248 PD06 Abstracts

CP1 applications are proposed. Efficient Optimization of Electrostatic Interactions Between Biomolecules Fejzi Kolaneci University of New York, Tirana We will present a novel PDE-constrained optimization [email protected] strategy for analyzing the optimality of electrostatic interactions between biomolecules. These interactions are important factors affecting binding affinity and specificity, CP1 and can be analyzed using linear continuum models. The Continuous Moran Process and the Diffusion of resulting optimization problems are well-posed but com- Genes in Infinite Populations putationally expensive. Our method uses an implicit representation of the Hessian, in combination with precondi- We consider the so called Moran process with frequency tioned Krylov methods, to dramatically reduce the com- dependent fitness given by a certain pay-off matrix. For putational expense. We will discuss several applications in finite populations, we show that the final state must be biomolecule analysis and design. homegeneous, and show how to compute the fixation prob- abilities. Next, we consider the infinite population limit, Jacob White and discuss the appropriate scalings for the drift-diffusion Research Lab. of Electronics limit. In this case, a degenerated parabolic PDE is formally MIT obtained that, in the special case of frequency independent [email protected] fitness, recovers the celebrated Kimura equation in population genetics. We then show that the corresponding initial Jaydeep Bardhan value problem is well posed and that the discrete model Electrical Engineering & Computer Science converges to the PDE model as the size of population goes Massachusetts Institute of Technology to infinity. We also study some game-theoretic aspects of [email protected] the dynamics and characterise the best strategies, in an appropriate sense. Bruce Tidor Fabio Chalub MIT Departamento de Matemática [email protected] Universidade Nova de Lisboa [email protected] Michael Altman Department of Chemistry Max O. Souza Massachusetts Institute of Technology Departamento de Matemática Aplicada [email protected] Universidade Federal Fluminense [email protected]ff.br CP1 Growth of a Spherical Tumor with a Necrotic Core: CP1 A Moving Boundary Simulation Computational Analysis Based on Mathematical Model for Biodegradation of Xenobiotic Polymers After vascularization, a tumor may develop a necrotic core, characterized by an outer rim of proliferating cells and an Biodegradation processes of xenobiotic polymers are stud- inner core of dead cells. Research shows that between these ied numerically. The weight distribution with respect to is a layer of quiescent cells which can begin dividing if en- the molecular weight before and after biodegradation is vironmental conditions change. What follows is an exten- introduced into analysis based on a mathematical model sion of previous work used to show the concentration of and an inverse problem is solved numerically to determine nutrient within the interior of a tumor over time. At each a biodegradation rate. Once the biodegradation rate is time step, the core boundary and the tumor boundary are found, an initial value problem can be solved numerically free to increase or decrease as appropriate when compared to simulate the transition of the weight distribution, and to certain threshold values, thereby simulating a moving the applicability of the numerical result can be tested. boundary problem. Masaji Watanabe Carryn Bellomo Graduate School of Environmental Science University of Nevada, Las Vegas Okayama University Department of Mathematical Science [email protected] [email protected] Fusako Kawai CP1 Research Institute for Bioresources Okayama University A Stochastic Model of the Tumor Growth for Dis- [email protected] persed Cells Regime

A stochastic model is developed to describe the growth of a CP1 heterogeneous tumor for dispersed cells regime. The mathematical model is a quasilinear stochastic partial diﬀeren- Multiple-Spike Ground State Solutions of the tial equation driven by a space-time white noise. The main Gierer-Meinhardt Equations feature of the model is that it takes into account random In many biological pattern formations and some biochemi- independent interactions between tumor cells, immune sys- cal reactions, an activator-inhibitor system of two reaction- tem cells and anticancer drugs. The existence of the weak diﬀusion equations serves as a mathematical model. The solutions and a comparison theorem are established. Some PD06 Abstracts 249

typical Gierer-Meinhardt equations suitable conditions on ϕ , the constants c1,...,cp, the functions aq,f1 and f2. A2 At = dΔA − A + , H Mahmoud M. El-Borai Prof.of Math. Dep. of Mathematics Faculty of Science 2 Alexandria University Alexandria Egypt Ht = DΔH − H + A , m m [email protected] where d/D << 1, A, H > 0 and A, H → 0as| x |→ ∞, feature two largely different diffusion coefficients and the essentially nonlocal nonlinearity. By the Lyapunov-Schmidt CP2 method, the maximum order estimate of the multiple spike On the Stability of Some Stochastic Differential numbers of a ground state solution is found and the con- Equations struction of such a solution is shown. Stochastic Volterra equations of the form: Yuncheng You University of South Florida t [email protected] dx(t)=f(x(t))dt + K(t − s)x(s)ds dt + g(x(t))dB(t), 0

CP2 are considered, where {B(t):t ≥ 0} is standard one - di- Particle Motion in the Circular, Restricted Three- mensional Brownian motion and the kernel K decreases to Vortex Problem zero non-exponentially. We study the convergence rate to zero of the stochastic solutions of the considered equation. Can a number of test particles traveling in different pe- It is proved under suitable conditions that : riodic orbits in the circular, restricted three-vortex realm be controlled such that they are placed in the same (equal |x(t)| lim = ∞, almost surely. period) orbit? Additionally, can the same or a different t→∞ K(t) controller be used to fix the relative positions of the test particles such that a virtual or dynamically natural formation can be established? It will be shown that the answer Khairia E. El-Nadi to both questions is, yes. Prof of Mathematical Statistics Dep. of Math. Faculty of Ralph R. Basilio Science Alexandria University Alexandria Egypt University of Southern California Khairia el [email protected] [email protected] CP2 Paul K. Newton Numerical Realizations of the Stochastic KdV Univ Southern California Equation With and Without Damping Dept of Aerospace Engineering [email protected] We investigate numerical simulations of an exact solution of a stochastic Korteweg deVries equation under Gaussian white noise. We compare the expectation values of the CP2 exact solutions to theoretical expectation values and to the On Some Stochastic Fractional Integrodifferential numerical simulations of the stochastic Korteweg deVries Equations equation with and without damping. We find that typically on average the diffused soliton vanishes long before the The purpose of this paper is to study the integro-partial typically reported asymptotic limit. differential equation of fractional order: Russell L. Herman α t ∂ u(x, t) q UNC Wilmington − aq(x)D u(x, t)=f1(u(x, t))+ f2(u(x, s))dW (s), ∂tα [email protected] |q|≤2m 0 with the nonlocal condition CP2 p First Passage Time Problem of a Time-Dependent Ornstein-Uhlenbeck Process to a Moving Bound- u(x, 0) = ϕ(x)+ cku(x, tk), ary k=1 In this paper we derive the closed-form formula for where 0 ≤ t1

Cho-Hoi Hui in the limit, gives the above classical solution until the Hong Kong Monetary Authority moment of contact and becomes the solution of the heat Research Department equation after the contact. We study the properties of this cho-hoi [email protected] approximation. In particular, we show that the velocities of the free boundaries become equal to each other in absolute value at the moment of contact and the temperature has a CP2 jump at the point of contact. The obtained analytical and Convergence Analysis of Operator Upscaling for numerical results are compared. the Acoustic Wave Equation Vladimir G. Danilov Wave propagation in a heterogeneous medium results in Moscow Technical University of Communications and models involving multiple scales. Operator-based upscal- Informatic ing solves the problem on a coarse grid, but still retains Professor, Head of Chair subgrid information and ﬁne-scale input data. First, the [email protected] problem is solved for the subgrid component deﬁned locally within each coarse block. Then, the subgrid solutions are used to augment the coarse-grid problem. We present con- CP3 vergence analysis for the method applied to the constant The Onset of Superconductivity at a Nor- density, variable sound velocity acoustic wave equation. mal/Superconducting Interface

Tetyana Vdovina We Study a modified model of Ginzburg and Landau that Department of Mathematics and Statistics considers superconducting electrons diffusing into a nor- University of Maryland Baltimore County mal material in contact with a superconductor. We assume [email protected] that each region occupy a half-space with a constant applied field parallel to the interface. we show, if the normal Susan E. Minkoff conductivity of the superconductor is less than the conduc- University of Maryland, Baltimore County tivity of the normal material then normal states are local Dept of Mathematics and Statistics minimizers for fields down to Hc2, which agrees with exper- sminkoff@math.umbc.edu imental observations that superconductivity is suppressed in this case. While when the conductivity of the superconductor is larger than that for the normal material the onset CP3 occur at fields larger than Hc2 and less than Hc3. Level Set Method Approaches to Dislocation Mod- els of Grain Boundaries Tiziana Giorgi New Mexico State Univeristy Grain boundaries in materials can be formally represented Department of Mathematics as arrays of dislocations lines. This perspective makes it tgiorgi@nmsu possible to study meso-scale processes, such as mechanisms for grain boundary motion and interactions of dislocations Hala Jadallah with grain boundaries. Using a level-set formalism for Western Illinois University dislocation dynamics pioneered by our group, we inves- Department of Mathematics tigate several interesting grain boundary problems using [email protected] dislocation models. The computational efficiency required to study these models is provided by a parallel level set method software library that we have recently developed. CP3 Homogenisation of Reaction, Diffusion and Interfa- David Srolovitz cial Exchange in Porous Media with Evolving Mi- Princeton University crostructure [email protected] Chemical degradation mechanisms of porous materials of- Yang Xiang ten result in a change of the pore geometry. These effects Department of Mathematics cannot be captured by the standard periodic homogenisa- HKUST tion method due to the local evolution of the microscopic [email protected] domain. A mathematically rigorous approach is suggested which allows the treatment of such problems. It makes use Kevin T. Chu of a transformation to a stationary (periodic) reference do- Mechanical & Aerospace Engineering main on which the homogenisation can be performed. A Princeton University physical interpretation also allows the direct modelling of [email protected] the transformed problem. Malte A. Peter, Michael Böhm CP3 Centre for Industrial Mathematics University of Bremen, Germany Free Boundary Interaction in Stefan Problem with [email protected], Undercooling [email protected] Suppose that the Stefan problem with kinetic undercooling has a classical solution until the moment of contact of free CP3 boundaries and the free boundaries have finite velocities until the moment of contact. Under these assumptions, we Solutions of the Smoluchowski Equation for Ne- construct a smooth approximation of the global solution of the Stefan problem with kinetic undercooling, which, PD06 Abstracts 251

matic Polymers under Imposed Fields [email protected]

We solve the Smoluchowski equation for rigid nematic poly- Stuart Antman mers under imposed elongational flow, magnetic or electric Mathematics Department & IPST fields. We show that: (1)The equation can be cast in a UMCP generic form. (2) The steady state solution is of the Boltz- [email protected] mann type, parametrized by material and two order parameters. (3) The external field must be parallel to one of the eigenvalues of the second moment tensor. A bifur- CP4 cation diagram of the order parameters is presented. The Floating Drops and Functions of Bounded Varia- solution method is extended to dilute solutions under im- tion posed electric field. The first moment of the steady state solution is shown parallel to the external field, under cer- Three fluids are in equilibrium under the action of grav- tain conditions. itational and surface tension forces. The fluid with intermediate density has a finite volume, and the motivat- Qi Wang ing example is that in which this volume forms a ’drop’ Florida State University floating on thhe most dense fluid wiith the least dense one Department of Mathematics above. In axisymmetric situations, for example when the [email protected] other two fluids have infinite volume, this problem can be formulated as a free boundary problem for axisymmetric Sarthok K. Sircar capillary surfaces. The drop is bounded by a sessile drop Department of Mathematics and an inverted sessile drop and the exterior interface is an Florida State University exterior capillary surface. This has been studied both the- [email protected] oretically and numerically, but the general existence theorem has proven difficult to obtain using these methods. Hong Zhou We formulate here a variational problem using functions of Department of Applied Mathematics bounded variation and prove the existence of a solution. Naval Postgraduate School, Monterey, CA [email protected] Ray Treinen University of Toledo [email protected] CP3 Dynamical Instability of Pinned Fundamental Vor- Alan Elcrat tices Wichita State University Kansas, USA We show dynamical instability of pinned fundamental ±1 [email protected] vortex solutions to the Ginzburg-Landau equations with external potentials in R2. For smooth and sufficiently small external potentials, there exists a perturbed vortex solu- CP4 tion centered near each critical point of the potential. Per- Explosive Instabilities In Rimming Flows turbed vortex solutions which are concentrated near min- ima (resp. maxima) are orbitally unstable (resp. stable). The linear stability of a thin film of viscous fluid on the in- We consider both gradient and Hamiltonian flows. side of a cylinder with horizontal axis, rotating about this axis is introduced in this paper. Both axial and azimuthal Stephen Gustafson components of the hydrostatic pressure gradient are taken Department of Mathematics into account, which yield solutions that collapse in both University of British Columbia dimensions. Despite the existence of these ‘explosive’ in- [email protected] stabilities, all solutions with harmonic dependence on the axial variable and time (normal modes) are neutrally sta- Fridolin Ting ble. This type of instability has been described in previous Lakehead University papers. However, no actual solution to describe the move- Department of Mathematical Sciences ment of the film of liquid through the cylinder has been [email protected] presented. This paper will rectify this and examine the properties of such a solution. CP4 Seán M. Lacey The Taylor-Couette Problem in a Deformable University of Limerick Cylinder [email protected]

The Taylor-Couette problem is a fundamental example in bifurcation theory, and has been the subject of over 1500 CP4 papers. This lecture treats a generalization where the rigid Ideal 2D Flow in a Domain with Holes and the outer cylinder is replaced by a deformable (viscoelastic) Small Obstacle Limit cylinder. A steady solution of this liquid-solid interaction problem can be found analytically; its linear stability is We study the limiting behavior of incompressible, ideal flow governed by a quadratic eigenvalue problem. The purpose in a 2D bounded domain with holes, when one of the holes of this lecture is to describe the analysis and computation becomes small. The difficulty lies in describing and con- of the spectrum. trolling the behavior of the potential part of the flow in terms of vortex dynamics. This work is a natural exten- David P. Bourne sion of [Iftimie et alli, CPDE 28 (2003) 349–370], which University of Maryland studied the same limit for flows in the exterior of a single 252 PD06 Abstracts

small obstacle. uids. Mathematically, flows of two liquids are described by Laplace equations, with (curvature-dependent) bound- Milton C. Lopes Filho ary conditions. Though the numerical simulations allow IMECC-UNICAMP one to solve the problem, there is no available technique [email protected] that would describe the time-changing fractal properties of the resulting flows. Such novel approach for time-varying description is proposed in the present work and well com- CP4 pared with experiment. Analytic Solution of Rayleigh-Taylor Problem Igor Kliakhandler, Sofya Chepushtanova We study the initial value problem for the classical Department of Mathematical Sciences Rayleigh-Taylor problems. If the initial data is analytic Michigan Technological University and satifies some other conditions, it is proved that unique [email protected], [email protected] analytic solution exists locally in time. The analysis is based on a Nirenberg Theorem on abstract Cauchy- Kovalevsky problem in properly chosen Banach spaces. CP5 An Approximated So- Xuming Xie lution to the Two-Dimensional Lid-Driven Cavity Morgan State University Flow, Using Adomian Decomposition Method and [email protected] the Vorticity-Stream Function Formulation

