CAMP/Nonlinear PDE's Seminar

Wednesdays at 3:30 PM in Eckhart 202.

Fall 2006 Schedule

September 20,
Francois Hamel, Universite Aix-Marseille III
Generalized bistable fronts passing an obstacle

In this talk, I will present the notions of generalized travelling fronts and some of their properties. Then I will explain the construction of generalized bistable reaction-diffusion fronts passing an obstacle. These fronts are almost-planar but because of the presence of the obstacle, they cannot be covered by the usual notions of fronts. Actually, the construction is possible when the obstacle is star-shaped. We will see that this condition plays a role in the determination of the large-time behavior of the solutions. It is related to a Liouville type result for the associated stationary problem. This talk is based on joint works with H. Berestycki, and H. Berestycki and H. Matano

September 27,
Gregory Lawler University of Chicago
Laplacian random motion

The Laplacian random walk with exponent b is the discrete process that grows at the tip with probabilities weighted by harmonic measure to the b power. The special case b=1 is also called the loop-erased random walk. We discuss possible continuum limits for this process in two dimensions. We show that, under the (unproved) assumption of a conformally invariant continuum limit, the limit must a Schramm-Loewner evolution (SLE). We also determine the parameter of the SLE as a function of b. In the case b=1, the existence of such a conformally invariant limit was proved a few years ago my Schramm, Werner, and me. Previous knowledge of SLE will not be assumed.

September 28, 3pm in E206
Boris Gershgorin RPI
The effects of Discretness and Resonances in beta-FPU chains

We study the statistical behavior of the 1D nonlinear \beta-FPU chain. When the nonlinearity is weak the thermal equilibrium state of the system can be described as weakly interacting linear waves, and the wave action n_k is proportional to 1/w_k with w_k - linear dispersion (Rayleigh-Jeans distribution). We demonstrate numerically that surprisingly even in the strongly nonlinear limit the system still can be effectively described like weakly interacting waves and hence the same distribution holds. This arises because strong nonlinearity effectively renormalizes the linear dispersion frequency. We find the theoretical prediction for the frequency renormalizing factor and show that it is in good agreement with numerical experiments. Moreover the presence of nonlinearity widens the resonance peaks which results in the occurrence of near-resonance wave-wave interactions. Such near-resonant interactions allow the application of the well developed Wave Turbulence formalism. We also employ Random Phase Approximation and use the Wiener-Khinchin formula and the perturbation methods to derive the form of the resonance width as a function of the wave number and the nonlinearity strength. The resulting analytical prediction for the resonance width is in good agreement with numerical experiments. Finally we numerically observed that in the thermodynamical equilibrium \beta-FPU exhibits spatially localized oscillations - Discrete Breathers.

October 4,
Laurent Demanet, Stanford University
Fast Computation of Fourier Integral Operators

Fourier integral operators are oscillatory integrals related to linear wave propagation. They are a central object in most imaging problems involving waves, like seismic or ultrasound imaging; and also in some tomography problems, like electron microscopy. In most cases, the quality of imaging hinges on the practicality of applying these oscillatory integrals on a large computational scale. In this talk, I will present a fast algorithm for the computation of large classes of Fourier integral operators. The algorithm is based on factorization and separation of the oscillatory kernel over adequate angular neighborhoods in the frequency variable, and inspired by modern constructions in applied harmonic analysis. We prove that the complexity of computing and applying a Fourier integral operator on an N-by-N grid, is O(N^{2.5} log N) in time, and as low as O(sqrt N) in storage. The constants in front of these estimates are small and depend weakly on the desired accuracy. We illustrate the properties and potential of the algorithm with several numerical examples.

