# Reception and poster session

Tuesday, September 25, 2012 - 4:05pm - 5:30pm

Lind 400

**Nonlinear Eigenvalues, Keldysh's Theorem, and the Evans function**

Yuri Latushkin (University of Missouri)

We show how to use an abstract Keldysh's Theorem (that gives a formula for the inverse of a holomorphic operator valued function) in order to look for the number and location of unstable eigenvalues of differential operators that appear when one linearizes partial differential equations about such special solutions as traveling waves. Relations with the Evans function are also described.**Traveling spots and rotating spirals of the wave front interaction model**

Hirokazu Ninomiya (Meiji University)

For example, we observe the spiral waves and target patterns in Belousov-Zhabotinskii reaction. In photosensitive BZ reaction with feedback, the locallized patterns are also observed. Many researchers have regarded these patterns by the curves without thickness and have treated such patterns mathematically. In this presentation, using the wave front interaction model proposed by Zykov and Showalter, we show the existence of traveling spots in the plane and the rotating waves in the disk as two dimensional patterns. I will also explain the existence of the rotating spirals.

This talk is based on several join works with Professors J.-S. Guo, J.-C. Tsai, C.-C. Wu and my Ph.D student Y.Y. Chen.**Vortex Crystals in Fluids: Exploring the Inviscid and Weakly Viscous $N$-Vortex Problems**

Anna Barry (University of Minnesota, Twin Cities)

Vortex crystals are configurations of vortices that maintain their basic shapes for long times, while perhaps rotating or translating rigidly in space. It is common to observe these patterns in nature and experiments, where the underlying dynamics are governed, for example, by the two-dimensional Euler or Navier-Stokes equations. One can often take advantage of the localized structure of vortices to derive equations of motion that are much simpler than the original PDE. In this work, we use a point vortex model and a generalized point vortex model (the latter allowing for positive core size) to study the existence and stability of a number of inviscid and weakly viscous vortex crystals, respectively. We complement this analysis with comparisons of our results to crystals observed in an electron column experiment as well as in hurricane eyes and related numerical simulations.**Computing Foliaitons, Inertial Manifolds, and Tracking Initial Conditions with FOIL8PAK**

Michael Jolly (Indiana University)

Foliations provide a nonlinear coordinate system that decouples

two variables in an ODE or PDE. Through each (base) point in

phase space there is a pair of leaves: one each in the (un)stable foliation.

If the base point is a steady state, the leaves are the

invariant stable and unstable manifolds. Under a spectral gap condition

the unstable manifold is an inertial manifold. The intersection of a

leaf in the stable foliation and the inertial manifold is the unique

initial condition through which the trajectory attracts at

an exponential rate the trajectory through the base point of the leaf.

We present a unified approach to compute all these objects.

A joint work with Yu-Min Chung.**Grain Boundaries in Swift-Hohenberg equations**

Qiliang Wu (University of Minnesota, Twin Cities)

We study grain boundaries in the Swift-Hohenberg equation. Grain boundaries arise as stationary

interfaces between roll solutions of dierent orientation. Our analysis shows that such stationary

interfaces exist near onset of instability for arbitrary angle between the roll solutions. The main difficulty stems from possible interactions of the primary modes

with other resonant modes. We develop a normal form analysis and a singular

perturbation approach to treat resonances**Pulse generators as a converter from time-periodic motion to**

spatially periodic structure

Heterogeneity is one of the most important and ubiquitous types of external perturbations. We

study a spontaneous pulse generating mechanism caused by the heterogeneity of jump type. Such

a pulse generator (PG) has attracted much interest in relation to potential computational abilities

of pulse waves in physiological signal processing. We investigate firstly the conditions for the onset

of PGs, secondly we show the bifurcational origin of their complex ordered sequence of generating

manners. To explore the global bifurcation structure of heterogeneity-induced ordered patterns

(HIOPs) including PGs, we devise the numerical frameworks to trace the long-time behaviors of PGs

as periodic solutions and detect the associated terminal homoclinic orbits those are homoclinic to

one type of HIOPs, which possesses a property of hyperbolic saddle. Such our numerical approaches

clarify a canditate for the organizing center producing a variety of PGs which is closely analogous

to the unfolding of homoclinic gluing structure in vector fields. This

is a joint work with Masaki Yodome and Takashi Teramoto.**Global Bifurcation Picture for Bound States in Nonlinear**

Schroedinger Equations

Eduard-Wilhelm Kirr (University of Illinois at Urbana-Champaign)

An open problem in

nonlinear dispersive equations is the asymptotic completness conjecture

which states that any initial data eventually converges to a

superposition of coherent states. A big obstacle in solving the

conjecture is the fact that the set of coherent states is not completely

known. I will present recent results which not only find all coherent

(bound) states in one dimensional nonlinear Schroedinger equation, and

analyse their stability but also makes significant progress in higher

dimensions. This is joint work with Vivek Natarajan (U. of Illinois).**Mathematical Model and Analysis for the Regulation of Kinase Activity on the Cell of Finite Cylinder**