In this work, we present a reliable algorithm to solve the CP5 two-dimensional lid-driven cavity flow, using the Adomian Equilibrium Ice Sheets Solve Variational Inequali- Decomposition Method (ADM). The vorticity-stream func- ties tion formulation is used for the incompressible flow considered here. The solution is calculated in the form of a series The standard model for steady (shallow) ice sheet flow on with easily computable coefficients. Also, numerical simu- an arbitrary bed with stress-dependent basal sliding and lation, using finite difference method (FDM), is performed temperature-dependent flow is shown to be a variational for comparison purposes. This comparison shows consid- inequality whose interior condition is the well-known and erably close agreements. highly nonlinear “ice sheet equation,” a nonlinear diffusion which generalizes the p-Laplacian equation. That is, Mahmoud Najafi such sheets solve a substantially-generalized p-Laplacian kent state university ashtabula obstacle problem for p ≥ 2. These abstract problems are drmnajafi@hotmail.com variational inequalities for strictly monotone nonlinear op- 1,p erators on the Sobolev space W0 (Ω). For these problems M. Taeibi-Rahni we establish existence, an a priori bound, and continuity Sharif University of Tech. Tehran Iran of the solution with respect to certain parameter fields. [email protected] Uniqueness remains unproven in some cases. Finite element methods are considered. Khashayark Avani Ed Bueler SUT, Tehran Iran Dept. of Mathematical Statistics [email protected] University of Alaska Fairbanks ff[email protected] CP5 Wave Propagation Through a Spatio-Temporal CP5 Checkerboard Structure A Mathematical Model for Electromagnetic Waves in Magnetized Media We consider propagation of waves through a spatio- temporal material structure with a checkerboard micro- The topic of this talk is the Landau-Lifschitz equation geometry. The squares of the checkerboard are filled with coupled with Maxwell’s equations describing the electro- alternating materials having equal impedance but different magnetic field in generally nonlinear magnetized medium. phase speeds. Within certain parameter ranges, we observe The Landau-Lifschitz equation is a first order ordinary dif- the formation of distinct and stable limiting characteristic ferential equation for the magnetization field coupled to paths that attract neighbouring characteristics after a few Maxwell’s equations for the electromagnetic field. The time periods. The average speed of propagation along the main subjects are the existence, uniqueness and asymptotic limit cycles remains the same throughout certain ranges of behavior of the solutions as well as certain small parameter parameters of the microgeometry. limits in this model. Suzanne L. Weekes Frank Jochmann Worcester Polytechnic Institute (WPI) Fakultaet II - Mathematik [email protected] - Technische Universitaet Berlin [email protected] CP5 Numerical Modeling of Fluid Mixing for Laser Ex- CP5 periments and Supernova Fractal Properties of Hele-Shaw Flow In collaboration with a team centered at U. Michigan and Hele-Shaw flow in circular geometry leads to well-known LLNL, we have conducted front tracking simulations for instability of the frontline between two immiscible liq- axisymmetrically perturbed spherical explosions relevant PD06 Abstracts 253

to supernovae as performed on NOVA laser experiments, [email protected] with excellent agreement with experiments. We have extended the algorithm and its physical basis for preshock Jacob White interface evolution due to radiation preheat. Our second Research Lab. of Electronics focus is to study turbulent combustion in a type Ia super- MIT nova (SN Ia) which is driven by Rayleigh-Taylor mixing. [email protected] We have extended our front tracking to allow modeling of a reactive front in SN Ia. Our 2d axisymmetric simulations show a successful level of burning. CP6 Existence of An Optimal Shape for a Pde System Yongmin Zhang in a Free Air/Porous Domain: Application to Hy- SUNY at Stony Brook drogen Fuel Cells [email protected] We consider a shape optimization problem related to a non- James G. Glimm linear system of PDE describing the gas dynamics in a free SUNY at Stony Brook air - porous domain, including gas concentrations, temper- Dept of Applied Mathematics ature, velocity and pressure. The velocity and pressure are [email protected] described by Stokes and Darcy laws, while concentrations and temperature are given by mass and heat conservation Paul Drake laws. The system represents a simplified dry model of gas University of Michigan dynamics in channel and graphite diffusive layers in hydro- [email protected] gen fuel cells. The model is coupled with the other part of domain through some mixed boundary conditions, involving nonlinearities, and pressure boundary conditions. CP6 Under some assumptions we prove that the system has a so- A PDE-Constrained Optimal Control Problem in lution and that there exists a channel domain in the class of Image Processing Lipschitz domains minimizing a certain functional measur- ing the membrane temperature distribution, total current, We present a control problem motivated by mammographic water vapor transport and channel inlet/outlet pressure image processing. The underlying model involves a degen- drop. erate parabolic PDE where the control enters in the diffusion coefficients and, implicitely, in the solution space. We Arian Novruzi discuss spaces of discontinuous functions adapted to this University of Ottawa type of control. We further show how a characterization [email protected] helps in obtaining existence results for the simultaneous solution of the control problem and the PDE. CP6 Kristian Bredies, Dirk Lorenz, Peter Maass Hydrodynamic Limit to Hamilton-Jacobi Equa- Center for Industrial Mathematics tions University of Bremen [email protected], [email protected] we study convolution-generated monotone cellular au- bremen.de, [email protected] tomata(CA) and ordered transition rules. It turns out that those CA satisfy translation invariance and finite propagation speed. We show that the hydrodynamic limit of the CP6 evolving occupied sets in such a CA is the level sets of the Accelerated Inverse Analysis Using Parameterized viscosity solution to a Hamilton-Jacobi equation, a first Model Order Reduction order geometric PDE

One alternative to directly coupling solver and optimizer Minsu Song for PDE constrained optimization problems is to use re- combinatorial and computational mathematics center cently developed parameterized model reduction methods Pohang Univ. of Science and Technology, South Korea to generate a parameterized low-order model and then use [email protected] the model in the optimizer. Such model-reduction based optimization methods have advantages in computational efficiency, particularly when the extracted model is used CP6 repeatedly. We demonstrate that the method is very ef- Numerical Study of Optimal Control for Nonlinear fective for an inverse scattering problem associated with Cahn-Hilliard Equations optical inspection. Cahn-Hilliard (CH) equations are described a continuous Luca Daniel model for phase transition in binary systems such as al- M.I.T. loy, glasses and polymer-mixtures. This work is to investi- Research Lab in Electronics gate numerical study for optimal control of CH equations [email protected] by the means of initial, boundary and distributed control. A semi-discrete algorithm is constructed for minimization Jung Hoon Lee, Dimitry Vasilyev of quadratic optimal criteria using finite element method. Massachusetts Institute of Technology The convergence and error estimate are deduced. Numer- [email protected], [email protected] ical demonstrations illustrated the efficiency of proposed paradigm. Anne Vithayathil Quan-Fang Wang Nanobiosym, Inc. Department of Automation and Computer-Aided 200 Boston Avenue, Suite 4700, Medford, MA 02055 254 PD06 Abstracts

Engineering state. The front stability results show that the regularized The Chinese University of Hong Kong equation (1) mirrors the physics of rarefaction and shock [email protected] waves in the Burgers equation. Razvan Fetecau CP7 Stanford University Nonoscillatory Central Schemes on Unstructured Department of Mathematics Triangular Grids for Hyperbolic Systems of Con- [email protected] servation Laws

An extension to unstructured triangulations of Jiang and CP7 Tadmor’s [SISC 19(6)] scheme will be proposed. Through Solutions to Coupled Systems of Conservation a direction-insensitive reconstruction, the proposed scheme Laws is of the correct order of accuracy in any direction, not just along coordinate axes. Moreover, the use of an adap- We discuss the construction of solutions to coupled sys- tive limiter ensures the sharp resolution of discontinuities tems of nonlinear conservation laws. These equations arise and convergence to the unique viscosity solution. Central- for example in models for traffic flow on road networks. upwind extensions, in the spirit of Kurganov et al [SISC The coupling is then due to the intersection of roads. We 23(3)], will be considered. derive necessary conditions including in particular the conservation of all moments of the hyperbolic equation. We Ivan Christov construct solutions to the coupled system based on consid- Department of Mathematics erations on special Riemann problems posed at the road Texas A intersections. &M University [email protected] Michael Herty Technische Universitaet Kaiserslautern Bojan Popov Fachbereich Mathematik Department of Mathematics [email protected] Texas A&M University [email protected] CP7 Transonic Regular Reflection for the Isentropic Gas Peter Popov Dynamics Equations Institute for Scientific Computation Texas A&M Universtiy We consider a two-dimensional Riemann problem for the [email protected] isentropic gas dynamics equations and derive regimes for which regular reflection can occur. We restrict our study to transonic regular reflection and write the problem in CP7 self-similar coordinates. This change of variables leads to A Hamiltonian Regularization of the Burgers a mixed type system and a free boundary problem for a Equation position of the reflected shock and a subsonic state behind the shock. We present preliminary ideas on analysis of this We consider the following scalar equation: problem using the theory of second order elliptic equations 2 2 and the fixed point theory. ut + uux − α utxx − α uuxxx =0, (1) with α>0. We may rewrite (1) as Suncica Canic Department of Mathematics vt + uvx =0, (2) University of Houston [email protected] where 2 v = u − α uxx, (3) Barbara Lee Keyfitz One can think of the equation (2) as the inviscid Burgers Fields Institute and University of Houston equation, [email protected] vt + vvx =0, where the convective velocity in the nonlinear term is re- Katarina Jegdic placed by a smoother velocity field u. This idea goes back University of Houston to Leray (1934) who employed it in the context of the [email protected] incompressible Navier-Stokes equation. Leray’s program consisted in proving existence of solutions for his modified CP7 equations and then showing that these solutions converge, as α ↓ 0, to solutions of Navier-Stokes. We apply Leray’s High Resolution Total Variation Diminishing ideas in the context of Burgers equation. We show strong Scheme for Linear System of Hyperbolic Conser- analytical and numerical indication that (2)-(3) (or equiv- vation Laws alently, (1)) represent a valid regularization of the Burgers A conservative high-resolution TVD scheme for linear hy- equation. That is, we claim that solutions uα(x, t)of(1) → perbolic systems is constructed using the flux limiter func- converge strongly, as α 0, to unique entropy solutions of tion. In the present work the numerical flux function of the inviscid Burgers equation. Interestingly, for all α>0, high-resolution scheme based on wave speed splitting is the regularized equation possesses a Hamiltonian structure. constructed in such a way that it respects the physical hy- We also study the stability of the traveling waves for equa- perbolicity property. Bounds are given for limiter function (1). These traveling waves consist of ”fronts”, which tions to satisfy TVD property. Numerical results are given are monotonic profiles that connect a left state to a right PD06 Abstracts 255

for both 1-D and 2-D systems, which show high-resolution [email protected] near points of discontinuity without introducing spurious oscillations. CP7 Ritesh Kumar Entropy Solutions for a Triangular System of Con- Indian Institute of Technology, Kanpur servation Laws Modelling Three Phase Flow in [email protected] Porous Media.

Mohan K. Kadalbajoo In this talk, we will consider a 2 × 2 triangular (hence res- Indian Institute of Technology Kanpur onant) system of conservation laws that models ”reduced” Kanpur-208016, India three phase flows in a porous medium where saturation of [email protected] one of the phases is independent of the other phases. A notion of entropy solutions is proposed and the entropy solutions are shown to form a L1 stable semi-group. Existence CP7 of solutions is proved using a vanishing viscosity argument A New Sticky Particle Method for Pressureless Gas and compensated compactness. Numerical schemes of the Dynamics Godunov type are proposed and shown to converge to the entropy solution. I will first present a new sticky particle method for the one- and two-dimensional systems of pressureless gas dy- Kenneth Karlsen, NilsHenrik Risebro namics. The method is based on the idea of sticky par- Centre of mathematics for applications ticles, which seems to work perfectly well for the models Department of Mathematics, University of Oslo with point mass concentrations and strong singularity for- [email protected], [email protected] mations. In this method, the solution is sought in the form of a linear combination of delta-functions, whose positions Siddhartha Mishra and coefficients represent locations, masses and momenta Center of Mathematics for Applications, of the particles, respectively. The locations of the par- University of Oslo, Norway. ticles are then evolved in time according to a system of [email protected] ODEs, obtained from a weak formulation of the system of PDEs. The particle velocities are approximated in a special way using a global conservative piecewise polynomial CP8 reconstruction technique over an auxiliary Cartesian mesh. A Comparison of AB/BDI2 and AB/CN Time This velocities correction procedure leads to a desired in- Stepping Schemes in a Chebychev-Tau Spectral teraction between the particles and hence to clustering of Navier-Stokes Solver particles at the singularities followed by the merger of the clustered particles into a new particle located at their cen- The semi-implicit time stepping schemes: Adams- ter of mass. Bashforth/Backward-Differencing (AB/BDI2) and Adams- Bashforth/Crank-Nicolson (AB/CN) were implemented in In the two-dimensional case, the convergence of the pro- a spectral methods based Navier-Stokes solver. The solver posed sticky particle method is still an open problem. I will was used for obtaining time dependent velocity and tem- show that our particle approximation satisfies the original perature fields for a natural convection in an inclined chan- system of pressureless gas dynamics in a weak sense, but nel problem. Contrary to the previous literature, for this only within a certain error, which is rigorously estimated. application, we have seen only a minor stability improve- I will also give an explanation why the relevant errors are ment when AB/BDI2 was used instead of AB/CN. The expected to diminish as the total number of particles in- improvement was not enough to allow for a time step size creases. reduction but it was enough to decrease initial oscillations on the global solution. The comparative results and the Finally, I will demonstrate the performance of the new comparisons with the previous literature are presented in sticky particle method on a variety of one- and two- terms of dimensionless groups. dimensional numerical examples and compare the obtained results with those computed by a high-resolution finite- Ilker Tari, Serkan Kasapoglu, Aykut Erdem volume scheme. The results obtained by the sticky particle METU method seem to be superior since the evolution of develop- [email protected], [email protected], ing discontinuities is accurately tracked and the developing [email protected] delta-shocks do not get smeared (this is the main advan- tage of using low-dissipative particle methods). CP8 Alina Chertock Diagonalizing Similarity North Carolina State University Transformations for Variable-Coefficient Differen- Department of Mathematics tial Operators [email protected] Let L be a second-order self-adjoint variable-coefficient differential operator on Cp(2π). We show that using symbolic Alexander Kurganov calculus and anti-differentiation operators, we can obtain Tulane University a sequence of unitary similarity transformations that ap- Department of Mathematics proximately diagonalizes L, yielding analytical representa- [email protected] tions of approximate eigenvalues and eigenfunctions. Each ∗ transformation has the form Lk+1 = Uk LkUk, where Uk Yurii Rykov is the exponential of a skew-symmetric pseudodifferential Keldysh Institute of Applied Mathematics operator. We will also discuss non-self-adjoint operators, Russian Academy of Sciences higher-order operators, other boundary conditions, and 256 PD06 Abstracts

higher spatial dimension. Numerical results will be pre- CP8 sented. Monotonicity and Stability of Numerical Solutions for Obstacle Problems Patrick Guidotti University of California at Irvine Monotonicity and L∞-stability theorems are established [email protected] for a discrete obstacle problem which is defined by a piecewise linear finite element discretization of a continuous James V. Lambers problem. These theorems extend the discrete maximum Stanford University principle of Ciarlet from linear equations to obstacle prob- Department of Petroleum Engineering lems. As their applications, we establish a convergence [email protected] theorem for an iterative algorithm to solve discrete obstacle problems and extend Baiocchi’s and Nitsche’s L∞- estimates to a conforming full discrete obstacle problem. CP8 Four Corners: Analytic and Numerical Solution of Todd Dupont Poisson’s Equation with Discontinuous Coefficients University of Chicago [email protected] We present closed-form formulae and an efficient numerical method for Kellogg’s solution of Poisson’s equation with Yongmin Zhang piecewise constant coefficient on each quadrant of the unit k SUNY at Stony Brook square. The solution has a singular behaviour, r ,k < 1, [email protected] r = distance from the junction point. We show how this model is a useful framework to analyze numerical methods for diffusion equations on Adaptive Mesh Refined (AMR), CP9 cell-centered, finite volume grids, common to CFD and oil Subspace Interpolation and Applications to Regu- reservoire simulations. larity for the Biharmonic Problem and the Stokes Systems Oren E. Livne Scientific Computing and Imaging Institute We consider the biharmonic Dirichlet problem on a polygo- University of Utah nal domain. New regularity estimates in terms of Sobolev- [email protected] Besov norms of fractional order are proved. The analysis is based on new subspace interpolation results. We apply our results to establish regularity estimates for the Stokes CP8 and Navier-Stokes systems on polygonal domains. Intergrid Operators for Optimal Convergence of Multigrid Algorithms Constantin Bacuta University of Delaware We study the effect of the interpolation and restriction op- Department of Mathematics erators on the convergence of multigrid algorithms for solv- [email protected] ing linear PDEs. Using a modal analysis of a subclass of these systems, we determine how two groups of the modal James H. Bramble components of the error are filtered and mixed at each step Texas A&M University in the algorithm. We then show how to determine optimal Cornel University tradeoffs between the convergence rate and computational [email protected] complexity for a given system.

