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PDE/Appl. Math. Seminar, Abstracts Fall 2005

(Sept. 8) Christoph Schwab (ETH Zürich) Sparse Adaptive FEM for Multiple Scale Problems

Elliptic homogenization problems in a domain $\Omega \subset \R^d$ with $n+1$ separated scales are reduced to elliptic one-scale problems in dimension $(n+1)d$. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work and memory requirement comparable to standard FEM for single scale problems in $\Omega$ while it gives numerical approximations of the correct homogenized limit as well as of all first order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters.

Anisotropic Besov regularity in the scales and adaptive wavelet FEM will be addressed. Numerical examples for model diffusion problems with two and three scales will be given. The complexity of the method will be compared to that of the recently proposed hierarchical multiscale methods.

References: [1] C. Schwab: High-dimensional finite elements for elliptic problems with multiple scales and stochastic data, Proc. Intern. Congress of Mathematicians Vol. III, Higher Education Press Beijing, P.R.C., (2002), pp. 727-734.

[2] V. Hoang and C. Schwab: High dimensional finite elements for elliptic problems with multiple scales, SIAM J. Multiscale Methods (2005)

(Sept. 15) Miguel Dumett (USC) Simulation of skin vapor alcohol and deconvolution of blood alcohol concentration

An inverse problem approach is followed to estimate parameters of a PDE model that describes the transport of alcohol throughout the skin. This techniques uses skin vapor alcohol concentration data coming from a transdermal device and data obtained from a breathalyzer. The simulations performed utilizing real data coming from human patients predict accurately skin vapor alcohol for a period of two weeks. In addition, blood alcohol concentration is deconvolved from the transdermal signal and the values predicted by this deconvolution process are compared to the breathalyzer data.

(Sept. 22) Pierre-Emmanuele Jabin N-particles approximation of the Vlasov equations with singular forces

I study the convergence in any time interval of a point-particle approximation of the Vlasov equation by particles initially equally separated for a force in $1/|x|^{\alpha}$, with $\alpha \leq 1$. Discrete versions of the $L^\infty$ norm and time averages of the force field are introduced. The core of the proof is to show that these quantities are bounded and that consequently the minimal distance between particles in the phase space is bounded from below.

(Sept. 30) Rober V. Kohn A New Perspective on Motion by Curvature

I'll present recent work with Sylvia Serfaty concerning motion by curvature. The level-set formulation of this interface evolution law is a degenerate parabolic equation. We show it can be interpreted as the value function of a deterministic two-person game. This result is unexpected, because the value function of a deterministic control problem is usually a first-order Hamilton-Jacobi equation.

(Oct. 6) Dianwen Zhu Global existence and stability of supersonic flows past a Lipschitz wedge

We systematically study two-dimensional steady supersonic Euler (i.e. non-potential) flows past a Lipschitz wedge and establish the existence and nonlinear stability of supersonic Euler flows when the total variation of the tangent angle functions along the wedge boundaries is suitably small. The core is to develop a modified Glimm difference scheme and identify a Glimm-type functional. By obtaining required $BV$ estimates, we establish the convergence of both approximate solutions to a global entropy solution and corresponding approximate strong shock fronts emanating from the vertex to the strong shock front of the entropy solution. The regularity of strong shock fronts emanating from the wedge vertex and the asymptotic stability of entropy solutions in the flow direction are also established.

(Oct. 13) Govind Menon Universality classes in Burgers turbulence

Mathematical theories of turbulence require the construction of stochastic processes that are also weak solutions to the equations of fluid mechanics. This is poorly understood. However we now have a good understanding of a vastly simplified model problem: the structure of shocks in Burgers equation with random initial data. I will describe a beautiful closure theorem of Bertoin for this problem. One consequence is a characterization of `universality' of shock statistics for large classes of initial data.

This is joint work with Bob Pego.

