AWM at JMM 2012

Special Session and Workshop Abstracts

Contents

Thursday January 5, 2012, 1:00 p.m.-3:50 p.m., Room 300, Hynes

AMS-AWM Special Session on Nonlinear Hyperbolic Partial Differential Equations, I

1:00 p.m. On the size of the Navier – Stokes singular set

Walter Craig*, McMaster University (1077-35-1222)

Consider the hypothetical situation in which a weak solution u(t, x) of the Navier-Stokes equations in three dimensions develops a singularity at some singular time t = T . It could do this by a failure of regularity, or more seriously, it could also fail to be continuous in the strong L2 topology. The famous Caffarelli Kohn Nirenberg theorem on partial regularity gives an upper bound on the Hausdorff dimension of the singular set S(T). We study microlocal properties of the Fourier transform of the solution in the cotangent bundle T ∗ (R3) above this set. Our first result is that, if the singular set is nonempty, then there is a lower bound on the size of the wave front set WF(u(T, .)), namely, singularities can only occur on subsets of T ∗ (R3) which are sufficiently large. Furthermore, if the solution is discontinuous in L2 we identify a closed subset S′(T) ⊆ S(T) on which the L2 norm concentrates at this time T. We then give a lower bound on the microlocal manifestation of this L2 concentration set, which is larger than the general one above. An element of the proof of these two bounds is a global estimate on weak solutions of the Navier-Stokes equations which have sufficiently smooth initial data.

1:30 p.m. Global well-posedness and decay for the viscous surface wave problem without surface tension

Ian T Tice*, Universite Paris-Est Creteil, LAMA (1077-35-1589)

We study the incompressible, gravity-driven Navier-Stokes equations in three dimensional domains with free upper bound- aries and fixed lower boundaries, in both the horizontally periodic and non-periodic settings. The effect of surface tension is not included. We employ a novel two-tier nonlinear energy method that couples the boundedness of certain high- regularity norms to the algebraic decay of lower-regularity norms. The algebraic decay allows us to balance the growth of the highest order derivatives of the free surface function, which then allows us to derive a priori estimates for solutions. We then prove local well-posedness in our energy space, which yields global well-posedness and decay. The novel LWP theory is established through the study of the linear Stokes problem in moving domains. This is joint work with Yan Guo.

2:00 p.m. Nonrelativistic Euler-Maxwell systems

Michael Sever*, The Hebrew University, Jerusalem, Israel (1077-35-141)

Construction of nonrelativistic Euler-Maxwell systems, candidates for MHD models, is reconsidered using previous results on characterization of Galilean symmetric approximations of Maxwell’s equations. In the context of a single fluid, the results are limited and disappointing. The Lundquist system, including the seemingly heroic expression for the electric field, all but necessarily results from the assumptions of Galilean symmetry and nonnegligible magnetic force on the fluid. However, the construction reveals an unexpected restriction on the applicability of the Lundquist model. At the expense of increased complexity, the difficulty is removed by consideration of a plasma model, including two fluids with charge per unit mass of opposite sign.

2:30 p.m. Broad Band Solitons in a Periodic and Nonlinear Maxwell System

Dmitry Pelinovsky*, McMaster University (1077-35-277)

We consider the nonlinear Maxwell equations with the small linear periodic refractive index. We show that the system of infinitely many coupled-mode equations for the Fourier amplitudes of counter-propagating waves cannot be truncated if no linear constant-coefficient dispersion is present. The new system of infinitely many coupled mode equations is analyzed for the existence of gap soliton solutions. We reduce it to an infinite system of coupled nonlinear Schro ̈dinger equations, for which we show the existence of coupled solitons by both Rayleigh-Ritz methods and numerical solution of the differential equations. Lifting the approximations of the coupled NLS solutions back to the coupled mode equations, we show that the broad band solitons are robust in the time-dependent computations. This is a joint work with Gideon Simpson (University of Toronto) and Michael Weinstein (Columbia University).

3:00 p.m. Blow Up of Solutions to the Generalized Proudman Johnson Equation

Alejandro Sarria, University of New Orleans;
Ralph Saxton*, University of New Orleans (1077-35-2274)

The inviscid Proudman Johnson equation provides a simple class of exact solutions for the incompressible, two-dimensional Euler equations and can be extended to allow similar classes of solutions to be constructed for higher dimensions. Further generalization leads to a rich variety in the evolution of solutions. In this talk, we will discuss background and new findings for the problem.