Pablo Navarrete, Edward Coyle CP9 Purdue University [email protected], [email protected] Some Nonlocal and Global Existence Results for a Nonlinear Heat Equation with Critical Exponent

CP8 In this study, we consider the nonlinear heat equation New Techniques for the Numerical Solution of El- p ut(x, t)=Δu(x, t)+u (x, t)inΩ× (0,T), liptic and Time Dependent Problems Bu(x, t)=0 on ∂Ω × (0,T), The unified transform method of A. S. Fokas for solving lin- u(x, 0) = u0(x)inΩ, ear and certain monlinear PDEs, has led to important new developments, regarding the analysis of various types of with dirichlet and mixed boundary conditions, where Ω ⊂ PDE problems. This work is based on these developments Rn is a smooth bounded domain and p =1+2/n is the 1 and presents new techniques for the numerical solution of critical exponent. For some initial condition u0 ∈ L ,we elliptic boundary value problems, as well as time depen- prove the nonlocal existence of the solution in L1 for the dent problems. A comparative evaluation with existing mixed boundary condition. We also establish the global well established methods will also be provided. existence in L1+ to the dirichlet problem, for any fixed >0 with u0 1+ sufficiently small. Theodore S. Papatheodorou, Anastasia N. Kandili HPCLAB - Comp. Engin. & Inf. Dept. Canan Celik University of Patras TOBB-ETU [email protected], [email protected] [email protected] CP9 Positive Solutions to Nonlinear p-Laplace Equa- PD06 Abstracts 257

tions with Hardy Potential in Exterior Domains can be consisting of defferent dimensional components and also non-smooth), and similar arguments lead to analogue We study the existence and nonexistence of positive (su- results in this case too. per) solutions to the singular quasilinear p-Laplace equation Vladimir B. Vasilyev μ p−2 C q−1 Department of Mathematics and Information −Δpu − |u| u = |u| u |x|p |x|σ Technologies Bryansk State University in exterior domains of RN (N ≥ 2). Here p ∈ (1, +∞) [email protected] and μ ≤ CH , where CH is the critical Hardy constant. We provide a sharp characterization of the set of (q, σ) ∈ R2 such that the equation has no positive (super) solutions. CP10 Fast Explicit Operator Splitting Method for The proofs are based on the explicit construction of ap- Convection-Diffusion Equations propriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the p– Systems of convection-diffusion equations model a variety Laplace operator with Hardy-type potentials, comparison of physical phenomena which often occur in real life. Com- principles and an improved version of Hardy’s inequality puting the solutions of these systems, especially in the in exterior domains. convection dominated case, is an important and challenging problem that requires development of fast, reliable and Sofya Lyakhova, Vitali Liskevich, Vitaly Moroz accurate numerical methods. We propose a second-order School of Mathematics fast explicit operator splitting (FEOS) method based on University of Bristol, UK the Strang splitting. The main idea of the method is to [email protected], [email protected], solve the parabolic problem via a discretization of the for- [email protected] mula for the exact solution of the heat equation, which is realized using a conservative and accurate quadrature formula. The hyperbolic problem is solved by a second- CP9 order finite-volume Godunov-type scheme. We provide a Lewy-Hörmander Nonexistence and Pseudospectra theoretical estimate for the convergence rate in the case of one-dimensional systems of linear convection-diffusion In 1957 Lewy showed that linear PDE with smooth equations with smooth initial data. Numerical convergence coefficients can have no solutions, and soon thereafter studies are performed for one-dimensional nonlinear prob- Hörmander produced a general theory of such problems lems as well as for linear convection-diffusion equations which was subsequenty generalized by Nirenberg, Treves, with both smooth and nonsmooth initial data. We finally and others. In 2001 Zworski pointed out that the con- apply the FEOS method to the one- and two-dimensional structions involved in this theory can be intrerpreted in systems of convection-diffusion equations which model the terms of pseudospectra. We will describe the connection polymer flooding process in enhanced oil recovery. Our re- between Lewy-Hörmander nonexistence and pseudospec- sults show that the FEOS method is capable to achieve a tra, and specifically, what are known as wave packet pseu- remarkable resolution and accuracy in a very efficient man- domodes of variable-coefficient differential operators. ner, that is, when only few splitting steps are performed. Lloyd N. Trefethen Guergana Petrova Oxford University Texas A&M University Oxford University Compting Lab [email protected] [email protected] Alina Chertock North Carolina State University CP9 Department of Mathematics Entropy Solutions for the p(x)-Laplace Equation [email protected] We consider a Dirichlet problem in divergence form with variable growth, modeled on the p(x)-Laplace equation. Alexander Kurganov We obtain existence and uniqueness of an entropy solution. Tulane University The proof is based on a priori estimates in Marcinkiewicz Department of Mathematics spaces with variable exponent. Joint work with Manel [email protected] Sanchón (CMUC, Portugal). José Miguel Urbano CP10 Universidade de Coimbra Global Existence and Uniqueness for a Model of the [email protected] Flow of a Barotropic Fluid with Capillary Effects We study the initial-value problem for a system of non- CP9 linear equations that models the flow of an inviscid, Some Distributions Related to Boundary Value barotropic fluid with capillary stress effects. The system Problems in Domains with Thin Non-Smooth Sin- includes a hyperbolic equation for the velocity, and an al- gularities gebraic equation (the equation of state). We prove the global-in-time existence of a unique, classical solution to 2 The concept of wave factorization for an elliptic symbol the system of equations. The key to the proof is a new L was introduced by the author. It is permitted to describe estimate for the density ρ. the structure of solutions (or solvability conditions) for a model pseudodifferential equation in canonical non-smooth Diane Denny domains of cone or wedge type. But canonical singularities Texas A&M University - Corpus Christi 258 PD06 Abstracts

[email protected] CP11 The Stability of Couette Flow in a Toroidal Mag- netic Field* CP10 Nondegeneracy, from the Prospect of Wave-Wave The stability of the hydromagnetic Couette flow is investi- Regular Interactions of a Gasdynamic Type gated when a constant current is applied along the axis of the cylinders. It is shown that if the resulting toroidal mag- A parallel is constructed between Burnat’s ”algebraic” ap- netic field depends only on this current, no linear instability proach [which uses a dual connection between the hodo- to axisymmetric disturbances is possible. *This material is graph and physical characteristic details] and Martin’s ”dif- based upon work supported in part by the U. S. Depart- ferential” approach [centered on a Monge-Ampère type ment of Energy under Grant No. DE-FG02-05ER25666 (to representation] regarding their contribution to describing I.H.) some nondegenerate gasdynamic [one-dimensional, multidimensional] regular interaction solutions. • Some specific Isom Herron, Fritzner Soliman aspects of the multidimensional ”algebraic” description are Rensselaer Polytechnic Institute beforehand identified and classified with an admissibility [email protected], [email protected] criterion − selecting a ”genuinely nonlinear” type where other (”hybrid”) types are formally possible. CP11 Marina Ileana Dinu Asymptotic Behavior Near Transition Fronts for Polytechnical University of Bucharest, Dept. of Math. III Equations of Cahn-Hilliard Type ROMANIA [email protected] We consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary PDE of Liviu Florin Dinu Cahn–Hilliard type in one space dimension. Under the as- Institute of Mathematics of the Romanian Academy sumption of spectral stability, described in terms of an ap- [email protected], [email protected] propriate Evans function, we develop detailed asymptotics for perturbations from standing wave solutions, establishing phase-asymptotic orbital stability for initial perturba- CP10 tions decaying with appropriate algebraic rate. Interconnection Between Quasilinear Differential Equations and Infinite Systems of Linear Pdes Peter Howard Department of Mathematics We establish an amusing connection between infinite linear Texas A&M University systems of PDEs and quasilinear differential equations, in- [email protected] cluding Hopf, Burger’s, Korteweg-de-Vriez equations, and other conservation laws. Such a connection enables to prove existence theorems and evaluate properties of solu- CP11 tions for infinite systems. Also, it provides the alterna- Decay of Solutions to Some Nonlinear Wave Equa- tive numerical method for solving quasilinear equations, tions with Dissipation based on numerical investigation of truncated linear systems. The uniqueness of solutions can be obtained in spe- We study asymptotic behavior of solutions to the Cauchy cial functional spaces generated by given infinite system. problem of some nonlinear wave equations of the form Pavel B. Dubovski Stevens Institute of Technology ut + P (u)x + K(u)=0, (∗) [email protected]

where P and u are real-valued functions, K is de- CP10 fined by Kv (ξ)=k(ξ)ˆv(ξ) and the circumflexes de- On the Reduced Description of Reactive Euler note Fourier transforms. The Heat equation, the gen- Equations of Detonation Theory eralized Burgers equations, the generalized Korteweg-de Vries-Burgers equations, the generalized regularized long We describe a method for reducing reactive Euler equations wave-Burgers equations, and the generalized Benjamin- governing the dynamics of one-dimensional detonations to Ono-Burgers equations are particular forms of equation a nonlinear ordinary differential equation for the lead shock (*). The asymptotical representations for large time are speed. The reduced equation contains a single term that studied in detail. needs to be approximated which depends on spatial derivatives of pressure or particle velocity behind the lead shock. Jerry Bona We discuss the properties of the reduced equation in con- University of Illinois at Chicago nection with pulsating instability of one-dimensional deto- [email protected] nation and also show how the equation can be incorporated into direct numerical simulation of the reactive Euler equa- Thanasis Fokas tions. Department of Applied Mathematics and Theoretical Aslan R. Kasimov Physics Department of Mathematics University of Cambridge, UK Massachusetts Institute of Technology [email protected] [email protected] Laihan Luo Richard Stockton College of New Jersey [email protected] PD06 Abstracts 259

CP11 in velocities are obtained by duality from bounds for the Stability and Attractors for Quasi-Steady Model of corresponding adjoint system. Cellular Flames David Hoff We discuss the recently introduced model of quasi-steady Indiana University development of cellular flames. The Quasi-Steady equa- hoff@indiana.edu tion (QSE) demonstrates the same basic instability mech- anism as its better-known counterparts, the Kuramoto- Sivashinsky equation (KSE), and the Burgers-Sivashinsky MS1 (BS) equation (the latter being just a linearly forced Burg- An Equation for the Formation of Step Bunches in ers equation). Namely, a linear stability analysis reveals the Morphological Evolution of Nano-scale Crystal the long-wave destabilization, which is suppressed, for the Surface Structures Below the Roughening Transi- small wavelengths, by the dominant dissipative principal tion term. In a sense, QSE is intermediate between BSE and KSE, as its dispersion relation coincides with that for BSE Lagrangian coordinates continuum equations for surface for short waves, and is identical to that of KSE (up to the evolution are obtained from a nano-scale description (step fourth order in k) for long waves. Note that, while BS interaction equations). The surface consists of interacting has trivial dynamics, similarly to KSE, QSE demonstrate steps, moving by adatom diffusion on terraces; and attach- very rich dynamical behavior. We demonstrate that QSE ment and detachment at steps. Diffusion-limited (DL) and possesses a universal absorbing set, and a compact attrac- attachment-detachment limited (ADL) kinetics, are con- tor of finite Hausdorff dimension, and give an estimate on sidered, for axially symmetric surfaces. The ADL equa- it. This is a joint work with Claude-Michel Brauner and tions capture step bunching. The bunches appear from Josephus Hulshof. the interaction between a (destabilizing) negative diffusion (step-line tension effects) and a stabilizing nonlinear diffu- Michael Frankel sion (step-step interactions). The local bunch-dynamics is Department of Mathematics described by a simple nonlinear equation combining these IUPUI effects. [email protected] Rodolfo R. Rosales Massachusetts Inst of Tech Victor Roytburd Department of Mathematics Department of Mathematical Sciences [email protected] RPI [email protected] Dionisios Margetis Department of Mathematics CP11 Massachusetts Institute of Technology Asymptotic Behavior of FitzHugh-Nagumo Equa- [email protected] tions

We study the asymptotic behavior of the FitzHugh- MS1 Nagumo system on unbounded domains. It is shown that On Phase Transition Dynamics the system has a compact global attractor which contains traveling wave solutions. We also investigate the limiting A multidimensional model is introduced for the dynamics behavior of the attractors as a parameter → 0. It is of a binary mixture of compressible fluids, which undergo proved that the limiting system has no global attractor, possible phase transition due to chemical reaction. The but all attractors for the perturbed system with positive model presented here can accommondate various physical but small are contained in a common compact set. contexts, namely liquid- liquid, gas-liquid, gas-gas phase equilibria, the evolution of gaseous stars, the dynamics of Bixiang Wang semiconductors and others. The model is formulated by Department of Mathematics, New Mexico Mining and the Navier Stokes equations in Euler coordinates, which Technology is now expressed by the conservation of mass, the balance [email protected] of momentum and entropy and the species conservation equation. This system takes now a new form due to the choice of rather complex constitutive relations, which are MS1 able to accommondate the binary character of the mixture. Uniqueness of Weak Solutions of the Navier-Stokes The existence of globally defined weak solution is obtained Equations of Multidimensional, Compressible Flow by the use of weak convergence methods and compactness arguments in the spirit of Feireisl and P.L. Lions. We prove uniqueness and continuous dependence on initial data of weak solutions of the Navier-Stokes equations Konstantina Trivisa of compressible flow in two and three space dimensions. University of Maryland The solutions we consider may display codimension-one Department of Mathematics discontinuities in density, pressure, and velocity gradient, [email protected] and consequently are the generic singular solutions of this system. The key point of the analysis is that solutions with minimal regularity are best compared in a Lagrangean MS1 framework; that is, we compare the instantaneous states On Compressible Navier-Stokes Equation of corresponding fluid particles in two different solutions rather than the states of different fluid particles instan- We consider the isentropic compressible Navier-Stokes taneously occupying the same point of space-time. Esti- equation with density dependent viscosity coefficients. We mates for H−1 differences in densities and L2 differences focus our study in the multivariable case when those coef- 260 PD06 Abstracts

ficients vanish on vacuum. In particular we prove the sta- viscoelastic, and anisotropic models will be discussed. Im- bility of weak solutions of this problem using new kind of ages with laboratory data will be presented. entropy inequalities which involve gradients of the density. This leads to new regularity on the density which, surpris- Kui Lin, Jens Klein ingly, holds only for degenerated viscosity coefficients. Rensselaer Polytechnic Institute [email protected], [email protected] Alexis Vasseur University of Texas at Austin Jeong-Rock Yoon [email protected] Department of Mathematical Sciences Clemson University Antoine Mellet [email protected] Department of Mathematics University of Texas-Austin Daniel Renzi [email protected] Department of Mathematical Sciences Rensselaer Polytechnic Institute [email protected] MS2 Controllability of a Coupled System that Models Joyce R. McLaughlin Cochlear Dynamics Rensselaer Polytechnic Inst The standard 2-dim cochlea model consists of a one-dim Dept of Mathematical Sciences elastic structure (modeling the basilar membrane) sur- [email protected] rounded by an incompressible 2-dim fluid within a 2-dim cochlear cavity. The dynamics are typically driven by a MS3 pressure differential across the basilar membrane transmit- ted through the round and oval windows (a portion of the Propagation of Fronts in Porous Media Combus- boundary of the cochlea). First we describe a highly ide- tion alized model in which the basilar membrane is modeled as Gaseous detonation is a phenomenon with a very compli- an infinite array of oscillators and the fluid is described cated dynamics. The mathematical study of the qualitative by Laplace’s equation. In this idealized setting we show behavior of solutions of the full problem is out of reach at that the coupled system is approximately controllable with the moment. Explosions in inert porous media provide a control acting on an arbitrary open set of the basilar mem- model that is realistic, rich and suitable for a mathemat- brane. If the basilar membrane has longitudinal membrane ical analysis. We will present some recent mathematical (string) elasticity, then exact controllability can be proved. results concerning the long time behavior of solutions of Scott Hansen this model. In particular, initiation of detonation, forma- Iowa State University tion of the detonation front, its stability and propagation Department of Mathematics limits will be discussed. [email protected] Peter Gordon NJIT Isaac Chepkwony [email protected] Iowa State University [email protected] MS3 A Variational Approach to Front Propagation in MS2 Infinite Cylinders Uniform Stability of a Nonlinear Plate-Beam Model Gradient reaction-diffusion-advection systems arise in the context of modeling the kinetics of phase transitions, popu- The question of uniform stability is considered for a model lation dynamics and combustion. These systems are known comprised of a nonlinear vonKarman plate coupled with to exhibit a variety of non-trivial spatio-temporal behav- a nonlinear beam equation. Stability properties of linked iors, most notably the phenomenon of propagation and structures composed of multiple elastic elements give rise to traveling waves. We introduce a variational formulation for an abundance of mathematical challenges. When a struc- the traveling wave solutions in cylindrical geometries with ture is composed of interconnected elastic elements of dif- transverse potential flow, which allows us to construct a ferent dimension, the behavior becomes much harder to certain class of traveling wave solutions and establish their both predict and to control. (Joint work with Guenter monotonicity, asymptotic decay and uniqueness. These so- Leugering.) lutions are special in a sense that they are characterized by a non-generic fast exponential decay ahead of the wave Mary Ann Horn and play an important role in propagation phenomena for Vanderbilt University the initial value problem. We also construct an area-type [email protected] functional that gives a matching upper and lower bound for the propagation speed in the sharp reaction zone limit MS2 for weakly curved fronts. Tissue Modeling and its Influence on Shear Stiff- Cyrill B. Muratov ness Imaging New Jersey Institute of Technology Department of Mathematical Sciences We discuss tissue properties and their importance in build- [email protected] ing an elastic model that will yield accurate algorithms and shear stiffness images. Experimental design, elastic, Matteo Novaga PD06 Abstracts 261

