include contributions due to elastic strain (stretching), flexural strain (bending), and hydrostatic pressure. We begin with a survey of some classical problems in differential geometry: Plateau's problem (find the soap film of least area that spans a given wire frame), soap bubbles (find the soap film of least area that encloses a fixed volume), and Willmore surfaces (find the surface with least bending energy). In a soap film, surface energy is simply proportional to surface area where the constant of proportionality depends on properties of the soap solution. The relation between stress and strain in a thin elastic wrinkled membrane leads to a more complicated expression for the elastic strain energy. In the case of a stratospheric balloon inflated with a lifting gas, the balloon skin is very thin and bending energy can be ignored. Shape determination is dominated by elastic strain and hydrostatic pressure due to the lifting. Theoretical and computational results for a model that we have developed for shape and stress analysis of NASA's stratospheric balloons will be presented. Finally, we consider cell membranes, structures whose surface energy involves stretching and bending. The lipid bilayer of a cell membrane provides a gateway in and out of the cell as well as a highway for events on and in it. A mathematical model that accurately describes the behavior of the cell as mechanical structure is essential to understanding cell functions. We will apply some of the lessons learned from differential geometry of surfaces and elastic balloons to the problem of modeling the equilibrium shape of a red blood cell. The balloon results include joint work with Bill Collier and Michael Barg. The work on cell membranes is an on-going project with Susan Gillmor/GW Chemistry Department and Jiuen Lee.
Barbara Niethammer, Oxford University.
Title: Effective theories for Oswald ripening
Junping Shi, College of William and Mary.
Title: Bistable dynamics in autocatalytic chemical reactions
Abstract: We consider a parabolic system that models an isothermal autocatalytic chemical reaction. If the spatial domain has dimension higher than 2 and the "order" of the reaction is high enough, then it is known that the system has a family of non-trivial steady states. We prove that each of these steady states is a "hair-trigger" for two types of long time behavior: if the initial value is below the steady state, then the solution of the system converges to a rest state of the system as time goes to infinity and so extinction occurs; if the initial value is above the steady state, then a wave front is developed and so we have the spread of "flame". We also supply some criteria on the initial value for spread/extinction of the reaction. We also consider the case of bounded reactor, and a S-shaped bifurcation diagram and bistable dynamical behavior are proved. The talk reports recent joint work with Xuefeng Wang of Tulane University, Yuwen Wang and Yuhua Zhao of Harbin Normal University, and Jifa Jiang of Tongji University.
Haitao Fan, Georgetown University.
Title: Recent Advances in Fluid Flows Involving Liquid/Vapour Phase Transitions
Abstract: Fluid flows involving liquid/vapor phase transitions occur very frequently, and have wide applications in science and engineering. Yet there are a lot of questions in this area left unanswered. In this talk, I shall concentrate on a model I proposed for the dynamic flow with liquid/vapor phase changes. Mathematical results for the model, especially those about the its traveling waves and their stability, are presented. Comparison with wave patterns observed in actual experiments are given. The (former) puzzling phenomenon of symmetry breaking and ring formation is explained.
Evelyn Sander, George Mason University.
Title: Two routes to chaos: Explosions and Cascades
Abstract: This talk consists of recent results on two types of bifurcations leading to complex dynamical behavior. These results are in the context of finite-dimensional iterated maps. An explosion is a global bifurcation in a family of iterated maps in which there is a discontinuous change in the size of the set of recurrent points as a parameter is varied. We describe explosions in dimensions one, two, and three. Explosions can be fully classified in dimension one. They lead to crises of chaotic attractors in two dimensions, and unstable dimension variability in three dimensions. A horseshoe has long been the hallmark of chaotic behavior. In one and two-dimensional maps, and in higher dimensional maps with at most one unstable eigenvalue, the path to a horseshoe is involves the celebrated cascade of period doubling bifurcations. We give a generalization of this result using index theory. This result guarantees that there are period doubling cascades for arbitrarily large weakly coupled systems of maps which each display horseshoe-like behavior.
Monica Musso, Pontificia Universidad Catolica de Chile.
Title: Bubbling along geodesics for some semilinear supercritical elliptic problem in bounded domains.