3:30 p.m. Two Phase Flow in Porous Media: the Saffman-Taylor Instability Revisited

Michael Shearer*, NC State University;
Kim Spayd, NC State University;
Zhengzheng Hu, NC State University (1077-35-1037)

Plane waves for two phase flow in a porous medium are modeled by the one-dimensional Buckley-Leverett equation, a scalar conservation law. We analyze linearized stability of sharp planar interfaces to two-dimensional perturbations, which involves a system of PDE. Numerical simulations of the full nonlinear system, including dissipation, illustrate the analytical results. We also discuss a modified Buckley-Leverett equation, in which the capillary pressure is rate-dependent, thereby adding a BBM-type dispersive term. This equation sustains undercompressive planar waves, but they are all unstable to two-dimensional perturbations.

Friday January 6, 2012, 8:00 a.m.-10:50 a.m, Room 300, Hynes

AMS-AWM Special Session on Nonlinear Hyperbolic Partial Differential Equations, II

8:00 a.m. Transonic flow in gas dynamics

Dehua Wang*, University of Pittsburgh (1077-35-296)

The mixed type problem of transonic flows past an obstacle will be considered. Recent results on the construction of global solutions will be presented. A connection with the isometric embedding problem in geometry will also be discussed.

8:30 a.m. A system of conservation laws with no classical Riemann solution. Existence of Dafermos profiles for singular shocks

Barbara Lee Keyfitz, The Ohio State University;
Charis Tsikkou*, The Ohio State University (1077-35-2501)

We consider a system of two equations derived from isentropic gas dynamics. We show that there is no classical Riemann solution and that singular shocks have Dafermos profiles.

9:00 a.m. Applications of Generalized Characteristics

Constantine M. Dafermos*, Brown University (1077-35-1049)

The lecture will present applications of the method of generalized characteristics to the study of the large time behavior of solutions of hyperbolic balance laws and to the theory of solutions of the Hunter-Saxton equation.

9:30 a.m. Singularity Formation in Nonstrictly Hyperbolic Equations

Katarzyna Saxton*, Loyola University, New Orleans (1077-35-2206)

We consider a 2 × 2 system for which, at some point (b, 0), the initial data intersect curves on which two characteristics coincide. Given that the system is genuinely nonlinear, one such curve on which the speed of both characteristics is zero will become the line x=b. We exam singularity formation along this line and prove that the solution breaks down in finite with or without damping. It is shown that, unlike the case for strictly hyperbolic systems, dissipation is not strong enough to preserve smoothness of small solutions globally in time. We will give an example of two further branches of curves starting at (b,0), in addition to the line x=b, where characteristics speeds are equal. The consequences of this phenomenon will be discussed.

10:00 a.m. On the transonic shocks of Euler-Poisson equations

Tao Luo*, Georgetown University (1077-35-582)

In this talk, I will present some results on the existence of unique and multiple transonic shock solutions and their stability for a system of Euler-Poisson equations. This is a joint work with Rauch, Xie and Xin.

10:30 a.m. Enhanced Lifespan of Smooth Solutions of a Burgers-Hilbert Equation

John K Hunter*, University of California at Davis;
Mihaela Ifrim, University of California at Davis (1077-35-2769)

We consider an initial value problem for an inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities in the two-dimensional flow of an inviscid, incompressible fluid. We use a normal form transformation, consisting of a near-identity transformation of the independent spatial variable, to remove the quadratic nonlinearity and prove the existence of small, smooth solutions on cubically nonlinear time-scales. For vorticity discontinuities, this result means that there is a cubically nonlinear time-scale before the onset of filamentation.