Department of Mathematics MS4 University of Pisa Transport in Molecular Motor Systems [email protected] The molecular motor systems which govern much of the transport in eukaryotes may be thought of as governed by MS3 a dissipation principle involving conformational changes in- Diffusion and Mixing in Fluid Flow duced by chemical reactions and potentials. The result in the simplest cases is a weakly coupled system of evolution We study enhancement of diffusive mixing on a compact equations. To the mind’s eye, the transport process is anal- Riemannian manifold M by a fast incompressible flow. ogous to a biased coin toss. We describe how this intuition We provide a sharp characterization of the class of flows may be confirmed by a careful analysis of the cooperative that make the deviation of the solution of an equation effect among the conformational changes and the poten- φt + Au ·∇φ =Δφ from its average arbitrarily small in tials. This is joint work with Stuart Hastings and Bryce an arbitrarily short time, provided that the flow amplitude McLeod. A is large enough. The necessary and sufficient condition on such flows is that the dynamical system associated with David Kinderlehrer the flow has no eigenfunctions in the Sobolev space H1(M). Carnegie Mellon University In particular, we find that weakly mixing flows always en- [email protected] hance dissipation in this sense. Leonid Ryzhik MS4 University of Chicago Surfactants in Foam Stability: A Phase Field Department of Mathematics Model [email protected] The role of surfactants in stabilizing the formation of bub- P. Constantin bles in foams is studied using a phase-field model. The Department of Mathematics model involves a van der Walls-Cahn-Hilliard-type energy University of Chicago with an added term accounting for the interplay between [email protected] the presence of a surfactant density and the creation of interfaces. In agreement with experimentation, it is proved A. Kiselev, A. Zlatos that the surfactant segregates to the interfaces. This is Department of Mathematics work in collaboration with Massimiliano Morini and Va- University of Wisconsin leriy Slastikov. [email protected], [email protected] Irene Fonseca Carnegie Mellon University MS3 Center for Nonlinear Analysis [email protected] Spreading of Combustion in the Presence of Strong Cellular Flows with Gaps Massimiliano Morini Recently Fannjiang, Kiselev and Ryzhik showed that cel- SISSA lular flows are capable of quenching large flames, provided Trieste,Italy the amplitude of the flow is also large. We show that this [email protected] phenomenon does not occur when flow cells are separated by narrow buffer zones where advection is absent. MS4 Andrej Zlatos Critical Points of Onsager Model of Nematic Phase Department of Mathematics Transition University of Wisconsin [email protected] We study Onsagers model of isotropic-nematic phase transitions with orientation parameter on a sphere. We consider symmetric Maier-Saupe potential and prove the ax- MS4 ial symmetry and derive explicit formulae for all critical The Impact of a Van Der Waals Type Interfacial points, thus obtaining their complete classification. Energy on Dimensional Reduction Valeriy Slastikov The asymptotic behavior of an elastic thin film penalized University of Warwick, Warwick, UK by a van der Wals type interfacial energy is investigated [email protected] when both its thickness and the magnitude of the additional energy vanish in the limit. Keeping track of both mid-plane and out of plane deformations (through the in- MS4 troduction of the Cosserat vector), the resulting behavior Limit of the Non Self-dual Chern-Simons-Higgs En- strongly depends upon the ratio between thickness and in- ergy terfacial energy. Non-locality is conjectured for a critical value of that ratio. We present some results on the gamma limit of the Chern- Simons-Higgs energy functional, far from self duality. We Gilles Francfort use a mixture of methods used in the study of related L.P.M.T.M., Université Paris-Nord, Villetaneuse, France Ginzburg-Landau energies. [email protected] Daniel Spirn University of Minnesota [email protected] 262 PD06 Abstracts

MS5 (1) either u is continuous and its modulus of continuity Wiener’s Criterion for Uniqueness of Solutions to can be given in a quantitative way, the Elliptic and Parabolic Problems in Domains β(s) with Noncompact Boundaries (2) or in cylinders Q of length s we have that In this talk I will introduce a notion of regularity of infinity β (s)s |{(x, t) ∈ Q such that u(x, t) ≤ s}| −1 ≤ C1 ≤ C2(log s) for the elliptic and parabolic equations and will formulate β(s) |Q| a necessary and sufficient condition for the existence of a unique bounded solution to the classical and parabolic This latter inequality implies the regularity of the solution Dirichlet problems in arbitrary open sets with noncompact for a large class of functions β and therefore we are able to boundaries. The criterion is the exact analogue of Wieners extend DeGiorgi’s theory to suitable non power-like singu- test for boundary regularity of harmonic and thermic func- larities. tions and demonstrates the thinness of the exterior of a Vincenzo Vespri domain at infinity. Department of Mathematics Ugur G. Abdulla University of Florence Department of Mathematical Sciences [email protected]fi.it Florida Institute of Technology abdulla@fit.edu Ugo P. Gianazza Department of Mathematics University of Pavia MS5 [email protected] Harnack Estimates for Non-negative Solutions of Quasi-linear Degenerate Parabolic Equations with Measurable Coefficients MS5 On a Boundary Harnack Inequality for p-Harmonic The classical Harnack estimates for non-negative solutions Functions of the heat equations do not hold for solutions of degener- Let D ⊂ Rn be a bounded Lipschitz domain, x ∈ ∂D, and ate equations (counterexample will be given). The estimate n holds however in an ”intrinsic form”. The novelty of the B(x, r)={y ∈ R : |y − x|

We consider local bounded solutions of Juan J. Manfredi n Department of Mathematics β(u)t = Di(aij (x, t)Dj (u)) University of Pittsburgh i,j=1 [email protected] where the coeﬃcients aij are uniformly elliptic, bounded and measurable functions and β is a C1-piecewise func- MS6 tion with at least a linear growth. If s0 is an irregular A Kinetic Transport and Reaction Model for Pro- point for β, we assume β to be locally concave in a proper cess Models in Microelectronics Manufacturing right neighborhood of s0 and locally convex in a proper left neighborhood of the same point. By using suitable De- The manufacturing of microelectronics devices involve gas Giorgi’s techniques, we have the following alternative: ﬂow at various total pressures, characterized by the Knud- sen number as relevant dimensionless group. We introduce PD06 Abstracts 263

a kinetic transport and reaction model and simulator based Li-Zhi Fang on a system of time-dependent linear Boltzmann equations Department of Physics that is valid over a wide range of Knudsen numbers. We University of Arizona present results that demonstrate the ability of the model to [email protected] analyze the behavior of the system and to provide unique insights into non-equilibrium effects. Jingmei Qiu Brown University Timothy Cale Division of Applied Mathematics Rensselaer Polytechnic Institute [email protected] Focus Center: Interconnections for Gigascale Integration [email protected] Chi-Wang Shu Division of Applied Mathematics Matthias K. Gobbert Brown University University of Maryland, Baltimore County [email protected] Department of Mathematics and Statistics [email protected] MS6 MS6 Deterministic Solver for Non-linear Boltzmann Dissipative Equations Discontinuous Galerkin Approximation of the Boltzmann-Poisson System We present a kinetic solver for Boltzmann energy dissipative equations based on spectral methods. We compute ex- We analyze the problem of numerically solving linear vari- ample of states associated to homogeneous cooling for gran- ants of the space inhomogeneous multidimensional linear ular flows and dynamically scaled solutions to elastic gases Boltzmann equation for charged transport. Determinis- models in the presence of a thermostat. These models have tic methods can be very accurate for problems in which stationary or self-similar states that correspond to non- the solution may or may not be far from thermodynamical equilibrium statistical states (NESS). In particular we show equilibrium and high accuracy is required. Additionally, that our calculations resolve long time dynamics to prob- they yield specific details of transient solutions and con- ability distribution functions with finite or infinity energy, sequently can be more efficient than the traditional prob- unbounded at the origin and power tails, which are self- abilistic Monte Carlo techniques for the computation of similar solutions to energy dissipative Maxwell Molecules transients. We employ a discontinuous Galerkin finite ele- models. ment method to preserve local conservation properties and exploit the advantages of these methods for the compu- Sri Harsha Tharkabhushana tations of Boltzmann-Poisson problems with large doping University of Texas, Austin profiles, such as in the modeling of semiconductor devices [email protected] or collisional plasma. Theoretical and numerical results are discussed. Irene M. Gamba Jennifer Proft Department of Mathematics and ICES The University of Texas at Austin University of Texas [email protected] [email protected]

MS6 MS7 A WENO Algorithm for the Radiative Transfer Extracting s-wave Scattering Lengths from BEC and Ionized Sphere at Reionization Dynamics

In this presentation, we will show that the algorithm based See MS1 in under annual meeting (AN06 abstracts.) on the weighted essentially nonoscillatory (WENO) scheme David S. Hall with anti-diffusive flux corrections can be used as a solver Department of Physics of the radiative transfer equations. This algorithm is highly Amherst College stable and robust for solving problems with both disconti- [email protected] nuities and smooth solution structures. We tested this code with the ionized sphere around point sources. It shows that the WENO scheme can reveal the discontinuity of the ra- MS7 diative or ionizing fronts as well as the evolution of photon Multi-Component Vortex Solutions in Coupled frequency spectrum with high accuracy on coarse meshes NLS Equations and for a very wide parameter space. This method would be useful to study the details of the ionized patch given by See MS1 in under annual meeting (AN06 abstracts.) individual source in the epoch of reionization. We demonstrate this method by calculating the evolution of theion- Dmitry Pelinovsky ized sphere around point sources in physical and frequency McMaster University, Canada spaces. It shows that the profile of the fraction of neutral [email protected] hydrogen and the ionized radius are sensitively dependent on the intensity of the source. MS7 Long-Long Feng Bose-Einstein Condensates in Optical Lattices and Purple Mountain Observatory China [email protected] 264 PD06 Abstracts

Superlattices MS8 A Mixed Type Problem for Two-dimensional Wave See MS1 in under annual meeting (AN06 abstracts.) Interactions

Mason A. Porter The wave interactions in two-dimensional Riemann prob- California Institute of Technology lems of gas dynamics will be discussed. A degenerate Gour- Department of Physics sat problem will be studied. The solution in the hyperbolic [email protected] region up to the sonic curve is obtained. Dehua Wang MS7 University of Pittsburgh Derivation of the Gross-Pitaevskii Equation for Ro- Department of Mathematics tating Bose Gases [email protected] See MS1 in under annual meeting (AN06 abstracts.) MS8 Robert Seiringer Department of Mathematics Periodic Solutions of Euler’s Equations Princeton University We consider the interactions of shocks in the p-system. We [email protected] obtain bounds for incident shocks of arbitrary strength, arbitrarily close to the vacuum. We obtain bounds for the MS8 states themselves, and for the wave strengths. We will discuss these in the context of global solutions with BV Variational Systems of Wave Equations initial data. We consider systems of wave equations that are governed Robin Young by variational principles whose Lagrangians are quadratic Department of Mathematics functions of the derivatives of the wave-field with coeffi- University of Massachusetts Amherst cients depending on the wave-field itself. We introduce [email protected] notions of genuine nonlinearity and linear degeneracy for such equations that are analogous to, but different from, the corresponding notions for hyperbolic systems of conser- MS9 vation laws, and we derive a new weakly nonlinear asymp- Homogenization of Strongly Local Dirichlet Forms totic equation for waves that lose genuine nonlinearity in a in Perforated Media system. We use this equation it to study the propagation of ‘twist’ waves in a massive director field. We investigate the homogenization for strongly local Dirichlet forms in perforated media giving an extension of Giuseppe Ali the results known for Laplace operator. The main difficul- Consiglio Nazionale delle Ricerche, Napoli, Italy ties arrive from the absence of a good notion of periodicity [email protected] and of extension properties for ”regular” domains. John Hunter Marco Biroli Department of Mathematics Dept. Mathematics UC Davis Polytechnic Institute Milano [email protected] [email protected]

Nicoletta Tchou MS8 1 University of Rennes 1 The L Well-Posedness for Steady Supersonic Euler [email protected] Flow Past a Curved Wedge by Wave Front Tracking Method

1 MS9 We study the L well-posedness for two-dimensional steady Undulations and Fluctuations supersonic Euler flows past a Lipschitz wedge. The wedge perturbs the flow and the waves reflect after interacting The issue of computing the flux of a fluctuating vector with the strong shock front or the boundary. After study- field across an undulating surface is commonly approached ing the Lyapunov functional between two solutions and by exploring the limits within which the formula estab- dealing with the boundary difficulty, we prove that the lished under blanket smoothness hypotheses is still valid, functional decreases in the flow direction. Therefore, the 1 the general wisdom being that the wigglier the surface, the L stability is established, so is the uniqueness of the solu- smoother the field has to be (and vice versa). I argue here tions obtained by the wave front tracking method. that the dim no man’s land lying just beyond these limits is well worth being penetrated, even though cautiously. Tianhong Li Stanford University Antonio DiCarlo [email protected] Dept. of Studies on Structures University “Roma Tre” [email protected] Gui-Qiang Chen Northwestern University USA MS9 [email protected] A Constructive Approach to Some Fractal Trans- PD06 Abstracts 265

mission Problems MS10 The Stability of Fronts in Cahn–Hilliard Type We describe a constructive approach to certain fractal Equations transmission problems for second order elliptic or parabolic operators. The transmission condition on the layer is of The stability of stationary fronts in Cahn–Hilliard type second order. The fractal layer is obtained in the limit of equations is important to the understanding of the evolu- polyhedral (pre-fractal) surfaces, by self-similar iteration tion of fronts in the sharp interface limit. In this talk we at small scales. From an analytic point of view, the main discuss a method for establishing the asymptotic stability problem to be studied is the convergence of the solutions that is based on a detailed analysis of dissipation at the obtained with the pre-fractal layers when the geometry be- fronts. This leads to a system of non-linear inequalities comes fractal. The main difficulty in this study is the jump which may be used to prove convergence. The method is of the geometric dimension of the layer, which is two for all fairly robust, and may be used to derive conditions for sta- pre-fractal approximations and intermediate between two bility in a range of setting. The work presented here is and three for the limit layer. In other words, while the joint work with M. Carvalho and E. Orlandi approximating layers have finite two-dimensional area, the limit layer is non rectifiable with infinite area. We first Maria Carvalho consider the elliptic case. We will describe the variational University of Lisbon and Georgia Tech approach consisting in proving the convergence of suitable [email protected] energy forms in the sense of homogeneization and we will also describe some extensions to the heat equation, by rely- Enza Orlandi ing on the convergence of the related semigroups. In order University of Rome III to obtain these convergence properties, we have to intro- [email protected] duce suitable renormalization factors. Maria Rosaria Lancia Eric A. Carlen Dept. Applied Mathematics Georgia Tech University of Roma La Sapienza [email protected] [email protected] MS10 MS9 Thin-film Equations: Blow-up, the Cauchy Prob- On Some Second Order Transmission Problems lem, Solutions and Countable Spectra of Asymp- totic Patterns We describe some transmission problems across fractal lay- × ers imbedded in Euclidean domains. There is a wide lit- We consider the fourth-order thin film equation in R erature dealing with transmission problems of this kind, (0,T) with an unstable second-order term which arise naturally in various fields: e.g., in electrostat- ics, magnetostatics, hydraulic fracturing and in the study of absorption or irrigation properties. In many of these ap- n p−1 3 ut = −(|u| ux)xxx − (|u| u)xx,n∈ [0, ),p>1. plications, one is interested in considering layers that pos- 2 sess much greater conductivity or permeability than the surrounding space. By referring, for example, to heat con- duction, the heat flow is then absorbed by the layer and We show that at the critical exponent p = p0 = n+3, there starts diffusing within it much more efficiently. This leads exists a countable spectrum of blow-up patterns, where the to a jump of the normal derivatives across the two sides first one is expected to be stable. We study both non- of the layer and this jump acts as a source term for the negative solutions of the free-boundary zero contact an- diffusion within the layer. In our talk we shall describe gle problem and oscillatory sign changing solutions of the some model examples of second order elliptic transmission Cauchy problem. Some other aspects concerning the cor- problems with highly conductive layers. The transmission rect functional setting of the Cauchy problem are discussed condition is also of second order, what is unusual in the case where the limit n → 0 plays a key role. We also investigate of second order operators. Moreover, the condition has an the uniqueness question for the free-boundary problem. implicit character, since the source term of the layer equation - the jump of the normal derivatives - is not among Viktor Galaktionov the data of the problem, but depends on the solution itself Dept. of Math. Sciences in the surrounding space. The layers we consider - say, in University of Bath a domain of R3 - have an additional unusual characteris- [email protected] tic: they are of fractal type, with a (Hausdorff) dimension intermediate between 2 and 3. Layers of this type can be expected to enhance the layer effects described above, by MS10 absorbing more energy from the surrounding space. We Blow-up for Quasilinear Parabolic Equations with present some regularity and numerical results, in the aim Thin-Film-Diffusion Like Operators of better understanding the analytical problems which arise when fractal and Euclidean structures mutually interact. We consider higher-order thin film equations with lower- order terms of reaction type. Using an extension of the Maria Agostina Vivaldi nonlinear capacity method, we calculate the critical Fujita Dept. Applied Mathematics exponents establishing precise parameter intervals where University of Roma La Sapienza no global nontrivial solutions are available. [email protected] Stanislav Pohozhaev Steklov Institute of Mathematics RAS, Moscow [email protected] 266 PD06 Abstracts