October 11,
Xavier Cabre, Universitat Politecnica de Catalunya
A priori bounds for stable solutions of nonlinear elliptic equations in low dimensions

Abstract: We consider the class of semi-stable solutions of semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain of $\R^n$ (sometimes convex). This class includes all local minimizers, minimal, and extremal solutions. In dimensions $n\le 4$, we establish a priori $L^\infty$ bounds which hold for every semi-stable solution and every nonlinearity $f$. This extends previous work of G.~Nedev for dimensions $n\le 3$. In relation with these results, A.~Capella and the author have studied the particular case of radial solutions, for which boundedness holds up to dimension $n\le 9$ for every~$f$. Some quasilinear analogues have also been established, both in the radial and nonradial cases, by A. ~Capella, M.~Sanch\'on, and the author.

October 18,
Scott Sheffield, New York University
Random geometry and SLE

We introduce and construct the "AC geometry" from the Gaussian free field and use it to prove various facts about Schramm-Loewner evolutions. (See this for a graphical description of what an AC geometry is.)

November 1,
Peter Constantin, University of Chicago
Transport and Dissipation: Statistical Solutions, Stochastic Representation

I will describe two results. The first one is about the absence of anomalous dissipation in damped, driven 2D NSE equations. The other is about decay of generalized relative entropies in Fokker-Planck equations. The title of the talk refers to the tools used to prove the two results. The first result is based on joint work with Fabio Ramos, the second on work with Gautam Iyer.

November 8,
Joseph Biello, University of California, Davis
PDE's and Waves for the tropical atmosphere: The Madden-Julian Oscillation and a new nonlinear wave theory

I shall discuss two new asymptotic regimes for the nonlinear PDEs governing the tropical atmosphere. Using systematic multiscale asymptotics, we arrive at an asymptotic closure for the ideal fluid equations governing dynamics on large scales in the tropical atmosphere. By selecting a plausible analytic model for smaller scale flows in the tropics, we predict the large scale structure of the Madden-Julian oscillation; this is a planetary scale organization of winds, the understanding of which has been called "the holy grail" of tropical meteorology. In the second problem, we study the same equations, but over longer time and spatial scales. The resultant coupled nonlinear dispersive equations for the amplitudes of interacting wave packets are novel both from the perspective of the atmospheric sciences and from a more general mathematical setting. These equations describe the influence of large scale tropical waves on midlatitude waves and, in particular, are relevant for understanding the effect of the Madden-Julian oscillation on midlatitude weather. Furthermore, the amplitude equations have a Hamiltonian structure and admit analytic solitary wave solutions.

November 9, THURSDAY
Tomasz Szarek, Silesian University, Katowice
Lower bound technique in the theory of random dynamical systems

Abstract: Lower bound technique appears to be a very useful tool in the theory of Markov operators. In particular A. Lasota and J. Yorke used it to prove the existence of an absolutely continuous measure for the Frobenius--Peron operators corresponding to piecewise monotonic transformations. Recently this technique has been extended to general Markov operators including those describing the evolution of measures for iterated function systems. Quite recently it has been shown its utility in the theory of SPDEs. In this talk, making use of lower bound technique, we are going to formulate criteria for the existence of an invariant measure on general complete separable metric spaces. Criteria for ergodicity of invariant measures will be given also.

November 15,
Chris Jones, University of North Carolina
Ocean Data Assimilation: Issues and Strategies for Lagrangian Observations

To accurately predict future ocean states, it behooves us to use all available observational data. Much of the sub-surface data in the ocean comes from Lagrangian instruments which are assumed to follow fluid particle trajectories. Model and observation errors are taken to be random processes and filtering methods are used to incorporate data into a re-initialization of the state. Special considerations arise for Lagrangian data as they are not given directly in terms of the state variables. Some methods will be discussed for dealing with this issue, and ideas coming from dynamical and stochastic systems will then be invoked to shed light on both the best approaches and the most useful data.

November 16, THURSDAY 3pm in E312
Weinan E, Princeton University
An overview of multiscale modeling

I will give an overview of a program on building a mathematical theory of crystalline solids, starting from atomistic models. I will discuss what the crucial issues are. I will start by reviewing the geometry of crystal lattices, the quantum as well as classical atomistic models of solids. I will then focus on a few selected problems: (1) The crystallization problem -- why the ground states of solids are crystals and which crystal structure do they select? (2) stability of crystals; (3) instability of crystals; (4) the generalized Peierls-Nabarro model for defects in crsytals.