Friday January 6, 2012, 1:00 p.m.-5:50 p.m., Room 300, Hynes

AMS-AWM Special Session on Nonlinear Hyperbolic Partial Differential Equations, III

1:00 p.m. Self-similar solutions for the diffraction of weak shocks

Allen M. Tesdall*, College of Staten Island, City University of New York;
John K. Hunter, University of California, Davis (1077-35-1177)

We formulate a problem for the unsteady transonic small disturbance equations that describes the diffraction of a weak shock near a point where its strength approaches zero and the shock turns into an expansion wave. Physically, this problem corresponds to the reflection of a weak shock wave by a semi-infinite screen at normal incidence. We formulate the equations in self-similar variables, and obtain numerical solutions using high resolution finite difference schemes. Our solutions appear to show that the shock dies out at the sonic line, a phenomenon which has not been previously observed.

1:30 p.m. On 2D viscous Boussinesq system on a bounded domain

Ronghua Pan*, Georgia Institute of Technology (1077-35-2375)

2D viscous Boussinesq’s system models atmospheric and oceanographic turbulence, and the field of Buoyancy driven flows. The system is one of the most commonly used simplified model equations for 3D incompressible Navier Stokes equation, sharing the same vortex stretching effect. In this talk, I will review some recent progress on the global well-posedness, and large time behavior of the system on a bounded domain. The talk is based on joint work with M. Lai, K. Zhao, and with S. Bianchini.

2:00 p.m. A semi-hyperbolic region for the pressure gradient system

Kyungwoo Song*, Yeshiva University and Kyung Hee University;
Yuxi Zheng, Yeshiva University (1077-35-2678)

We consider solutions in regions including a semi-hyperbolic area in a self-similar plane. We give a talk on new results or developments on semi-hyperbolic patches using a pressure gradient system, which is a simple model of the two-dimensional compressible Euler system

2:30 p.m. Conservation laws with prescribed eigencurves

Helge Kristian Jenssen, Penn State University
Irina A Kogan*, North Carolina State University (1077-35-204)

We consider systems of hyperbolic conservation laws for n unknown functions in one space and one time variable. There is a local frame on ℝn, called eigenframe, associated with each system (consisting of the eigenvectors of the Jacobian matrix of the flux). The integral curves of such frame, called eigencurves, contain rarefaction curves and play an important role in solving the Cauchy problems for such systems.
In this talk, we explore the properties of a conservative system that are determined by the frame alone. Given a local frame on ℝn, what degree of freedom do we have, if we want to construct a system of conservation laws with this eigenframe? To what extent does a frame determine the number of companion conservation laws (entropies) associated with a system? A broader goal of this project is to obtain geometric classification of hyperbolic conservation laws that would lead to a better understanding of the properties of their solutions.

3:00 p.m. Some exact solutions to nonlinear hyperbolic PDE

Robin Young*, University of Massachusetts, Amherst (1077-35-632)

I shall present some interesting exact solutions to hyperbolic systems. These demonstrate some of the different phenomena that can occur as a result of interactions of waves with O(1) strength. I’ll present solutions of the compressible Euler equations which shed light on the vacuum as well as shock formation and cancellation, and includes a nontrivial space- periodic solution. I’ll also present an example of nonuniqueness of solutions without shocks for a system which is not in conservative form.

3:30 p.m. Shock Wave Stability for Conservation Laws with Physical Viscosities

Tai-Ping Liu, Academia Sinica and Stanford University;
Yanni Zeng*, University of Alabama at Birmingham (1077-35-1040)

We study the nonlinear stability of shock waves for conservation laws with physical viscosities. Suppose that the initial data is a small perturbation of a weak shock. We show that the solution to the Cauchy problem converges to a translated shock profile. Detailed pointwise estimates on the convergence are obtained. The strength of the perturbation and that of the shock are assumed to be small, but independent. Our assumptions on the viscosity matrix are general so that our results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. Our analysis depends on accurate construction of the approximate green’s function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that we can close the nonlinear term through the Duhamel principle.

4:00 p.m. Global dynamics of a diffuse interface model for solid tumor growth

John Lowengrub, University of California-Irvine;
Edriss S. Titi, University of California-Irvine and the Weizmann Institute of Science;
Kun Zhao*, University of Iowa (1077-35-228)