MS10 estimates for the transition region separating phases. In The Degenerate Cahn-Hilliard Equation and Wet- particular we derive analytic estimates proving that chiral- ting ity lowers the transition temperature regime above which minimizers are nematic and below which minimizers are in We consider a smectic phase. n β ut +(u (uxxx + u ux))x =0,x∈ (0,L),t>0, Sookyung Joo n ux = u uxxx =0,x=0,L,t>0, Institute for Mathematics and its Applications, UMN [email protected] which generalizes the (one-sided) deep quench Cahn- Hilliard equation and constitutes also a thin film equation in the unstable regime, when β<2 and n is suitably MS11 restricted. Using energy estimates corresponding to the Steady Poiseuille Flows of Nematic Liquid Crystals equation under consideration as well as thin film entropy at Large arguments, we prove global existence of non-negative entropy solutions. Using a generalised Stampacchia lemma, We consider a system of nonlinear second order ordinary estimates on the rate of propagation of the degeneracy set differential equations modeling Poiseuille flow of liquid have also been obtained crystals with variable degree of orientation, at the limit of large Ericksen number. The system is singularly per- Andrey Shishkov turbed and degenerate, and as a result the solutions are Institute of Applied Math. and Mechanics of NAS of highly oscillatory. We obtain the relations satisfied by the Ukraine, Young measures generated by sequences of weak solutions, Donetsk, Ukraine and show that the persistent oscillations are encoded in [email protected] the Young measure generated by the molecular alignment variable φ. The effective equations consist of the isotropic Amy Novick-Cohen Newtonian flow equation, supplemented by algebraic mo- Department of Mathematics mentum relations for the Young measure generated by φ. Technion, Haifa, Israel The latter relations impose restrictions on admissible mi- [email protected] crostructures in the effective flow. We conclude that the configuration of the flow is effectively isotropic with an ordering remnant. MS11 Telephone-cord Instabilities in Thin Smectic Cap- Alexander Panchenko illaries Washington State University [email protected] Telephone-cord patterns have been recently observed in smectic liquid crystal capillaries. In this talk we analyse the effects that may induce them. As long as the capil- MS11 lary keeps its linear shape, we show that a nonzero chiral Molecular Dynamics Models of Liquid Crystal cholesteric pitch favors the SmA*-SmC* transition. How- Elastomers ever, neither the cholesteric pitch nor the presence of an intrinsic bending stress are able to give rise to a curved Liquid crystal elastomers are a fascinating class of mate- capillary shape. The key ingredient for the telephone-cord rials that combine the elasticity of rubber with the ne- instability is spontaneous polarization. The free energy matic or smectic order of liquid crystals. These materials minimizer of a spontaneously polarized SmA* is attained have been called “artificial muscles” because they rapidly on a planar capillary, characterized by a nonzero curva- change shape under heating/cooling, applied fields, or op- ture. More interestingly, in the SmC* phase the combined tical illumination, with induced strains as high as 400%. effect of the molecular tilt and the spontaneous polarization We have developed a finite element elastodynamics tech- pushes towards a helicoidal capillary shape, with nonzero nique to model shape change in these materials, with ex- curvature and torsion. plicit coupling between liquid crystalline order and elastic strain. We model several geometries, including a nematic Paolo Biscari elastomer artificial “earthworm” that moves via expan- Dipartimento di Matematica sion/contraction combined with anisotropic friction. Once Politecnico di Milano validated, such models will serve as a bridge between fun- [email protected] damental soft condensed matter theory and practical engineering design. MS11 Robin Selinger Phase Transition from Chiral Nematic Toward Liquid Crystal Institute, Kent State University Smectic Liquid Crystals [email protected] The Chen-Lubensky energy is used to investigate phase transitions from chiral nematic to smectic C* and smectic MS12 A* liquid crystal phases. We consider a liquid crystalline Homogenization of Free Boundary Velocity material confined between two parallel plates, where the dimensions of the material are assumed to be large rela- We study the homogenization limit of some (Hele- tive both to the width of a smectic layer and the mate- Shaw/Stefan type) free boundary problems with oscillating rial’s chiral pitch. We take boundary conditions so that free boundary velocity. We first prove the existence of the the smectic phase melts at the plates’ surfaces and prove limiting free boundary velocity, which yields the uniform the existence of energy minimizers. Then under the phys- convergence of the free boundaries in the homogenization ically observed assumption that the Frank elasticity constants become large near a phase transition, we establish PD06 Abstracts 267

limit. the numerically observed pulsating waves. Inwon Kim Regis Monneau Department of Mathematics CERMICS, Paris UCLA [email protected] [email protected] Georg S. Weiss Graduate School of Mathematical Sciences MS12 University of Tokyo On the Contact-angle Condition for Stranski- [email protected] Krastanow Islands

In this talk we present a rigorous derivation of the zero MS13 contact-angle condition for Stranski-Krastanow islands in On Vortices in a Spinor Ginzburg–Landau Model strained solid films. Recent papers in the physics literature have introduced Irene Fonseca spin-coupled (or spinor) Ginzburg–Landau models for com- Carnegie Mellon University plex vector-valued order parameters in order to account Center for Nonlinear Analysis for ferromagnetic or antiferromagnetic effects in high- [email protected] temperature superconductors and in optically confined Bose–Einstein condensates. These models give rise to new N. Fusco types of vortices, with fractional degree and non-trivial core Dipartimento di Matematica structure. By studying the associated system of equations Università di Napoli ”Federico II” in R2 which describes the local structure of these vortices, [email protected] we show some new and unconventional properties of these vortices. We illustrate the various possibilites with some M. Morini specific examples of Dirichlet problems in the unit disk. SISSA These results are obtained in collaboration with L. Bron- [email protected] sard, P. Mironescu, and E. Sandier.

Giovanni Leoni Stanley Alama Carnegie Mellon University McMaster University [email protected] Department of Mathematics and Statistics [email protected]

MS12 MS13 Flame Propagation in Periodic Media The Boundary Distribution of Surface Supercon- I will describe some results on the homogenization of a ductivity free boundary problem arising in the modeling of flame propagation in premixed gas. The asymptotic regime cor- It is well-known that when the applied magnetic field is responds to a situation in which the flame width is smaller in the range HC2

of degree 1 near ∂Ω and a vortex of degree −1 near ∂Ω space. Examples are inelastic BTE modeling granular which rapidly converge to ∂A. In this work it was conjec- gases or Elastic BTE for gas mixtures. We study self- tured that the global minimizers do not exist when κ>κ0. similar solutions, usually referred as homogeneous cooling In a subsequent joint work with D. Golovaty and V. Ry- states, their asymptotic behavior by analyzing their spec- balko this conjecture was proved. The proof is based on tral properties in Fourier space. We show self-similar so- an introduction of a related auxiliary linear problem which lutions with initial finite moments of all orders will have allows for an explicit energy estimate. power like tails, and so they can not have finite moments for all orders. This fact posses new challenges and issues Leonid Berlyand on hydrodynamic fluid limit models. In addition, we show Penn State University the long time asymptotics is qualitatively similar to classi- Eberly College of Science cal non-linear wave equations. This work has been carried [email protected] out in collaboration with S. Bobylev and C. Cercignani. Irene M. Gamba MS13 Department of Mathematics and ICES Vortices for a Rotating Toroidal Bose-Einstein University of Texas Condensate [email protected] We construct local minimizers of the Gross-Pitaevskii energy, introduced to model Bose–Einstein condensates MS14 (BEC) which are subject to a uniform rotation. We con- Traveling Waves in Thin Liquid Films Driven by sider the case where the condensate occupies a solid torus Surfactant of revolution in R3 with starshaped cross-section. For large enough angular speeds we produce local minimizers of the In the lubrication approximation, the motion of a thin liq- energy which exhibit vortices, for small enough values of uid film is described by a single fourth-order partial differ- the length-scale parameter . These vortices concentrate at ential equation that models the evolution of the height of one or several planar arcs which minimize a line energy, ob- the film. When the fluid is driven by a Marangoni force tained as a Γ-limit of the Gross-Pitaevskii functional. The generated by a distribution of insoluble surfactant, the thin location of these limiting vortex lines can be described un- film equation is coupled to an equation for the concentra- der certain geometrical hypotheses on the cross-sections of tion of surfactant. In this talk, I show the basic structure of the torus. These results are obtained in collaboration with this system, and begin an analysis of wave-like solutions in S. Alama and J.A. Montero. the specific context of a thin film flowing down an inclined plane. Numerical simulations reveal an array of traveling Lia Bronsard waves, which persists when capillarity and surface diffusion McMaster University are neglected. The analysis of the limiting system has some [email protected] surprises, and in this talk, I show how far we have come in understanding the numerical results analytically, and the analytical results numerically. MS14 Fluid-Structure Interaction in Blood Flow Michael Shearer North Carolina State Univ. Fluid-structure interaction in blood flow will be discussed. [email protected] Suncica Canic Department of Mathematics MS15 University of Houston Control Problems in Fluid-Structure Interactions [email protected] Fluid structure interaction coupled system comprising of a three dimensional Navier Stokes equation coupled to a MS14 dynamic system of elasticity will be considered. The cou- Stability of Compressible Vortex Sheets pling takes place on the boundary – fixed interface between the solid and the fluid. The following new results will be Compressible vortex sheets are important in multidimen- presented: 1) wellposedness of weak and strong solutions; sional compressible Euler flows. In this talk we will dis- 2) boundary feedback stabilization in the neighborhoods of cuss some recent developments in the theoretical analysis unstable equilibriums. of compressible vortex sheets and related free boundary problems in multidimensional fluid dynamics and MHD. Irena Lasiecka Some further trends and open problems on compressible University of Virginia vortex sheets will also be addressed. [email protected] Gui-Qiang Chen Northwestern University MS15 USA Control of Elastic Systems with Nonlinear Consti- [email protected] tutive Law

m We consider an elastic body in a region R0 ⊂ R ,m=2 MS14 or 3. Our purpose in this talk is to explore the control Non-conservative Boltzmann Equations implications of the use of nonlinear constitutive laws obtained from adding quartic terms to the quadratic potential We consider energy dissipative space homogeneous Boltz- energy expressions of the theory of linear elasticity. We in- mann Transport Equations (BTE) for interactions of dicate the eﬀect of incorporating those terms and explore Maxwell type, which have a good representation in Fourier consequences for both static and dynamic control applica- PD06 Abstracts 269

tions. MS16 Evans Function Computation for Large Systems David Russell Virginia Tech We describe our new approach to Evans function compu- [email protected] tation for large systems, which avoids the spatial and temporal blow up of exterior-product methods. We introduce a polar-coordinate shooting method, for which the angu- MS15 lar equation is a variation of continuous orthogonalization Mathematical Analysis of Aircraft Wing–Airflow and the radial equation is easily computable and restores Interaction analyticity. We then use this new method to explore the shock wave stability problem for large one-dimensional vis- A model describing interaction of a long slender aircraft cous and viscous-dispersive conservation laws. wing with subsonic inviscid isentropic potential air flow will be presented. It is governed by a system of hyper- Jeffrey Humpherys bolic PDEs with time convolution integral forcing terms. Brigham Young University Asymptotic formulas for aeroelastic modes (wing vibra- jeff[email protected] tion frequencies) will be given. Expansion of a solution of the model equations with respect to the aeroelastic mode Kevin Zumbrun shapes will be presented. Application to flutter instability Indiana University and its suppression will be discussed. USA Marianna Shubov [email protected] University of New Hampshire Department of Mathematics and Statistics MS16 [email protected] Multi-pulses in PDEs with Reflection and Phase Invariance MS15 We consider parameter-dependent dynamical systems with On the Developments of Galin’s Stick-Slip Contact reflection and SO(2) symmetry which possess a homoclinic Problem orbit to a saddle focus. The reflection symmetry is broken In this talk, an analytical solution procedure developed by the wave speed. To address the existence of N-pulse for the the problem of indentation with friction of a rigid solutions and their PDE stability we use Lin’s method and indenter into an elastic half-space is considered. The corre- Lyapunov-Schmidt reduction. We discuss standing and sponding mixed boundary-value problem is formulated in traveling 2 and 3-pulse solutions. Motivation comes from planar bipolar coordinates, and reduced to a singular inte- the complex cubic-quintic Ginzburg-Landau equation. gral equation with respect to the unknown normal stress in Bjorn Sandstede the slip zones. An exact analytical solution of this equation University of Surrey is constructed using the Wiener-Hopf technique, which al- [email protected] lowed for a detailed analysis of the contact stresses, strain, displacement, and relative slip zone sizes. Also, a simple analytical solution is furnished in the limiting case of full Vahagn E. Manukian stick between the cylinder and half-space. North Carolina State University [email protected] Olesya Zhupanska University of Florida [email protected]fl.edu MS16 The Saddle-node of a Homogeneous Oscillation in Spatially Extended Reaction-diffusion Systems MS16 Invariant Manifolds and the Stability of Traveling We study the saddle-node bifurcation of a spatially homo- Waves in Burgers Equation geneous oscillation in a reaction-diffusion system posed on the real line. Beyond stability of the primary homogeneous Invariant manifolds are constructed in the phase space of oscillations created in the bifurcation, we investigate exis- perturbations to the traveling wave solution of Burgers tence, bifurcation and stability of wave trains with large equation and used to determine the asymptotic rate of con- wavelength that accompany the homogeneous oscillation. vergence to the wave. This provides a geometric explanation of existing results demonstrating that the decay rate Jens Rademacher of perturbations can be increased by requiring that initial Weierstrass Institute for Applied Analysis and Stochastics data lie in appropriate algebraically weighted spaces. [email protected]

Margaret Beck Arnd Scheel University of Surrey School of Mathematics/IMA Department of Mathematics University of Minnesota [email protected] [email protected]

C. Eugene Wayne Boston University MS17 Department of Mathematics On the Existence of Stokes Waves of Extreme Form [email protected] In this talk we present an alternative proof of the existence of (weak) Stokes waves of extreme form in the ﬁnite depth 270 PD06 Abstracts

case. We also show the existence of a family of regular Leipzig, Germany waves. The approach is based on free boundary techniques [email protected], [email protected] of Alt, Caffarelli, and Friedman. Danut Arama MS18 Carnegie Mellon University A New Phase Space Based Formulation for Trans- Pittsburgh, PA 15213 mission Tomography [email protected] A new formulation is developed for travel time tomography. Our new method uses multiple arrival times and phase MS17 information between the source and the measurement on Some Problems in the Modeling and Analysis of the boundary of the domain. The formulation can also deal Quasi-static Evolution in Brittle Fracture with anisotropic medium property. Numerical results show that the unknown wave speed can be recovered with a good Some problems in the modeling and analysis of quasi-static accuracy with fast convergence. This is a joint work with evolution in brittle fracture Based on Griffith’s criterion Jianliang Qian, Gunther Uhlmann, and Hongkai Zhao. for crack growth, a method was recently proposed for de- termining crack paths by taking continuous-time limits of Jianliang Qian discrete-time variational problems. This has been success- Department of Mathematics fully carried out, but an important difference remains be- Wichita State University tween these solutions and Griffith’s model. I will explain [email protected] the method and the main issues in its implementation, and then describe the remaining gap and some efforts to remove Eric Chung it. University of California Irvine Department of Mathematics Christopher Larsen [email protected] WorcesterPolytechnic Institute Math Dept [email protected] Hongkai Zhao University of California, Irvine Department of Mathematics MS17 [email protected] Relaxed Energies for H1/2–Maps with Values into the Circle Gunther Uhlmann University of Washington We consider, for maps in H1/2(R2; S1), a family of energies [email protected] related to seminorms equivalent to the standard one. These seminorms are associated to measurable matrix fields in the half space. For maps having finitely many prescribed topo- MS18 logical singularities, we show that the infimum of the en- A Lax-Friedrichs Sweeping Method for Static ergy is equal to the lenght of a minimal connection relative Hamilton-Jacobi Equations to a natural geodesic distance on the plane, connecting the singularities. As a consequence, we compute the relaxed In this talk, we will present a Lax-Friedrichs fast sweep- energy functionals on smooth maps. This is a joint work ing method to approximate the viscosity solution of static with A. Pisante. Hamilton-Jacobi equations. By solving for the value of a specific grid point in terms of its neighbors and incorporat- Vincent Millot ing a group-wise causality principle into the Gauss-Seidel Carnegie Mellon University iteration, we have an easy-to-implement and fast conver- Pittsburgh, PA 15213 gent numerical method. For computational boundary con- [email protected] dition, we give a simple recipe which enforces a version of discrete min-max principle. Some convergence analysis is done for the one-dimensional eikonal equation. Extensive MS17 2-D and 3-D numerical examples illustrate the efficiency Convergence of Equilibria of Thin Elastic Beams and accuracy of the approach.