November 22
Jiahong Wu, Oklahoma State University
Nonlinear Partial Differential Equations with Fractional Diffusion

Nonlinear partial differential equations with fractional diffusion arise in numerous seemingly diverse fields such as fluid flow, material viscoelastic theory, electromagnetic theory, astrophysics, control theory of dynamical systems, economics and so on. This talk focuses on two important nonlinear PDEs with fractional diffusion: the 2D dissipative quasi-geostrophic (QG) equation and the generalized incompressible magnetohydrodynamics (MHD) equations. We will summarize some of the recent progress made by Kiselev, Nazarov and Volberg and by Caffarelli and Vasseur on the QG equation with the critical index and discuss the applicability of the approach of Caffarelli and Vasseur to the supercritical case. For the generalized MHD equations, we present a regularity criterion that imposes conditions on the velocity alone. This criterion indicates that the global regularity issue concerning the MHD equations may not necessarily be more difficult than the same issue on the Navier-Stokes equations.

November 29,
Phil Morrison, University of Texas
Vlasov Plasma and Fluctuation Spectra

November 30, THURSDAY 4pm in E312
Jean-Michel Roquejoffre, Universite Paul Sabatier, Toulouse
Oscillatory dynamics of a reaction-diffusion system modelling a burner flame

A burner flame can be described by the usual one-dimensional thermo-diffusive model for flame propagation. The model comprises an equation for the temperature, posed on the half-line, and an equation for the mass fraction of the reactant, posed on the whole real line. This peculiarity entails~instability properties of the steady solutions. In the large activation energy framework, G. Joulin (1982) derives, in a formal fashion, a differential-delay equation accounting for the unstable~dynamics of the flame front. The goal of the talk is to discuss a mathematically rigorous proof of this result.

December 5, TUESDAY 3pm in E202
Vered Rom-Kedar, The Weizmann Institute, Israel
G-CSF control of neutrophil dynamics in the blood

White blood cell neutrophil is a key component in the fast initial immune response against bacterial and fungal infections. In oncological practice it is often necessary to predict and prevent the infectious complications that follow chemotherapy induced neutropenia (dangerously low levels of neutrophils in the blood). Granulocyte colony stimulating factor (G-CSF) which is naturally produced in the body, controls both the neutrophils production in the bone marrow and the neutrophils delivery into the blood. G-CSF injections are widely used to prevent and treat neutropenia. However, the optimal schedule and intensity of the G-CSF application has not been fully determined. We develop a robust two-dimensional ordinary differential equation model which accurately mimics the clinically observed G-CSF - neutrophil dynamics on time scales of several days.The resulting model is structurally stable, and thus its dynamical features are independent of the precise form of the various rate functions. Choosing a specific form for these functions, three complementary parameter estimation procedures are examined. Fitting the 6 emerging parameters on one clinical data set (training set), the model supplies good predictions of the other available clinical data sets. The simplicity and robustness of the model are key ingredients to its application in the management of individual oncological patients under relevant clinical conditions.      Joint work with  E. Shochat  and Prof. L. Segel (Late)  

December 6,
Steve Shkoller, University of California, Davis
Well-posedness for the free-boundary 3D Euler equations with or without surface tension

The free-boundary Euler equations describe the motion of perfect fluids with free surfaces or with moving material interfaces. The nonlinear a priori estimates for this system rely heavily on the geometric structures present in these equations, and it turns out that finding approximations which do not destroy this structure, and for which existence and uniqueness can be proven, is somewhat difficult. I will discuss the geometric a priori estimates for Euler, as well as a new method for well-posedness that relies on a "convolution by horizontal layers" smoothing operator as well as artificial viscosity in the case of surface tension. No irrotationality assumptions are made so that we can study general fluid motion problems, and in particular, the effect of vorticity on boundary shape, and vice versa.

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