In this talk I will report recent progress on the rigorous analysis of a diffuse interface model which arises in modeling of spinodal decomposition in binary fluid in a Hele-Shaw cell, tumor growth and cell sorting, and two phase flows in porous media. We consider the system of partial differential equations in bounded domains in 2D or 3D. The system is supplemented by initial data and no-flux boundary conditions. The first part of the results is contributed to the existence, uniqueness and regularity of solutions to the initial-boundary value problem. First, it is shown that, for large data, strong solutions are globally (locally resp.) well-posed in 2D (3D resp.). Second, it is shown that strong solutions indeed possess the same regularity as regular solutions. Moreover, it is shown that solutions enjoy the Gevrey regularity within their life-spans. In the second part, the long-time asymptotics of the solutions is studied. It is shown that, in 2D and 3D, strong solutions converge to constant equilibria exponentially as time goes to infinity provided that the initial perturbations are small. On the other hand, for large initial perturbations, it is shown that the constant states are still global attractors of the model under mild conditions on the volume of domain.

4:30 p.m. Vegetative Pattern Formation Model Systems: Comparison of Turing Diffusive and Differential Flow Instabilities

Bonni J Kealy*, Washington State University;
David J Wollkind, Washington State University (1077-35-244)

A particular interaction-diffusion plant-surface water model system for the development of spontaneous stationary vegeta- tive patterns in an arid flat environment is investigated by means of a weakly nonlinear diffusive instability analysis. The main results of this analysis can be represented by closed-form plots in the rate of precipitation versus the specific rate of plant loss parameter space. From these plots, regions corresponding to bare ground and vegetative patterns consisting of tiger bush, labyrinth-like mazes, pearled bush, irregular mosaics, and homogeneous distributions of vegetation may be identified in this parameter space. Then those Turing diffusive instability predictions are compared with both relevant observational evidence and existing numerical simulations involving differential flow migrating stripe instabilities for the associated interaction-dispersion-advection plant-surface water model system.

5:00 p.m. A new result in blow-up for long-wave unstable thin film equations

Marina Chugunova, University of Toronto
Mary C. Pugh*, University of Toronto
Roman Taranets, University of Nottingham (1077-35-2217)

This talk will provide an introduction to long-wave unstable thin film equations of the form

ut = −(unuxxx)x − B(umux)x.

The exponents n and m determine whether or not finite-time blow-up of the solution might occur. In this talk, we present new results for the critical (n = m + 2) and supercritical cases (m > n + 2) on the line.

5:30 p.m. Structure of solutions near sonic line for pressure gradient equation

T. Zhang*, Penn State University
Y. Zheng, Yeshiva University (1077-35-2520)

We consider the two-dimensional pressure gradient system in the self-similar plane and show that a local smooth solution exists extending from a sonic line towards the hyperbolic region with given boundary conditions on the sonic line. Characteristics are only Ho ̈lder continuous at the sonic line. We introduce a new coordinate system, that involves the state variable pressure p as well as the self-similar ones, so that the characteristics become smooth and terms of singularity are shifted to within the lower order terms in the characteristic form of the equation. We establish the existence of a local smooth solution by showing that an iteration sequence converges under a new metric.

Saturday January 7, 2012, 8:30 a.m.-10:20 a.m., Room 306, Hynes

AWM Workshop: Research Presentations by Recent Ph.D.s, I

8:30 a.m. On Legendrian Graphs

Danielle O’Donnol, Smith College
Elena Pavelescu*, Occidental College (1077-57-263)

We investigate Legendrian graphs in (R3, ξstd). We extend the classical invariants, Thurston-Bennequin number and rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with tb = −1 and rot = 0 if and only if it does not contain K4 as a minor. There are many examples of knots and links which are characterized up to Legendrian isotopy by the pair (tb,rot). We ask what graphs are characterized up to Legendrian isotopy by the pair (tb, rot) and give several examples.

9:00 a.m. Springer’s representation via sl(3) webs

Heather M. Russell*, University of Southern California;
Julianna S. Tymoczko, University of Iowa;
Matthew Housley, BYU (1077-57-240)

Springer varieties are a special collection of flag varieties whose homology carries an action of the symmetric group with the top-dimensional homology an irreducible representation. For a fixed integer n, there is a Springer variety for each partition of n. In the case of two-element partitions, Springer varieties have important connections to Khovanov’s theory of categorified tangle invariants. In past work, we use this to explicitly construct the Springer representation for two-element partitions of n. This construction operates skein-theoretically on crossingless matchings.
This simple, knot-theoretic approach can be used once again to construct the Springer representation in the case of three-element partitions of n. In this setting embedded, trivalent, directed graphs called sl(3) webs replace crossingless matchings, and the skein relation is more interesting. We will show how certain combinatorial data about symmetric group representations are naturally and intuitively encoded in this knot-theoretic framework.