In a series of recent papers a hierarchy of lower dimensional Chiu-Yen Kao theories for nonlinearly elastic thin beams has been rigor- University of Minnesota ously derived,starting from three-dimensional elasticity, by Institute for Mathematics and its Applications means of Gamma-convergence. This approach guarantees [email protected] convergence of minimizers of the 3d elastic energy to minimizers of the Gamma-limit problem.In this talk the convergence of (possibly non-minimizing) stationary points of MS18 the 3d elastic energy is discussed. These are joint works On the Complexity of Fast Sweeping Algorithms with S. Müller and M. Schultz. We explore—both theoretically and empirically—the com- Maria Giovanna Mora putational complexity of fast sweeping algorithms. While SISSA there are no surprises in two dimensions, the situation for Trieste, Italy three dimensions is less encouraging than it might first ap- [email protected] pear. On the empirical side, we also present a flexible new code base for approximating the solution of stationary Stefan Müller, Maximilian Schultz Hamilton-Jacobi equations, and discuss some of the trade- Max-Planck Institute PD06 Abstracts 271

offs in the choice of Python as an implementation language. cost is roughly the same as solving a classical barrier. In the one-dimensional case, we use a finite-volume method that Ian M. Mitchell extends the Hamiltonian-preserving scheme introduced by University of British Columbia Jin and Wen for a classical barrier. In the two-dimensional Department of Computer Science case, we consider a mesh-free particle method that can be [email protected] computed efficiently and that may be extended to higher- dimensions. MS18 Shi Jin, Kyle Novak Fast Sweeping Methods for Eikonal Equations on University of Wisconsin at Madison Triangular Meshes [email protected], [email protected] The original fast sweeping method relies on natural ordering provided by a rectangular mesh. We propose novel or- MS19 dering strategies so that the fast sweeping method can be Simulating the Wigner Equation with the Discon- extended efficiently and easily to any unstructured mesh. tinuous Galerkin Method and a Non-polynomial To that end we introduce multiple reference points and Approximation Space order all the nodes according to their distances to those reference points. We show extensive numerical examples The Wigner equation, with a suitable collision operator, to demonstrate the accuracy and efficiency of the new al- models the interaction between a quantum system and its gorithm on triangular meshes. environment. It is a useful model for semiconductor devices, and current simulations are typically based on oper- Hong-Kai Zhao ator splitting methods. We demonstrate simulation via the University of California, Irvine Discontinuous Galerkin method, using an oscillatory basis. [email protected] The use of a non-polynomial approximation space for DG calculations has recently been proposed by Yuan and Shu, Jianliang Qian and promises greater accuracy on a coarser mesh. Department of Mathematics Wichita State University Richard Sharp [email protected] University of Texas at Austin [email protected] Yong-Tao Zhang University of California, Irvine MS20 [email protected] Optical Manipulation of Matter Waves

MS19 See MS16 in under annual meeting (AN06 abstracts.) Nonlinear Fourth-order Parabolic Equation and Ricardo Carretero-Gonzalez Related Entropy Functionals Department of Mathematics San Diego State University A nonlinear fourth-order parabolic equation is presented; [email protected] the equation appears in quantum semiconductor modeling and also in the study of interface ﬂuctuation in spin system. Several Lyapunov functionals are used in order to show ex- MS20 istence results and long-time asymptotics for bounded and Soliton Dynamics and Scattering for the Gross- unbounded domains in one and more dimensional space. Pitaevskii Equation Exponential decay in time of solutions to the steady state is proved by the entropy-entropy production method using See MS16 in under annual meeting (AN06 abstracts.) logarithmic Sobolev inequalities. Stephen Gustafson Maria Gualdani Department of Mathematics The University of Texas at Austin University of British Columbia [email protected] [email protected]

MS19 MS20 A Semiclassical Model for Thin Quantum Barriers Topological Effects with Matter Waves: A New Twist on BEC We present a time-dependent semiclassical transport model for mixed state scattering with thin quantum barriers. The See MS16 in under annual meeting (AN06 abstracts.) idea is to use a multiscale approach as a means of connecting regions for which a classical description of the system Chandra Raman dynamics is valid across regions for which the classical de- Physics Department scription fails, such as when the gradient of the poten- Georgia Tech. tial is undefined. We do this by first solving a stationary [email protected] Schrödinger equation in the quantum region to obtain the scattering coefficients. These coefficients allows us to build the interface condition to the particle flux that bridges MS20 the quantum region, connecting the two classical domains. High Temperature Superfluidity and the Super- Away from the barrier, the problem may be solved by tradi- fluid - Mott Insulator Transition in Ultracold tional numerical methods. Therefore, the overall numerical 272 PD06 Abstracts

Atomic Gases MS21 Computing Multi-valued Solutions for Euler- See MS16 in under annual meeting (AN06 abstracts.) Poisson Equations

Christian Schunck In this talk we first review the critical threshold phenom- MIT ena for Euler-Poisson equations, which arise in the semi- sch classical approximation of Schrödinger-Poisson equations and plasma dynamics. We then present a novel level MS21 set method for the computation of multi-valued velocity and electric fields of one-dimensional Euler-Poisson equa- Exact Solutions to Supersonic Flow onto a Solid tions. This method uses an implicit Eulerian formulation Wedge in an extended space—called field space, which incorpo- We present a problem arising in supersonic flow onto a solid rates both velocity and electric fields into the configura- wedge and show how to construct exact solutions for the tion space. Multi-valued velocity and electric fields are compressible potential flow model. captured through common zeros of two level set functions, which solve a linear homogeneous transport equation in the Tai-Ping Liu field space. The evaluation of density is also discussed. Stanford University Taiwan/USA Zhongming Wang [email protected] Iowa State University [email protected] Volker W. Elling Brown University Hailiang Liu [email protected] Iowa State University USA [email protected] MS21 Traffic Congestion-An Instability in a Hyperbolic MS22 System Homogenization of Integral Energies via the Peri- In this talk I’ll discuss second order traffic models and show odic Unfolding Method they support stable oscillatory traveling waves typical of those seen on a congested roadway. These solutions arise We consider the periodic homogenization of nonlinear in- when the trivial solution of uniformly spaced cars traveling tegral energies of the type at constant velocity is unstable. Wave profiles and speeds 1,p m x will be exhibited. (1) u ∈ W (Ω; R ) → f , ∇u dx, Ω εh James M. Greenberg where Ω is a bounded open subset of Rn with a Lipschitz Carnegie Mellon University boundary, and f is a Carathéodory energy density satisfy- Dept of Mathematical Sciences ing [email protected] f :(x, z) ∈ Rn × Rnm → f(x, z) ∈ [0, +∞[, f( · ,z) Lebesgue measurable,Y-periodic for every MS21 z ∈ Rnm,f(x, · ) continuous a.e. x ∈ Rn. Hyperbolic Conservation Laws on Manifolds (4) with Y the reference cell ]0, 1[n. Assume that f(x, ·) is The theoretical work on discontinuous solutions of nonlin- n quasiconvex for a.e. x ∈ R , and furthermore that, for ear hyperbolic conservation laws in several space variables 1 p ∈ [1, +∞[, M>0, and an Y -periodic a ∈ L (Y ), it has been restricted so far to problems set in the Euclid- satisfies the following growth conditions: ian space. Motivated by numerous applications, to geophysical fluid flows and general relativity in particular, I f(x, z) ≤ a(x)+M|z|p for a.e. x ∈ Rn, and every z ∈ Rnm, will attempt to set the foundations for a study of entropy vertz|p ≤ f(x, z) for a.e. x ∈ Rn, and every z ∈ Rnm. solutions to hyperbolic conservation laws on a manifold. (5) Understanding the interplay between the geometry of the → manifold and the behavior of discontinuous solutions of It is then known that as εh 0, one has partial differential equations is particularly challenging. In x inf lim infh→+∞ f , ∇uh dx : this lecture, I will discuss the derivation of the L1 contrac- Ω εh tion property and the total variation diminishing property, 1,p m p m which are known to play a central role in the theory and {uh}⊂W (Ω; R ),uh → u in L (Ω; R ) the numerical analysis of hyperbolic problems. Next, I will present the generalization to manifolds of, both, Kruzkov’s x = inf lim suph→+∞ f , ∇uh dx : and DiPerna’s well-posedness theories. Finally, I will es- Ω εh tablish the convergence of the finite volume schemes on 1,p m p m manifolds, with a special attention for the case of the 2- {uh}⊂W (Ω; R ),uh → u in L (Ω; R ) sphere. This a joint work with M. Ben-Artzi (Jerusalem). = fhom(∇u)dx, Philippe G. LeFloch Ω University of Paris 6 where the homogenized energy density fhom is defined by Laboratoire Jacques-Louis Lions lefl[email protected] nm 1 fhom : z ∈ R → lim inf f(y, z + ∇v)dy : t→+∞ n t tY PD06 Abstracts 273

1,p m [email protected] v ∈ W0 (tY ; R ) .

MS23 This convergence result was established in [1] and [2] by Weak and Strong Global Attractors for the 3D sophisticated Γ-convergence arguments. Since then, many Navier-Stokes Equations attempts at simplifying the proof (for example by using two-scale convergence), seem not to have borne fruit. In Often the evolution of an autonomous dissipative dynam- [5], a joint paper with Alain Damlamian and Riccardo De ical system can be described by a semigroup of solution Arcangelis, we apply the tool of periodic unfolding from [3]. operators. If this semigroup is asymptotically compact, This gives a direct proof of the convergence result (under then the classical theory of attractors yields the existence slightly weaker assumptions than in [1] and [2]), making use p of a compact global attractor. However, for some dynami- of simple weak convergence arguments in L -type spaces. cal systems the semigroup is not asymptotically compact. This paper is part of a series of ongoing works concerning Some dynamical systems do not even possess the semigroup the applications of the periodic unfolding method to ho- due to the lack of uniqueness (or proof of uniqueness). We mogenization. The first one [4], treated the same problem will define and study such objects as an omega-limit, in- but in the case of convex densities. variant set, and global attractor for a dynamical system that is not well-posed. Then we will apply the obtained Doina Cioranescu results to the 3D incompressible Navier-Stokes equations. Université Pierre et Marie Curie [email protected] Alexey Cheskidov Department of Mathematics University of Michigan MS22 [email protected] The Unfolding Approach for Periodic Homogeniza- tion of Non Linear Elliptic PDE’s and for the Mosco-convergence of Convex Integrals MS23 The Weak Attractor of the 3-D Navier-Stokes In this presentation, we show how the use of the periodic Equations unfolding method greatly simplifies and clarifies the homogenization of non-linear elliptic operators (in the mul- We study the pojection of the weak attractor of the 3D tivalued framework generalizing the Leray-Lions type, as 1,p periodic Navier-Stokes in the normalized, dimensionless a map from the space W0 (Ω) to its dual space). In the energy-enstrophy plane, obtaining restrictive bounds on case of subdifferentials of convex functions, the result can the attractor’s location. These bounds imply that the con- be expressed in terms of the Mosco-convergence of the cor- ditions necessary for the direct energy cascade can occur responding sequence of integral functionals on the space 1,p either near the origin of the phase space or near a singu- W0 (Ω). larity. Alain Damlamian Ciprian Foias Université Paris 12 Val de Marne Texas A&M University 94010 Créteil Cedex, France fi[email protected] [email protected] Michael S. Jolly, Radu Dascaliuc MS22 Indiana University Department of Mathematics Asymptotics of a Spectral Problem Associated with [email protected], [email protected] the Neumann Sieve

We analyze the asymptotic behavior of a Steklov spectral MS23 problem associated to the Neumann Sieve model, i.e a three dimensional set Ω, cut by a hyperplan Σ where each of the The Attractor of the 2-D Navier-Stokes Equations two-dimensional holes, − periodically distributed on Σ, in the Energy-enstrophy Plane have diameter r. Depending on the asymptotic behavior r We examine how the global attractor of the 2-D Navier- of the ratios 2 we ﬁnd the limit problem of the spectral Stokes equations projects in the normalized, dimensionless problem and prove that the sequences λn, formed by the energy-enstrophy plane. A particular motivation is to see n-th eigenvalue of the problem, converge to λn, the n- ∈ whether a condition for a direct energy cascade can be th eigenvalue of the limit problem, for any n N.We achieved. For any force, the attractor must lie between also prove the weak convergence, on a subsequence, of the a line (the Poincare inequality) and a parabola. An op- associated sequence of eigenvectors un, to an eigenvector timization of the bounding parabola results in a sharper associated to λn. When λn is a simple eigenvalue, we show bound, yet preliminary computations suggest that an even that the entire sequence of the eigenvectors converges. As sharper, linear estimate holds. a consequence, similar results hold for the spectrum of the DtN map associated to this model. Ciprian Foias TexasA&M, Daniel Onofrei Indiana University Dept. Mathematical Sciences [email protected] Worcester Polytechnic Institute [email protected] Radu Dascaliu Department of Mathematics Bogdan M. Vernescu Indiana University Worcester Polytechnic Inst [email protected] Dept of Mathematical Sciences 274 PD06 Abstracts

Michael S. Jolly MS24 Indiana University Solute Transport in Porous Media with Network Department of Mathematics Microgeometry [email protected] We study solute transport in porous media with periodic microstructures consisting of collections of interconnected MS23 by thin channels. We identify a physical mechanisms that Forced Two Layer Beta-plane Quasi-geostrophic promotes mixing of the solute with the host liquid and ob- Flow tain the macroscopic transport equation that the concentration of solute satisfies. We discuss examples motivated We consider a model of quasigeostrophic turbulence that by the transport of nutrients in bones. has proven useful in theoretical studies of large scale heat transport and coherent structure formation in planetary Guillermo Goldsztein atmospheres and oceans. The model consists of a cou- Georgia Tech pled pair of hyperbolic PDE’s with a forcing which repre- Atlanta, GA sents domain-scale thermal energy source. Although the [email protected] use to which the model is typically put involves gathering information from very long numerical integrations, little of a rigorous nature is known about long-time properties MS24 of solutions to the equations. First we define a notion of Optimal Lower Bounds on the L∞ Norm of the weak solution, and show using Galerkin methods the long- Stress in Random Elastic Composites time existence and uniqueness of such solutions. Then we show that the unique weak solution produces, via the in- Consider a cube containing two isotropic linear elastic ma- verse Fourier transform, a classical solution for the system. terials. The only information given on the configuration Moreover, we prove that this solution is analytic in space of the two materials is the measure of the set occupied by and positive time. each material. Subject to this limited geometric information one is interested in the L∞ norm of a specified stress Lee Panetta invariant when the domain is subject to a uniform force Texas A&M University on the boundary. We provide a methodology for deriving [email protected] optimal lower bounds on the L∞ norm for various stress invariants when only the volume fraction of each material Constantin Onica is known. Indiana University Institute for Sci. Comp. and Appl. Math. Robert P. Lipton [email protected] Department of Mathematics Louisiana State University [email protected] MS24 Optimal Structures of Multiphase Composites and MS24 their Fields Fluctuating Fractal Surfaces The translation bounds for multiphase mixtures are sharp only in a range of volume fractions of components. We We construct a family of fractal surfaces with fluctuating analyse the fields in translation-optimal structures for two- geometry at all scales and describe some metric and an- and three-dimensional conductive composites and describe alytic properties of these sets from the point of view of the largest possible classes of multimaterial structures that composite media. achieve the bounds as well as the limits of attainability of Umberto Mosco these bounds. The two schemes of synthesis of optimal Department of Mathematical Sciences structures are suggested. One is an ”inverse lamination Worcester Polytechnic Institute scheme” in which the rank-one connected fields in lami- [email protected] nates are fixed but the geometry is varied, the second is an adjusted differential scheme. Our results bring together and expand classes of optimal microgeometries found ear- MS25 lier by Milton, Milton and Kohn, Lurie and Cherkaev, Shocks in Driven Liquid Films Gibiansky and Cherkaev, Gibiansky and Sigmund. We also discuss new supplementary bounds for multicompo- Driven contact line problems in thin liquid films are an ac- nent mixtures and structures that realize these bounds. tive area of research. The mathematical theory of shock Some results were obtained in collaboration with Vincenzo waves has recently been shown to play an important role Nezi. in our understanding of basic properties of the contact line motion. I will present the theory for two recently studied Andrej V. Cherkaev experimental systems: (1) Thermally driven films coun- Department of Mathematics terbalanced by gravity are described by a scalar conserva- University of Utah tion with a non-convex flux. Such systems are known to [email protected] produce ‘undercompressive shocks’ in which characteristics emerge from the shock on one side. (2) A related problem Nathan Albin is that of particle laden flow driven by gravity. The dif- University of Utah ferential settling rate of the particles with respect to the na fluid results in the formation of double shock fronts which are solutions of a system of two conservation laws for the motion of the species. Comparison between theory and experiment will be discussed, along with open mathematical PD06 Abstracts 275

problems directly related to the experiments. Purdue University [email protected] Andrea L. Bertozzi UCLA Department of Mathematics Marianne Korten [email protected] Department of Mathematics Kansas State University MS25 [email protected] The Hele-Shaw Problem as a ”Mesa” Limit of Ste- fan Problems: Existence, Uniqueness, and Regu- MS25 larity of the Free Boundary A Deterministic Control Based Approach to Cur- We study a Hele-Shaw problem as a ”Mesa” type limit of vature Flows one phase Stefan problems in exterior domains. We study In a joint work with Bob Kohn, we give a new control-type the convergence, determine some of qualitative properties interpretation on the level-set approach to motion by cur- of the unique limiting solution, and prove regularity of the vature and related interface motion laws. More precisely, free boundary of this limit under very general conditions we give a family of discrete-time, two-person games whose on the initial data. Indeed our results handle changes in value functions converge in the continuous-time limit to the topology and multiple injection slots. This project is joint solution of the motion-by-curvature PDE. The value func- work with Marianne Korten and Charles Moore. tion of a deterministic control problem is normally a ﬁrst- C. N. Moore order Hamilton-Jacobi equation, while the level-set formu- Department of Mathematisc lation of motion by curvature is a second-order parabolic Kansas State University equation. [email protected] Robert V. Kohn Courant Institute Ivan Blank New York University Department of Mathematics [email protected] Worcester Polytechnic Institute [email protected] Sylvia Serfaty Courant Institute, NYU Marianne Korten New York, NY Department of Mathematics [email protected] Kansas State University [email protected] MS26 Lawrence-Doniach System for High-temperature MS25 Superconductors Free Boundary Measures and Partial Regularity of Solutions in a Mixed Type Evolution Equation We consider the Lawrence-Doniach model for layered superconductors, which describes a large class of high- Solutions of the mixed type equation temperature superconductors as a ﬁnite number of parallel superconductors, with Josephson coupling between them, ut =Δxα(u) + divxF (u), including nonlinear Maxwell’s equations that describe the