Puchen Liu (University of Houston)

Regulatory network process cellular signals in time and space enabling cell self-organization. Kinase cascades can emerge from receptors and transmit signals from the cell membrane to the nucleus. Usually, for the simplicity, the biology analyst assumes that the kinase activity happens on the cell with shape of the ball. But as biology shows, in reality, there are many cells with the shape of finite cylinder. In this work, we just propose the model of kinase activity on the cell of finite cylinder. ONE QUENSTION is that how to describe the kinase activity on the cell surface i.e mathematically how to impose the boundary conditions. Inspired by the case of neuron and its long axon, we only give the nonzero flux of the kinase on one of circular faces of the cylinder. Based on the model, we provide the steady state solution, i.e the concentration of the kinase and the analysis on how the kinase concentration varies according to the change of the parameters (diffusion coefficient, reaction rate etc.). We demonstrate the critical value (bifurcation value) of some parameter and the bifurcation from the steady state zero solution. We also give the axially symmetric solution (concentration) of the steady state equation in series form. Our partial parameter analysis shows that the concentration of kinase on the finite cylinder is probably not the monotone function of the parameter (dephosphorylation rate) as that the case with the cell of the ball shows.**Viscous Profiles for Shock Waves in Isentropic Magnetohydrodynamics**

Andreas Klaiber (Universität Konstanz)

Standing planar waves in isentropic magnetohydrodynamics are governed

by a five-dimensional gradient-like system, which depends

parametrically on two constants of integration and three viscosity

coefficients. Depending on the constants of integration, there exist

up to four rest points. A heteroclinic orbit connecting two such rest

points of this system corresponds to a viscous profile for the

standing shock wave with these rest points as end states.

In the typical case that there are four rest points we show how the

respective ratios of the viscosity coefficients determine which of the

rest points are connected by heteroclinic orbits. We have used the

Conley index and geometric singular perturbation theory to prove the

results.**Oscillons near Hopf bifurcations**

Kelly McQuighan (Brown University)

Oscillons are planar, spatially localized, temporally oscillating, radially symmetric structures. They have been observed in various experimental contexts, including fluid systems, granular systems, and chemical systems. Oscillons often arise near forced Hopf bifurcations, which are modeled mathematically with the forced complex Ginzburg-Landau (FCGL) equation. We present a proof of the existence of oscillons in the forced planar complex Ginzburg-Landau equation through a geometric blow-up analysis. Our analysis is complemented by a numerical continuation study of oscillons in the forced Ginzburg-Landau equation using Matlab and AUTO.**Large time existence of strong solutions to the micropolar fluid equations in cylindrical domains**

Bernard Nowakowski (University of Warsaw)

Micropolar fluid equations, which were introduced by Eringen in 1966, are a significant step toward generalization of the standard Navier-Stokes model. By their very nature, they take into account the structure of the media they describe, which plays crucial role in modeling for some well-known fluids, e.g. animal blood or liquid crystals. In particular, the deviancy from the Navier-Stokes model becomes highly apparent in micro-scale when at least one dimension of the domain is only a few times larger than the size of the molecules (e.g. blood vessels, lubricants). Although not all the aspects of physical experiments have been fully explained, it is justified to assume that the surface stresses and the internal degrees of freedom of particles are the deciding factors for properties of fluid motion.

In this work we show the existence of strong solutions to micropolar fluid equations in cylindrical domains. We do not assume any smallness on the initial or the external data but only for their derivative alongside the axis of the cylinder.**A correspondence between the dynamics of parabolic PDE's and the ones of ODE's**

Romain Joly (Université de Grenoble I (Joseph Fourier))

This poster introduces a striking correspondence between the dynamics of the scalar parabolic equations and the ones of ODE's. It presents a list of dynamical properties of the PDE, depending of the domain and symmetries of the non-linearity, and draws a parallel with the dynamical properties of some classes of ODE's. This correspondence is helpful for predicting which dynamical behaviours can be expected in some classes of scalar parabolic equations. Several recent results of this poster have been obtained in joint works with G. Raugel and P. Brunovsky.**Global existence and uniqueness of a three-dimensional model of cellular electrophysiology**

Hiroshi Matano (University of Tokyo)Yoichiro Mori (University of Minnesota, Twin Cities)

We study a three-dimensional model of cellular electrical activity,

which is written as a pseudodierential equation on a closed surface \Gamma

in R^3 coupled with a system of ordinary dierential equations on \Gamma.