Abstract: In this talk we present some new results concerning solvability of the classical elliptic problem Delta u + u^p = 0, u > 0 under zero Dirichlet boundary conditions in a bounded domain in R^N , when p>1 is above the Sobolev critical exponent (N+2)/(N-2). The critical exponent in one dimension less p=(N+1)/(N-3) turns out to play a very interesting role in this problem. We show that, if p=(N+1)/(N-3)-epsilon, for some small and positive epsilon, the problem admits a positive solution concentrating along a curve. This result corresponds to joint work with Manuel del Pino and Frank Pacard.
Prof. Musso will visit the GWU math department from Dec 12 to 15.
Manuel del Pino, Universidad de Chile.
Title: Multiple bump-lines and transition layers for planar autonomous elliptic problems.
Abstract: In this talk I will describe a construction of new solutions to some classical autonomous semilinear elliptic equations in the plane. These solutions constitute a "gluing" of one-dimensional profiles with a single transition, located very far apart one to each other. In the case of the Allen-Cahn equation, solutions with a finite number of nearly parallel transition layers are built, while for the stationary nonlinear Schrodinger equation multiple bump-line patterns are found. The Toda system is shown to rule the asymptotic shape of these transition lines. This is joint work with Michal Kowalczyk, Frank Pacard and Juncheng Wei.
Prof. Del Pino will visit the GWU math department from Dec 12 to 15.
Magdalena Musielak, The George Washington University.
Title: A Computational Model of Nutrient Transport and Acquisition by Diatom Chains in a Moving Fluid.
Abstract: The role of fluid motion in the transport of solutes to and away from cells and aggregates is a fundamental question in biological and chemical oceanography. However, little is known about behavior of phytoplankton cells in well-defined flow fields. Experimental data to test the contribution of advection to nutrient acquisition by phytoplankton are scarce, mainly because of the inability to visualize, record and thus imitate fluid motions in the vicinities of cells in natural flows. Nutrient fluxes on the scale of interest are difficult to detect, and experimental errors in the measurements of enhancement of flux due to flow may often be comparable in magnitude to the predicted values, especially if the test organisms are very small. Thus, computational experiments are needed to analyze the contribution of advection to mass transfer and nutrient acquisition by phytoplankton. We present in this talk a mathematical model that describes the flexible diatom chain, the surrounding fluid, and the nutrient, by a coupled mechanical system. The chain is modelled as a collection of neutrally-buoyant cylinders connected by filaments. The motion of the fluid is governed by the incompressible Navier-Stokes equations. We use the immersed boundary method to couple the interaction of non-motile diatom chains with the viscous, incompressible, moving fluid, and with the nutrient that is advected by and diffusing in the fluid and also consumed by the cells. We apply our model to investigate the behavior of diatom chains in various flow regimes. We examine the impact of shape, length, and flexibility of chains on nutrient uptakes in a turbulent environment. Our numerical solutions for nutrient mass transfer to diatom cells fall within the bounds of the known analytic solutions for limiting cases. These results confirm intuitive predictions, and open the door to possible experimental work to measure the nutrient transport and acquisition for chains with different elasticities.
Title: A biomechanical single-cell-based model of morphological transformations in epithelial tissues.
Abstract: The proper structure and function of multi-cellular organisms is a result of interactions of individual cells in the body, and is controlled and guided by various signals interchanged between the neighbouring cells or sensed from the cell local microenvironment. We are interested in building a biomechanical model that can be used to investigate cell collaborative or competitive behaviour within the tissue that may lead to the development and maintenance of normal tissues, or to the formation of various tumours. Our computational model of individual cells is based on the immersed boundary method [C.~S. Peskin Acta Numerica (2002)] and couples the continuous description of a viscous incompressible cytoplasm and the extracellular matrix, with the dynamics of separate elastic deformable cells, containing their own elastic plasma membrane, fluid cytoplasm and individually regulated cell processes, such as cell growth, division, epithelial polarisation, apoptosis, and exchange of signals with the surrounding microenvironment. This approach allows for modelling various multicellular phenomena by focusing on biomechanical properties of individual cells and on communication between them and between the cell and their microenvironment. It also allows to investigate how individual cells contribute to the formation and maintenance of the whole complex system. Several computer simulations of the formation of various tissues will be presented, including the development of hollow epithelial acini, formation of ductal carcinomas and growth of solid and invasive tumours.