n induced magnetic field. Assuming an applied magnetic where x ∈ IR ,α(u) is a continuous piecewise linear func- field which is nontangential to the layers, we prove that tion such that α(u)=0for|u|≤1 and having slope 1 oth- n a change of phase from superconducting to normal occurs, erwise, and F (u) is a Lipschitz from IR to IR , combine the and we estimate the upper critical field in terms of the ma- features of conservation laws, and of free boundary prob- terial parameters, the Josephson constant, and the angle lems. We will discuss the regularity of weak solutions to of the nontangential applied field. Our estimates extend this equation, in particular, we will show that for solutions 2 to the nonlinear regime predictions by physicists based on locally in L ,α(u) satisfies local energy estimates, and that the linearized system. u is continuous in the set {|u| > 1}. With this in hand we will identify the free boundary measures supported on the Patricia Bauman boundary of {|u| > 1}. This allows us to study separately Purdue University the jumps in u at the free boundary and the discontinu- [email protected] ities occuring within the region {|u| < 1} governed by the consevation law. Yangsuk Ko University of California, Bakersfield G.-Q. Chen yangsuk ko@firstclass1.csubak.edu Department of Mathematics Northwestern University [email protected] MS26 Bifurcation Structure of a Ginzburg-Landau Model C. N. Moore in a Ring Department of Mathematisc Kansas State University We deal with the one-dimensional Ginzburg-Landau energy [email protected] functional which is a simplified model of the superconductivity in a thin tubular ring. The Euler-Lagrange equation Monica Torres of this functional, called the Ginzburg-Landau equation, Department of Mathematics has two physical parameters. We show a global bifurcation 276 PD06 Abstracts

structure in the parameter space if the tubular domain has Department of Mathematics uniform thickness. Moreover we exhibit a local bifurcation [email protected] diagram around a singularity for the case of nonuniform thickness. MS27 Yoshihisa Morita A Particular Large Data Solution of the 1-D Euler Ryukoku University System Department of Appl. Math. and Informatics [email protected] It is known from explicit examples of finite time blowup in BV and L∞ that the class of strictly hyperbolic systems is too large to allow a general global existence result MS26 for large data. On the other hand it is hoped that physi- Transitions of Liquid Crystals and Critical Values cal systems, and especially the Euler system, have enough of Elastic Coefficiants structure to admit global weak solutions, even for large amplitude/variation data. We present an example of a so- We use the Landau-de Gennes model to examine the effects lution to the Euler system which exhibits the same type of twist and bend to liquid crystals. In the case of equal of interaction pattern for which blowup has been shown in elastic coefficients (Ki = K2 = K3 = K) we show that other systems. The data giving rise to these particular so- there exists a critical value Kc such that the liquid crystal lutions may be arbitrarily large in BV or L∞. By carefully is in the smectic state if KKc. Under some non-degeneracy and regularity for these particular solutions. assumptions on the nematic states we find the precise value of Kc and obtain the asymptotic behavior of minimizers of Erik Endres EK as K approach Kc. Pennsylvania State University Department of Mathematics Xingbin Pan [email protected] East China Normal University [email protected] Kris Jenssen Pennsylvania State University MS26 [email protected] Something About Ginzburg-Landau Vortices in the High-Kappa Limit MS27 I will talk either about vortex collisions in the Ginzburg- Self-similar Solutions for Transonic 2D Riemann Landau heat flow or about some refined Gamma- Problems convergence results for Ginzburg-Landau with applied We discuss recent developments of transonic self-similar 2D magnetic field Riemann problems. Sylvia Serfaty Eun Heui Kim Courant Institute, NYU California State University at Long Beach New York, NY [email protected] [email protected]

MS27 MS27 Transonic Shock Solutions for a System of Euler- Dependence of Large Entropy Solutions to the Eu- Poisson Equations , 1-d Case ler Equations on Physical Parameters

μ A boundary value problem for a 1-d system of Euler- We compare the entropy solution U to a hyperbolic sys- Poisson equations is considered. The boundary conditions tem of conservation laws for which the ﬂux function de- are supersonic on the left end and subsonic on the right pends on a parameter vector μ =(μ1,...,μk) with the end. The solutions with transonic shocks are constructed, Standard Riemann Semigroup S generated when μ ≡ 0 1 and the shock locations are shown to be uniquely and ex- and establish L error estimates pointwise in time with re- plicitly determined by the boundary data. spect to the ﬂux parameter μ. We apply our results to the isentropic and relativistic Euler equations with large Tao Luo oscillations for which the parameters are the adiabatic ex- Georgetown University ponent γ>1 and the speed of light c<∞ in comparison [email protected] with the isothermal Euler equations.

Yongqian Zhang MS28 Fudan University Artiﬁcial Compressibility Method for the Incom- – pressible Navier Stokes Equations

Gui-Qiang Chen We study an hyperbolic approximation of the Leray solu- Northwestern University tion to the 3D Navier Stokes equation. In particular we de- USA scribe an hyperbolic version of the so called compressibilty [email protected] method investigated by J.L.Lions, Temam. This approximation is motivated by numerical analysis applications Cleopatra Christoforou where, in order to overcome the diﬃculty related the di- Northwestern University vergence free condition, an artiﬁcial compressibilty is introduced. We will study the problem on unbounded domains PD06 Abstracts 277

by replacing Sobolev compact embedding with the use of namics of a compressible, viscous, and heat conducting dispersive estimates of Strichartz type combined with bi- fluid. In particular, we concentrate on the case when the linear estimates. Mach number tends to zero. We shall show that in the limit we arrive at the Oberbeck-Boussinesq approximation. Donatella Donatelli Several generalizations of this result are also discussed, in University of L’Aquila -Italy particular the case of a chemically reacting fluid. [email protected] Eduard Feireisl Pierangelo Marcati Mathematical Institute ASCR, Zitna 25, 115 67 Praha 1 Dipartimento di Matematica, Università di L’Aquila Czech Republic [email protected] [email protected]

MS28 MS29 The Existence of Weak, Small Energy Solutions for Lagrangean Structure and Propagation of Singu- the Equations of Motion of 3D Compressible, Vis- larities in Multidimensional Compressible Flow cous Fluid Flows with the No-slip Boundary Con- ditions Not available at time of publication.

We consider the equations of motion of a compressible, vis- David Hoff cous, isentropic fluid in a bounded domain in R2orR3 with Indiana University the no-slip boundary conditions. Given a constant, equi- hoff@indiana.edu librium state we construct a global in time, regular weak solution, provided that initial data are close to the quilib- Marcelo dos Santos rium when measured by weak norms and discontinuities in University of Campinas, Brazil the initial density decay near the boundary. [email protected] Mikhail Perepelitsa University of Massachusetts MS29 [email protected] Navier’s Slip and Incompressible Fluids with Tem- perature, Pressure and Shear Rate Dependent Vis- cosity MS28 Boundary Layer Analysis for Isentropic Gas Dy- There is compelling experimental evidence for the viscos- namics with Real Viscosity and Related Systems ity of a fluid depending on the temperature, the shear rate as well as the mean normal stress (pressure). Moreover, We study the problem of boundary layer formation for vis- while the viscosity can vary by several orders of magnitude cosity approximations of the isentropic gas dynamics, in- the density suffers very minor variation, when the range cluding source terms as friction and topography, for phys- of the pressure is sufficiently large, thereby providing jus- ical viscosity models. The case of Saint-Venant equations tification for considering the fluid as being incompressible for shallow water is also discussed. while at the same time possessing a viscosity that is dependent on the pressure. We investigate the mathematical Chiara Simeoni properties of internal unsteady three-dimensional flows of Universite de Nice - Sophia Antipolis such fluids subject to Navier’s slip at the boundary. We es- Laboratoire J. .A. Dieudonne tablish the long-time existence of a weak solution for large [email protected] data provided that the viscosity depends on the temperature, the shear rate and the pressure in a suitably specified Debora Amadori manner. This specific relationship however includes the University of L’Aquila (Italy) classical Navier-Stokes-Fourier fluids and power-law fluids [email protected] as special cases. This talk is based on joint work with M. Bulicek and K.R. Rajagopal. MS28 Josef Malek Homogenization in Parabolic Equations Charles University, Prague Mathematical Institute We are interested in homogenization of parabolic equa- [email protected]ff.cuni.cz tions including diffusion equation, Incompressible Navier- Stokes equation in reticulated domain or perforated domain, which are arose from cell biology, rheology. In the MS30 limit, we identify the effective parabolic equations. The Control-theoretic Approach to Construction of Fast Iterative Solvers Bo Su Iowa State University - USA We outline the dynamical systems/optimal control per- [email protected] spective on the design of numerical approximations. In particular, we describe a natural class of process level discretizations of a continuous time/continuous state deter- MS29 ministic optimal control problem, when the goal is that On Singular Limits for the Full Navier-Stokes- the related iterative solver for the dynamic programming Fourier System equation should converge with a number of iterations that is independent of the discretization. An extension to de- We discuss some singular limits of global-in-time solutions to the full Navier-Stokes-Fourier system describing the dy- 278 PD06 Abstracts

terministic diﬀerential games is also described. Degenerate Elliptic Equations Paul Dupuis We will present a useful class of wide stencil schemes for Brown University degenerate elliptic equations. These types of schemes have Division of Applied Mathematics been used to build convergent schemes for several types [email protected] of second order equation, including: Level Set Motion by Mean Curvature, the Monge-Ampere equation, the Inﬁnity Laplacian, the equation for the convex envelope. These MS30 schemes are easy to build, and are provably convergent. An Application of Fast Sweeping Methods to We will discuss adapting these schemes to build schemes Transmission Traveltime Tomography for Hamilton-Jacobi equations on unstructured grids.

Traditional transmission travel-time tomography hinges on Adam Oberman ray tracing methods. We propose a new PDE-based ap- Simon Fraser University proach which avoids using the cumbersome ray-tracing Department of Mathematics technique. We start from the eikonal equation, define a [email protected] mismatching functional and then derive the gradient of the functional using an adjoint state method. These computations can be efficiently done using the fast sweeping MS31 method. We demonstrate the robustness of the method Existence and Stability of Traveling Waves of the with examples including the Marmousi synthetic velocity Regularized Short Pulse and Ostrovsky Equations model. We consider the question of existence and stability of trav- Shingyu Leung eling waves for the regularized short pulse and Ostrovsky University of California Los Angeles equations. The regularization term in the short pulse equa- Department of Mathematics tion arises naturally as a higher order expansion term in [email protected] the derivation of the original short pulse equation. The existence of single bump traveling waves for both equations Jianliang Qian is proved via the Fenichel theory for singularly perturbed Wichita State University ODEs and a Melnikov type transversality calculation. We Department of Mathematics then consider the spectral stability of the waves via topo- [email protected] logical arguments based on the augmented Evans bundle, which is constructed from the fast-slow decomposition of the underlying traveling wave. MS30 A Fast Sweeping Method Based on Discontinu- Christopher Jones ous Galerkin Methods for Static Hamilton-Jacobi University of North Carolina Equations Department of Mathematics [email protected] We will report on our preliminary results on combining the discontinuous Galerkin methods, a properly cho- Nicola D. Costanzino sen numerical Hamiltonian, and Gauss-Seidel iterations Brown University with alternating-direction sweepings to design a high or- UNC-Chapel Hill der fast sweeping method for static Hamilton-Jacobi equa- [email protected] tions. The main new feature of the method is to explore the advantages of the local compactness of the discontinuous Galerkin methods which fits well in the fast sweeping MS31 framework. A description of the algorithm and preliminary On Subsonic Detonation Waves in Inert Porous numerical results for Eikonal equations will be presented. Medium

Hong-Kai Zhao We consider Sivashinsky’s model of subsonic detonation University of California, Irvine that describes propagation of combustion fronts in highly [email protected] resistable media. It is known that there exists a traveling wave asymptotically connecting the unburnt and burnt Fengyan Li states, which is unique if thermal diffusivity is neglected. University of South Carolina Using geometric singular perturbation theory we study the fl[email protected] uniqueness and stability of the wave in the case of positive thermal diffusivity. Chi-Wang Shu Anna Ghazaryan Division of Applied Mathematics University of North Carolina at Chapel Hill Brown University [email protected] [email protected] Chris Jones Yong-Tao Zhang Department of Mathematics University of California, Irvine University of North Carolina [email protected] [email protected]

MS30 Peter Gordon Wide Stencil Schemes for First and Second Order New Jersey Institute of Technology [email protected] PD06 Abstracts 279

MS31 dynamics is then shown to be asymptotically close to the Rigorous Asymptotic Expansions for Critical Wave one governed by the 2-D “pressureless” rotational Euler Speeds in a Family of Scalar Reaction-diffusion equations with subcritical initial data, which in turn yields Equations the increasingly long time existence at this singular regime. The novelty of our approach is the use of an approximate We investigate traveling waves in the family of reaction- system that is linear while still capturing both the singular- diffusion equations given by ity and the advection dynamics of the underlying nonlinear − m − system. The near periodic dynamics shown here is closely ut = uxx 2u (1 u). related to the circular fluid motions observed in geophysical For m ≥ 1 real, there is a critical wave speed ccrit(m) sciences. which separates waves of exponential structure from those that decay only algebraically. We derive rigorous asymp- Bin Cheng Department of Mathematics totic expansions for ccrit by perturbing off two solv- able cases, the classical Fisher-Kolmogorov-Petrowskii- University of Maryland at College Park Piscounov equation (m = 1) and a degenerate cubic equa- [email protected] tion (m = 2). In addition, we study the asymptotic critical speed in the limit as m →∞. Our approach uses geometric Eitan Tadmor singular perturbation theory, as well as the blow-up tech- University of Maryland nique, and confirms results previously obtained through USA asymptotic analysis. [email protected]

Tasso J. Kaper Boston University MS32 Department of Mathematics Weakly Nonlinear Approximation of Multi-D [email protected] Hyperbolic-Parabolic System with Entropy

Freddy Dumortier We consider weakly nonlinear approximation around a con- University of Hasselt stant state of general multi-D hyperbolic-parabolic system [email protected] with entropy. Using method of multiple time scales, we derive the averaged system. We give a sufficient condition under which global weak solution to the averaged system Nikola Popovic exists. We apply the results to the multi-D Navier-Stokes- Boston University Fourier system of compressible gas dynamics, which sat- Center for BioDynamics and Department of Mathematics isfies the structure condition and derive a new averaged [email protected] system on the fast mode. We apply the global solutions to the averaged system to construct local Maxwellian to MS31 approximate solutions to the Boltzmann equation Exchange Lemma for Nontrivial Slow Flows Ning Jiang Department of Mathematics The Exchange Lemma of Jones and Kopell deals with University of Maryland at College Park the passage of solutions near a slow manifold of a geo- [email protected] metric singular perturbation problemx ˙ = f(x, y), y˙ = g(x, y), (x, y) ∈ Rm × Rn.For = 0 the slow manifold is given by g(x, y) = 0 and consists of normally hyperbolic David Levermore equilibria. For small >0 the slow manifold is nearby, and University of Maryland (College Park) on it, after a change of variables, the differential equation USA [email protected] becomesx ˙ 1 = , x˙ 2 = ··· =˙xm = 0. However, if normal hyperbolicity fails somewhere, then some fast direc- tions must be added to the slow manifold, and usually the MS32 blowing-up technique of Roussarie, Dumortier, Szmolyan, et. al. must be used to analyze the flow on the enlarged On the Conditional Global Regularity of the 1D slow manifold. I shall discuss generalizations of the Ex- Euler-Poisson Equations with Pressure change Lemma needed to study the flow past the enlarged We prove that the one-dimensional Euler-Poisson system slow manifold. The motivation for this work is rarefaction- driven by the Poisson forcing together with the usual γ- like solutions of the Dafermos regularization of a system of law pressure admits global solutions for a large class of ini- conservation laws. tial data. Thus, the Poisson forcing regularizes the generic × Stephen Schecter finite-time breakdown in the 2 2 p-system. Global reg- North Carolina State Univ ularity is shown to depend on whether the initial config- Department of Mathematics uration of the Riemann invariants and density crosses an [email protected] intrinsic critical threshold. Dongming Wei MS32 University of Maryland, College Park [email protected] On the Long Time Regularity of the Shallow Water Equations Eitan Tadmor We study the long time existence of classical solutions to University of Maryland the Shallow Water (SW) equations with pressure gradient USA and rotational forcing that are both singular within certain [email protected] scaling regime of the Froude and Rossby numbers. The SW 280 PD06 Abstracts

MS32 Department of Mathematics Non-selfsimilar Global Solutions and Their New Princeton University Structures of Multi-dimensional Conservation [email protected] Laws