We prove global existence/uniqueness for this system for a large class

of nonlinearities including the FitzHugh-Nagumo and the Hodgkin-

Huxley kinetics. One of the main diculties is the absence of a max-

imum principle, which we overcome in part by introducing the notion

quasipositivity. We also discuss asymptotic smoothing and the depen-

dence of the solution on the size of the domain.**Invariance of the Gibbs measure for the periodic quartic gKdV.**

Geordie Richards (University of Minnesota, Twin Cities)

We prove invariance of the Gibbs measure for the (gauge transformed) periodic quartic gKdV. The Gibbs measure is supported on H^s for s less than 1/2, and the quartic gKdV is analytically ill-posed in this range. In order to consider the flow in the support of the Gibbs measure, we combine a probabilistic argument with the second iteration and construct local-in-time solutions to the (gauge transformed) quartic gKdV almost surely in the support of the Gibbs measure. Then, we use Bourgain's idea to extend these local solutions to global solutions, and prove the invariance of the Gibbs measure under the flow. Finally, Inverting the gauge, we construct almost sure global solutions to the (ungauged) quartic gKdV below $H^{1/2}$.**Slowly Non-Dissipative PDEs and their Non-Compact Global Attractors**

Nitsan Ben-Gal (University of Minnesota, Twin Cities)

This poster will present recent results in the area of global attractor theory for two types of slowly non-dissipative PDEs. We use a variety of PDE and dynamical systems techniques to obtain explicit decompositions of the non-compact global attractor structures for classes of asymptotically symmetric and asymptotically asymmetric grow-up equations. We present a number of these techniques, and address how inertial manifold theory, the Fučik spectrum, Poincaré compactification and the Conley index at infinity allow us to analyze behavior approaching and within the realm at infinity.**Accelerated Fronts in a two stage invasion problem**

Matt Holzer (University of Minnesota, Twin Cities)

We study wavespeed selection in a staged invasion process. That is, we study a model in which an unstable homogeneous state is replaced via an invading front with a secondary state. This secondary state is also unstable and, in turn, replaced by a stable homogeneous state via a secondary invasion front. In particular, we are interested in the selected wavespeed of the secondary front and under what conditions does the in uence of the primary front increase this speed. We nd three regimes: a locked regime where both fronts travel at the same speed, a pulled regime where the secondary front travels at the linear spreading speed associated to the intermediate state, and an accelerated regime where the selected speed is between these two speeds. We show that the transition to locked fronts can be described by the crossing of a resonance pole in the linearization about the primary front. In addition, we use properties of this resonance pole to derive an estimate for the selected wavespeed in the accelerated case and to determine the transition between accelerated and pulled fronts**Linear Stability of Subsonic Solitary Waves for the one-dimensional Benney-Luke Equation**

Milena Stanislavova (University of Kansas)

The linear stability for the subsonic solitary waves of the one-dimensional Benney-Luke equation is investigated rigorously and the critical wave speeds are computed explicitly. This is achieved via the abstract stability criteria recently developed by Stanislavova and Stefanov.**Traveling waves in high Lewis number combustion model**

Anna Ghazaryan (Miami University)

High Lewis number combustion model is a model that is used to describe

propagation of fronts in the process of burning of high density liquid

fuels. Both existence and stability of such fronts are important

issues in applications and, at the same time, are challenging

mathematical problems. I will present new results that prove the

existence of fronts and show that, depending on the value of the

exothermicity parameter, these fronts can be either convectively or

absolutely unstable.**On models of short pulse type in continuous media**

Yannan Shen (University of Minnesota, Twin Cities)

We consider short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. Assuming general Kramers-Kronig or Sellmeier formulas for the permittivity and permeability, we derive two short-pulse equations (SPEs) for the high- and low-frequency band gap respectively in one space dimension. Then we generalize this model into two dimensional case, and give two 2D SPEs, we will discuss robustness of line breather solutions, and evolution of localized 2D initial data.**Maximum Principle and Symmetry results for Viscosity Solution of Fully Nonlinear Equations**

Our paper is concerned about maximum principle and radial symmetry of viscosity solution for fully nonlinear equation. We obtain the radial symmetry for non-proper fully nonlinear elliptic equation in Euclidean space under certain asymptotic rate at infinity. New maximum principles for fully nonlinear equation are established. Our results apply to Pucci's extremal operators, Bellman or Isaccs equations. Radial symmetry and the corresponding maximum principle in a punctured ball are shown. We also study the symmetry of viscosity solution for fully parabolic equations.**KAM theory for conformally symplectic systems**

Renato Calleja (University of Minnesota, Twin Cities)

We present a KAM theory for systems that transport a symplectic form into a multiple of itself. The KAM theorem is written in an a-posteriori format and has many practical consequences. The theorem allows to produce numerical schemes that are remarkably efficient and work close to the breakdown of analyticity.**Invariant Measures for Dissipative Dynamical Systems:**

Abstract Results and Applications

Nathan Glatt-Holtz (University of Minnesota, Twin Cities)

We establish the existence of certain invariant probability measures that can be associated with the time averaged observation of a broad and generic class of dissipative dynamical systems via the notion of a generalized Banach limit. This work extends a framework originally developed for the study of statistically stationary solutions of the Navier-Stokes equations.

We apply these abstract results by considering some concrete dynamical systems where the semigroup is known to be non-compact thus addressing examples not covered under previous results in this direction.