9:30 a.m. Categorification of the polynomial ring

Radmila Sazdanovic*, University of Pennsylvania;
Mikhail Khovanov, Columbia University (1077-18-212)

We introduce a categorification of the one-variable polynomial ring Z[x], based on the geometrically defined graded algebra and show how to lift various operations on polynomials to the categorified setting. This construction generalizes to cate- gorification of various orthogonal polynomials, including Chebyshev polynomials and the Hermite polynomials.

10:00 a.m. Promotion and rowmotion

Jessica Striker*, University of Minnesota;
Nathan Williams, University of Minnesota (1077-05-256)

We present an equivariant bijection between two actions—promotion and rowmotion—on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and recent work of D. Armstrong, C. Stump, and H. Thomas on root posets and noncrossing partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to certain objects under rotation. In particular, we obtain an equivariant bijection between plane partitions of height two under rowmotion and noncrossing partitions with a fixed number of blocks under rotation.

Saturday January 7, 2012, 2:30 p.m. − 4:20 p.m., Room 306, Hynes

AWM Workshop: Research Presentations by Recent Ph.D.s, II

2:30 p.m. Algebraic theory for discrete models in systems biology

Franziska Hinkelmann*, Mathematical Biosciences Institute, Ohio State University;
Reinhard Laubenbacher, Virginia Bioinformatics Institute, Virginia Tech (1077-92-203)

Systems biology aims to explain how a biological system functions by investigating the interactions of its individual components from a systems perspective. Modeling is a vital tool as it helps to elucidate the underlying mechanisms of the system. Many discrete model types can be translated into the framework of polynomial dynamical systems (PDS), that is, time- and state-discrete dynamical systems over a finite field where the transition function for each variable is given as a polynomial. This allows for using a range of theoretical and computational tools from computer algebra, which results in a powerful computational engine for model construction, parameter estimation, and analysis methods.

3:00 p.m. Theories without the independence property

Lynn C. Scow*, University of Illinois at Chicago (1077-03-246)

A first-order theory has the independence property (IP) if there is some formula in the language of the theory that defines a random graph relation on tuples from any saturated enough model of the theory. In [Laskowski, 1992], it was shown that a theory fails to have the independence property just in case any uniformly definable class of subsets is a Vapnik-Chervonenkis class. Given a set X, a class of subsets ? is a Vapnik-Chervonenkis class if for any finite set F ⊂ X, intersections C ∩ F with C ∈ ? give less than the maximal number of subsets of F . We will use very homogeneous sequences indexed by ordered graphs in order to characterize this property.

3:30 p.m. A Functional Central Limit Theorem for Empirical Processes

Cristina Tone*, University of Louisville (1077-60-247)

We introduce a functional central limit theorem for empirical processes endowed with real values from a strictly stationary random field satisfying an interlaced mixing condition. We proceed by using a common technique from Billingsley (1999), by first obtaining the limit theorem for the uniformly distributed case. We then generalize the result to the case where the absolutely continuous marginal distribution function is no longer uniform. In this case we show that the empirical process endowed with values from the ρ′-mixing stationary random field, due to the strong mixing condition, doesn’t converge in distribution to a Brownian bridge, but to a continuous Gaussian process with mean zero and the covariance given by the limit of the covariance of the empirical process.

4:00 p.m. A fluid-structure interaction problem for a supercavitating elastic curvilinear foil

Anna Zemlyanova*, Texas A&M University (1077-76-253)

A problem of a fluid-structure interaction is considered for a curvilinear supercavitating thin elastic foil in a stream of ideal fluid. The Tulin single-spiral-vortex model is used to describe the closure of the cavity. The problem is decoupled with the help of the method of successive approximations. The conformal mapping method together with the Riemann-Hilbert approach is employed for the solution of the fluid mechanics problem. The equations of the thin shell theory are used to describe the deformation of the curvilinear foil. The covergence of the method of successive approximations is investigated. The numerical results including the cavity profile and the displacements of the elastic foil are obtained.