In this talk, we will discuss the multi-dimensional conser- MS33 vation laws whose Riemann data just contain two different Modulation Equations: Stochastic Bifurcation in constant states which are separated by a smooth curve or a Large Domains surface. Non-selfsimilar structures and properties are dis- covered. Furthermore, global solutions of a class of 2-D We consider the stochastic Swift-Hohenberg equation on a systems of conservation laws will be also presented and are large domain near its change of stability. We show that, formulated by implicit function, their structure combining under the appropriate scaling, its solutions can be approxi- 2-D non-selfsimilar elementary waves and non-constant in- mated by a periodic wave, which is modulated by the solu- termediate states will be shown. tions to a stochastic Ginzburg-Landau equation. We then proceed to show that this approximation also extends to Xiaozhou Yang the invariant measures of these equations. Shantou University [email protected] Greg Pavliotis Imperial College, London [email protected] MS33 Coherence in SPDE MS33 There has been much recent work in the phenomenon of Rare Events in Large Systems stochastic systems which become coherent in certain scaling regimes. In this talk, we will present examples of PDE Large deviation theory analyzes the probability of a rare or their corresponding discretizations which act coherently event in a system of bounded size, in the limit of vanish- but non-trivially under the application of small stochastic ing noise. Other limits are also mathematically interesting forcing. and physically relevant. In particular, rare events in systems that are large (exponentially large in the inverse noise Lee Deville strength) raise new questions. Courant Institute of Mathematical Sciences [email protected] Maria G. Reznikoff Princeton University Mathematics Department MS33 [email protected] Noise-induced Target Pattern Formation in Ex- citable Media MS34 Spatially distributed excitable systems, or “excitable me- Thin Films of Martensitic Materials: Can We Pre- dia”, are an important class of excitable systems whose dict Their Microstructure? main biological function is long-range signal transmission through self-sustained waves of activity. The stan- Because of their technological applications as microactu- dard modeling approach to excitable media is via sys- ators, martensitic thin films utilizing the shape memory tems of nonlinear deterministic partial differential equa- effect have been intensively studied in recent years. The tions. These ignore the effect of small random fluctuations key analytical issue for these applications is to provide a inherent in such media. Here, we show that under rather rigorous derivation of effective thin film models from Non- generic conditions the inclusion of the noise may lead to linear Elasticity via dimension reduction. I will describe qualitatively new dynamical behaviors, which, neverthe- a new mathematical theory of martensitic thin films and less, remain essentially deterministic. Specifically, we ana- discuss its connections with several recent results in this lyze patterns of recurrent activity in a prototypical model direction. of an excitable medium in the presence of noise. Without noise, this model robustly predicts the existence of spiral Marian Bocea waves as the only recurrent patterns in two dimensions. North Dakota State University Remarkably, we found that, in addition, small noise is ca- [email protected] pable of generating coherent target patterns, another type of recurrent activity which is widely observed experimen- MS34 tally. Our findings demonstrate the need to re-examine current modeling approaches to active biological media. Atomistic Potential Energy Surfaces and Solid Me- chanics Cyrill B. Muratov New Jersey Institute of Technology How does collective “solid mechanics” behaviour emerge Department of Mathematical Sciences from (accurate quantum-mechanical or approximate) [email protected] many-atom potential energy surfaces? I will discuss basic open problems, scope and limitation of known methods (e.g. homogenization theory), and recent mathematical re- Eric Vanden Eijnden sults. One such result (joint with F.Theil, Warwick) is Courant Institute identification of the 3D fcc (face-centered cubic) lattice as a New York University local minimizer of the celebrated Lennard-Jones pair inter- [email protected] action model. For a long time any progress in this direction was beyond reach; a key ingredient is a recent rigidity the- Weinan E orem from nonlinear PDE theory (due to Friesecke, James, PD06 Abstracts 281

and Müller). poses in a sparse and separable way. As a result, it becomes realistic to build the full matrix exponential up to some Gero Friesecke time which is much bigger than the usual CFL timestep. University of Warwick Once available, this new representation can be used to per- Mathematics Institute form giant ’upscaled’ time steps to solve the wave equation [email protected] faster than with a spectral method. We will show numerical examples as well as complexity results based on a priori estimates of sparsity and separation ranks. MS34 Mathematical and Computational Methods for Laurent Demanet Coarse-graining Applied and Computational Mathematics Caltech In this talk we discuss recent progress in obtaining coarse- [email protected] grained stochastic approximations of microscopic lattice dynamics. We prove rigorous error estimates both at equilibrium and non-equilibrium in terms of weak convergence MS35 and relative entropy calculations and obtain a posteriori es- Nonlinear Wave Equations on a Curved Back- timates that can guide adaptive coarse-grained simulations. ground We also demonstrate, with direct comparisons between microscopic Monte Carlo and coarse-grained simulations, that We will discuss a local well-posedness result for a semilinear the derived mesoscopic models provide a substantial CPU wave equation with variable coefficients in 4+1 dimensions. reduction in the computational effort. As part of the proof we will present a new definition for the Xs,b spaces in the variable coefficient case. Markos A. Katsoulakis UMass, Amherst Daniel Tataru Dept of Mathematics University of California, Berkeley [email protected] [email protected]

Dan A. Geba MS34 University of California, Berkeley/ University of Inclusions for Which Constant Magnetization Im- Rochester plies Constant Field, and Their Applications [email protected] In this paper, we find a class of special regions that have the same property with respect to second order linear partial MS35 differential equations that ellipsoids have in infinite space. Combining Finite Elements and Geometric Wave That is, in physical terms, constant magnetization of the Propagation in 1-D inclusion implies constant magnetic field on the inclusion. The inclusions apparently enjoy many interesting proper- We show that the initial value problem for a strictly hy- ties with respect to homogenization and energy minimiza- perbolic partial differential equation on the circle can be tion. In particular, we use them to give new results on a) solved at cost O(N). To obtain this the equation is ap- homogeneous and inhomogeneous inclusion problems sub- proximately decoupled, into first order equations that can ject to periodic boundary conditions; b) optimal bounds be solved using the method of characteristics. An ODE for of the effective moduli of two-phase composites; c) energy- the error is of the form dv/dt = R(t)v, with R(t) bounded, minimizing microstructures; and d) the characterization of which, with an appropriately designed scheme, leads to the the Gθ-closure of two well-ordered composites. stated complexity. Liping Liu Christiaan C. Stolk University of Minnesota University of Twente [email protected] [email protected]

Richard James University of Minnesota, Minneapolis MS35 [email protected] Evolution Equations, Curvelet Transform, and Curvelet-curvelet Interaction Perry Leo University of Minnesota We discuss how to solve certain evolution equations with [email protected] the curvelet transform. The analysis presented here yields an approach that requires solving, scale-wise, for the geometry reminiscent of the propagation of singularities on the MS35 one hand, and solving a matrix Volterra integral equation Wave Atoms and Time Upscaling of Wave Equa- (of the second kind) on the other hand. The Volterra equations tion can be solved by recursion, in a step-by-step manner with reﬁnement – as in the computation of certain multiple We present a geometric algorithm for solving the 2D wave scattering series; this process reveals the curvelet-curvelet equation in smooth inhomogeneous periodic media. We interaction. It is shown that the representation (matrix) of introduce a tight frame of ’wave atoms’, a family of wave the kernel in the Volterra equation is optimally sparse. packets interpolating between wavelets and Gabor, whose key property is a precise balance between oscillations and Gunther Uhlmann, Hart Smith support called parabolic scaling. In that frame, the time- University of Washington dependent Green’s function of the wave equation decom- [email protected], [email protected] 282 PD06 Abstracts

Maarten V. de Hoop [email protected] Purdue University [email protected] MS36 Capacity of a Multiply-connected Domain and MS36 Nonexistence of Ginzburg-Landau Minimizers with Numerical Simulation of Vortex Dynamics in Prescribed Degrees on the Boundary Ginzburg-Landau-Schrödinger Equation Suppose that ω ⊂ Ω ⊂ R2. In the annular domain In this talk, we will study numerically vortex dynamics in A =Ω\ ω¯ we consider the class J of complex valued the Ginzburg-Landau-Schrödinger equation (GLSE) in two maps having the same degree d on ∂Ω and on ∂ω. The dimensions (2D) existence of minimizers of Eκ for all κ when cap(A) ≥ π (domain A is “thin”) and for small κ when cap(A) <π 1 2 (α − iβ)∂tψ(x,t)=Δψ + V0(x) −|ψ| ψ, (domain A is “thick”) was established by Berlyand and ε2 Mironescu (’04). Here we provide the answer for the re- 2 x ∈ R ,t>0, maining case of large κ when cap(A) <π. We prove that, 2 when cap(A) <π, there exists a finite threshold value κ1 ψ(x, 0) = ψ0(x), x ∈ R , of the Ginzburg-Landau parameter κ such that the mini- with nonzero far field conditions mum of the Ginzburg-Landau energy Eκ is not attained in J when κ>κ1 while it is attained when κ<κ1. imθ |ψ(x,t)|→1 (e.g. ψ → e ),t≥ 0, Volodymyr Rybalko when r = |x| = x2 + y2 →∞; Institute for Lower Temperature Physics and Engineerin [email protected] where ψ(x,t) is the complex-valued wave function, V0(x)is a given real-valued external potential satisfying V0(x) → 1 | |→∞ Dmitry Golovaty when x , ε is a positive constant, α and β are two The University of Akron nonnegative constants satisfying α+β>0. We will present Department of Theoretical and Applied Mathematics an efficient and accurate numerical method for numerical [email protected] simulation of and apply it to study quantized vortex dynamics in GLSE. By formulating the equation in 2D polar coordinate system, the nonzero far field conditions which Leonid Berlyand are highly oscillating in the transverse direction will be ef- Penn State University ficiently resolved in phase space in a natural way. This Eberly College of Science allows us to develop a time-splitting method which is un- [email protected] conditionally stable, efficient and accurate for the problem. Moreover, the method is time reversible when it MS36 is applied to the nonlinear Schrödinger equation (NLSE) with nonzero far field conditions. We also apply the nu- On a Minimization Problem with a Mass Con- merical method to study issues such as the stability of straint Involving a Potential Vanishing on Two quantized vortex, interaction of a few vortices, dynamics Curves of the quantized vortex lattice and motion of vortex with We study a singular perturbation type minimization prob- an inhomogeneous external potential in GLSE. Based on lem with a mass constraint over a domain or a manifold, our numerical results, we find that the quantized vortex involving a potential vanishing on two curves in the plane. with winding number |m| = 1 is dynamically stable, and | | In the case of the problem on the sphere we give a precise resp., m > 1 dynamically unstable, in GLSE. Further- description of the limiting behavior of both the minimizers more, different interaction patterns between a few vortices ± and their energies when the small parameter epsilon goes with winding number m = 1 as well as dynamics of quan- to zero. tized vortex lattices in GLSE are reported. This talk is based on joint works with Qiang Du and Yanzhi Zhang. Nelly Andre University of Tours Weizhu Bao [email protected] Natl. Univ. of Singapore Department of Mathematics [email protected] Itai Shafrir Techion [email protected] MS36 Composite Superconducting Systems PP0 The talk will deal with analytical results regarding surface Spectral-Element Simulations of Electrokinetic nucleation in superconducting samples in contact with a Fluid Flows in Nanochannels semiconductor or normal material, in the context of the Ginzburg-Landau theory. In particular, we will show that Micro- and nano-fluidic systems hold tremendous promise for a superconductor coated with a properly chosen semi- for separation and mixing of biochemical samples. Many conductor one should expect enhanced surface supercon- simple devices can be modeled using a two dimensional ductivity. In contrast, for an appropriate metal coating, approximation to the governing PDEs for electrokinetic the surface sheet can be completely suppressed. fluid flows. Modeling more sophisticated devices, however, will require the simulation of three-dimensional problems, Tiziana Giorgi which are computationally challenging to solve using tra- Department of Mathematics ditional finite-element methods. Spectral elements offer an New Mexico State University attactive alternative, especially because many commonly PD06 Abstracts 283

encountered problem geometries are highly regular. We ation at an Austenite-Martensite Interface present spectral-element simulations of several electrokinetic devices. A pseudo-spectral Fourier-Chebyshev collocation method for simulating a scalar two-dimensional viscoelastic dy- Jaydeep Bardhan namic model of microstructure creation with capillarity, 2 2 Electrical Engineering & Computer Science utt − βΔut =(3ux − 1)uxx + uyy − Δ u, is developed Massachusetts Institute of Technology and implemented in Matlab. By using an integration ma- [email protected] trix formulation for the Chebyshev modes, the numerical instability associated with large dimension Chebyshev dif- Sumita Pennathur ferentiation matrices is overcome. The integration matrices Sandia National Laboratories are sparse and so allow a large number of modes to be com- [email protected] puted, and hence for small values of the spatially regular- izing capillarity parameter to be used. The computational Kevin T. Chu results are compared to previous simulations without cap- Mechanical & Aerospace Engineering illarity. Princeton University Benson K. Muite [email protected] University of Oxford [email protected] PP0 An Ant Algorithm for the Multi-Row Layout Prob- PP0 lem in Automated/flexiable Manufacturing Sys- Symbolic Integration and Summation tems The homotopy algorithm is a powerful method for indefi- Multi-row layout problem is one of commonly used layout nite integration of total derivatives, and for the indefinite patterns in which each machine or facility is located in a summation of differences. By combining these ideas with coordinate system as (x,y). This layout pattern can used in straightforward Gaussian elimination, we construct an al- flexible/automated manufacturing systems. In this paper, gorithm for the optimal symbolic integration or summation this problem is formulated as a q.a.p. programming model. of expressions that contain terms that are not total deriva- An ant algorithm has been developed to solve this problem. tives or differences. The optimization consists of minimiz- An improvement heuristic called local search technique is ing the number of terms that remain unintegrated or are developed to obtain a betters solution. The performance not summed. Further, the algorithm imposes an ordering of the proposed heuristic is tested over a number of prob- of terms so that the differential or difference order of these lems selected from the literature. Solving these problems remaining terms is minimal. with proposed algorithm yields better solution compared to existing algorithm in this area. Bernard Deconinck, Michael A. Nivala University of Washington Amir Jafari [email protected], Urmia University [email protected] Departman of Mechanical Engineering AMIR JAFARI [email protected] PP0 PP0 The Billiard Problem in Nonlinear and Nonequi- librium System Mathematical Analysis to An Adaptive Network of the Plasmodium System We consider about the billiard problem in nonlinear and nonequilibrium system, whose behavior is similar to the Physarum polycephalum contains a tube network by means usual billiard problem at a glance. However it has some of which nutrients and signals circulate through the body. remarkable properties quite different from the usual one in When the organism is put in a maze and food sources are view of dynamical system. We reveal some of them by use presented at two exits, the tube network changes its shape of a reduction equation. We finally show that the strange to connect two exits through the shortest path. Recently, behavior comes from intermittent type Chaos possessed by a mathematical model for this adaptation process of path- the system by use of computer simulation. finding was proposed by Tero et al. We analyze this model for some special cases mathematically rigorously. Isamu Ohnishi Hiroshima University Isamu Ohnishi isamu [email protected] Hiroshima University isamu [email protected] Tomoyuki Miyaji Dept. of Math. and Life Sci., Tomoyuki Miyaji Hiroshima University Dept. of Math. and Life Sci., [email protected] Hiroshima University [email protected] PP0 PP0 Oil Production Linked to Stochastic Differential Equation Viscoelastic Dynamics for a Scalar Two Dimen- sional Continuum Model for Microstructure cre- Oil barrel price is a random variable that can be modeled by a stochastic differential equation. Here we illustrate a simple model for the net cash of the oil production based on 284 PD06 Abstracts

the oil well production, oil barrel price, production costs, initial costs, and the decline production in order to calculate the production to maximize the net cash. The optimization problem is solved by two approaches: a discrete method and one based on Markov controls. David E. Rivera-Recillas Instituto Mexicano del Petroleo [email protected]

Francisco Rocha-Legorreta Instituto Mexicano del Petr´oleo [email protected]

Israel Castillo-Nu˜nez Instituto Mexicano del Petroleo [email protected]

PP0 Double Reduction from the Association of Symme- tries with Conservation Laws: Applications to the Heat, Bbm, and One Dimensional Gas Dynamics Equations

A method to find a double reduction of partial differential equations (PDEs) with two independent variables is derived. The method is based on the definition of the association of Lie point symmetries of PDEs with conservation laws [Kara A.H. and Mahomed F.M.,The relationship between symmetries and conservation laws, Int. J. Theo- retical Phys. 39(1)(2000) pp24-30].

Applications to the linear heat equation,the BBM equation and a system of diﬀerential equations from one dimensional gas dynamics illustrate the new theory. Astri Sjoberg University of Johannesburg, Department of Applied Mathematics [email protected]

PP0 A Quasistationary Analysis of a Stochastic Chem- ical Reaction: Keizer’s Paradox Revisited

The deterministic and stochastic models of an autocat- alytic biochemical reaction are compared. The two models show distinctly different steady state behavior. To resolve this “paradox” proposed by J. Keizer, we compute the expected time to extinction and a quasistationary probability distribution in the stochastic model. Mathematically, we identify that exchanging the limits of infinite system size and infinite time is problematic. An appropriate system size for numerical computations is discussed. Hong Qian Department of Applied Mathematics University of Washington [email protected]

Melissa M. Vellela University of Washington Applied Mathematics [email protected]