Lia Bronsard, McMaster University.
Title: Global minimizers for anisotropic models of superconductivity.
Prof. Bronsard will visit the GWU math department from Nov 8 to 11.
Chongchun Zeng, Georgia Institute of Technology.
Title: Approximately invariant manifolds and dynamic spike solutions of a singular parabolic equation.
Abstract: Consider a nonlinear parabolic equation $u_t = \epsilon^2 Delta u - u + f(u)$ on a smooth bounded domain with the Neumann boundary condition. In the past years, there had been extensive studies on steady spike solutions. Here a spike solution $u$ is one which is almost equal to zero everywhere except on a ball of radius $O(\epsilon)$ where the $u=O(1)$. In this talk, we show that there exist dynamic spike solutions which maintain the spike profile for all time with the spike moving on the boundary of the domain.
Prof. Zeng will visit the GWU math department from Oct 24 to 26.
Padmanabhan Seshaiyer, George Mason University.
Title: Mathematical modeling and solution methodology for biological and bio-inspired applications
Abstract: Biological and bio-inspired applications arising in science and engineering often involve complex nonlinear interactions characterized by fluid dynamics, structural mechanics or in general a fluid-structure interaction. Efficient solutions to such complex coupled problems has remained a challenging problem in computational mathematics. In this talk, an overview of a variety of analytical, computational and experimental techniques to model, analyze and solve such coupled problems efficiently will be presented.
Yves van Gennip, Technische Universiteit Eindhoven.
Title: A variational model for patterns in polymer melts.
Abstract: In nature one encounters patterns everywhere, f rom the stripes on zebras and the sand patterns in a desert, to the beating of your heart and the fingerprints on your fingers. A pattern forming system that has received extensive attention from experimentalists in recent decades is the diblock copolymer melt. Two types of polymer molecules are chemically bonded together to form a diblock copolymer. These molecules mutually repel each other, but due to the chemical bond they are restricted in their movement away from each other. These competing influences lead to pattern formation on a length scale between that of the system and that of the molecules. In recent years also mathematicians began studying models for diblock copolymer melts. We are investigating a model for the simplest extension of this system, a blend of diblock copolymers and a third type of polymer, called homopolymer, which is not bonded to the copolymer molecules. This blend model, which is variational in nature, is now well-understood in one dimension. We know how minimisers of the energy look and under which conditions they are non-unique. In higher dimensions we have bounds on the energy and we investigated the stability of some specific morphologies in two dimensions. Our study shows an interesting dependence of stability in these cases on the strength of the mutual repulsion between the different types of polymer molecules.
Mr. van Gennip will visit the GWU math department from Oct 3 to 6.
Xiaofeng Ren, The George Washington University.
Title: A singular limit arising from some pattern formation phenomena in physics and biology.
Abstract: Similar patterns, such as stripes and spots, are often observed in very different problems in physics and biology. Examples include the Ohta-Kawasaki theory for diblock copolymers and the Gierer-Meinhardt theory for morphogenesis. The first is modeled by a nonlocal variational problem and the second by a reaction diffusion system. I will present two techniques that reduce the two problems, in a suitable parameter range, to one free boundary problem. The first technique is the formal asymptotic analysis, and the second is the rigorous Gamma-convergence theory. Some results on the free boundary problem will also be presented without proofs.
Theodore Kolokolnikov, Dalhousie University.
Title: Stability of curved interfaces in the perturbed Allen-Cahn model.
Abstract: We consider the existence and stability of an interface in the singular limit of a perturbed Allen-Cahn model in two dimensions. It is well known that in the unperturbed Allen-Cahn model, the interface boundary evolves according to the mean curvature law which minimizes its perimeter. Therefore the only non-trivial equilibrium state for the unperturbed model consists of a straight interface, typically located at the "neck" of a domain. The perturbation of the Allen-Cahn model by a small term has a large effect on the shape and stability of the interface. In particular, the equilibrium solution now consists of a curved interface. We fully characterize the stability of such an equilibrium in terms of a certain geometric eigenvalue problem, and give a simple geometric interpretation of our stability results.
Prof. Kolokolnikov will visit the GWU math department from Sept. 12 to 16.