(Oct. 20) Stuart Antman (UMD) Analytic consequences of incompressibility

A material body is incompressible if every deformation of it locally preserves its volume, in particular, if the Jacobian determinant of every continuously differentiable deformation of it is identically 1. (Rubber and much living tissue (which is composed mostly of water) are examples of incompressible materials.) Since the nonlinear PDEs of evolution for such 3-dimensional bodies have largely resisted analysis, it is useful to have effective theories for slender bodies (like worms, snakes, and eels) governed by equations with but one independent spatial variable. This lecture shows that the actual construction of one such very attractive theory requires the solutions of a sequence of first-order PDEs (by the method of characteristics). Although the resulting equations are more complicated than those for bodies not subject to the constraint of incompressibility, they admit some tricky a priori bounds and they have novel regularity properties not enjoyed by the latter. The governing equations for an elastic body can be characterized by Hamilton's Principle. The ODEs governing travelling waves for these equations can also be characterized by Hamilton's Principle, but the kinetic and potential energies for these ODEs do not correspond to those of the PDEs. These ODEs, which have a nonstandard structure, admit, under favorable assumptions, perioperiodicdic travelling waves with wave speeds that are are supersonic with respect to some modes of motion and subsonic with respect to others.

(Oct. 20) Thierry Goudon (Lille) Kinetic equations, diffusion asymptotics, flux-limited closure

It is a well known fact that, in small mean free path regime, kinetic equations can lead to diffusion equations. Besides, kinetic equations can be approached by a finite number of moments equations. Spefically, we will be interested in closure based on entropy minimization principle. We investigate the asymptotic behavior of the resulting nonlinear hyperbolic system in the diffusive regime. We show that the system is globally well-posed and admits global smooth solutions. We also show that it is consistent with the diffusion limit.

(Nov. 10) Dionisios Margetis (MIT) Evolution of crystal surfaces: From microscopic models to continuum laws

The description of evolving crystal surfaces requires mathematical modeling across the scales. In this talk I describe progress towards the derivation and analysis of continuum laws for crystal-surface motion. First, microscopic laws are formulated for nanoscale objects, ``steps'', which compose crystal surfaces. Step motion laws describe motion by step curvature and interactions, and are viewed as discrete, numerical schemes for continuum laws. Second, macroscopic laws are derived from step models. The surface height satisfies a fourth-order, nonlinear PDE that describes the anisotropic effect of surface fluxes on evolution. Third, particular PDE solutions are invoked to plausibly unify experimental observations of decaying profiles. Fourth, free-boundary problems are studied for the evolution of crystal shapes with a flat surface region (``facet''). The appropriate boundary conditions are nonlocal with time. The continuum predictions compare favorably with numerical simulations for individual steps.

(Nov. 29) Alex Kiselev (Wisconsin) Diffusion and Mixing in Fluid Flow

Enhacement of diffusion by advection is a classical subject that has been extensively studied by both physists and mathematicians. In this work, we consider enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and sufficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular, we find that weakly mixing flows always enhance the relaxation speed in this sense. The proofs are based on a new general criterion for the decay of the semigroup generated by a dissipative operator of certain form. They employ ideas from quantum dynamics, in particular the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian (and related to a theorem of Wiener on Fourier transforms of measures).

(Dec. 1) Maria Reznikoff (Princeton) Coarsening Rates for Gradient Flows

Sometimes physical systems exhibit ``metastability,'' in the sense that states get drawn toward so--called metastable states and are trapped near them for a very long time. A familiar example is the one--dimensional Allen Cahn equation: initial data is drawn quickly to a ``multi--kink'' state and the subsequent evolution is exponentially slow. The slow coarsening has been analyzed by Carr & Pego, Fusco & Hale, and Bronsard & Kohn.

In general, what causes metastability? Our main idea is to convert information about the energy landscape (statics) into information about the coarsening rate (dynamics). We give sufficient conditions for a gradient flow system to exhibit metastability. We then apply this abstract framework to give a new analysis of the 1--d Allen Cahn equation. The central ingredient is to establish a certain nonlinear energy--energy--dissipation relationship. One benefit of the method is that it shows that exponential closeness to the multi--kink state is not only propagated, but also generated.

This work is joint with Felix Otto, University of Bonn.

(Dec. 6) Chun Liu (Penn State) Macro-Micro Models in Viscoelastic Materials: An Energetic Variational Approach

I will describe an unified energetic variational framework for elastic complex fluids. It highlights the competition of the kinetic energy and the elastic energies, through the transport of the internal elastic variables. As applications, I will discuss some macro-micro scale coupling effects in these materials. Corresponding analytical and numerical issues will be addressed, in particular, I want to present some recent well-posedness results and a new moment closure algorithm that preserves the energy law of the original system.

(Dec. 8) Willi Jäger (Heidelberg) Reactive flow through porous media

Consider flow, diffusion, transport and reactions through porous media consisting of a fluid and a solid phase. Assuming periodic media consisting of a fluid and a solid phase. Assuming periodic structures and a scale parameter \epsilon (measuring e.g. the pore size) and properly scaled model equations (micro-system) the scale limit \epsilon\to0 is studied under different assumptions. The aim is to derive macroscopic laws (macro-system) reducing the complexity of the model. The main mathematical problem consists in deriving the proper scale dependent estimates for the solutions to the given micro-system needed to apply the functional analytic tools of multi-scale convergence. Here, as usual nonlinear terms are posing obstacles in passing to the limit. Using localized coordinate systems and introducing macro- and microscopic variables, the arising compactness problems can be reduced essentially. In this seminar, the underlying concepts are introduced and applied to selected examples. Open problems arising in important applications will be discussed.

(Dec. 9) Willi Jäger (Heidelberg) Multi-scale modelling in biosciences - ion transport through membranes

Modelling and simulation of processes in biosciences lead in general to complex model systems This complexity is caused by the system size, the arising nonlinearities, the range of scales involved, the stochastic nature of the processes and of the underlying geometry. It is a main challenge for Mathematics to reduce the complexity e.g. by analytic or numerical multi-scale methods. Already in setting up models these techniques are necessary to include information about the real processes on different scales and to link the corresponding models. The effect of microscale processes on the macro-scale behaviour has to be analysed. As an important example we consider the transport of ions through membranes. Mathematical modelling and simulation of ion concentrations inside and outside living cells, in their cyto-plasma and their nucleus, separated by membranes, are decisive for a better understanding of the bio-system "cell". Due to that fact that there is more and more information available about the processes on the micro-scale it is necessary to link model equations on micro-scale to a macroscopic description. It is important that model data can be computed using micro-scale information.

Here two domains are considered separated by a membrane perforated by channels placed in periodically distributed cells. The thickness of the membrane and the diameter of the cells are of order ?. The transport of ions is modelled by the Nernst-Planck equations, properly scaled in the channels. Charges fixed to the channels are included modelling the influence of the channels and the changes of its conformation. Effective laws for the ion transport trough membranes are derived performing an asymptotic analysis with respect to the scale parameter ?. The effective model consists in the Nernst-Planck equations on both sides of the membrane together with appropriate transmission conditions for the ion concentrations and the electric potential across the cell membrane. These conditions are determined solving micro-problems for cell problems in the membrane.

New methods of homogenization have to be developed and applied in order to deal with the nonlinear model equations and the reduction of the membrane to a two dimensional interface.

(Dec. 13) Alexei Novikov (Penn State) A homogenization approach to large-eddy simulation of incompressible fluids

In the development of large-eddy simulation one makes two primary assumptions. The first is that a turbulent flow can be categorized by a hierarchy of lengthscales. The second assumption states that the small scales have universal properties, characterized by, e.g. a spectral power law. This motivated a number of physical models that attempt to account for the presence of small scales by suitably modifying the corresponding partial differential equations (PDE), the Navier-Stokes equations. Homogenization theory addresses rigorously the issue of modification of PDE in the presence of small scales. The goal of this talk is to apply homogenization methods to LES modeling of fluid flows. We will start with the phenomenon of eddy viscosity in two-dimensions: in the presence of small-scale eddies the transport of large-scale vector quantities can be accompanied with depleted, and even ``negative" diffusion, when the Reynolds number is sufficiently large.

Georg Dolzmann

Last modified: Thu Dec 1 09:30:41 EST 2005