UIUC Dept. of Mathematics Seminar Calendar

      July 2009             August 2009           September 2009   
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                        30 31

Tuesday, September 1, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, September 1, 2009

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Submitted by dhundert.

Johannes Zimmer (Department of Mathematics, University of Bath)
Moving interfaces in solids: from conservative lattice models forphase transitions to macroscopic dissipation
Abstract: Moving interfaces in solids can dissipate energy, but are commonly described by conservative (Hamiltonian) equations on the lattice scale. How can conservative lattice models generate dissipation on the continuum scale? To understand this, we consider a model problem, namely that of a moving phase boundary in a solid (the talk will start with a short survey on phase transitions in solids). A well-accepted microscopic model is then that of a one-dimensional chain of atoms with nearest neighbor interaction. To describe phase transitions, the elastic potential is chosen to be nonconvex; we will consider a piecewise quadratic energy with two wells. A simple solution class describing the motion of an interface is then the class of travelling waves. A solution which explores both wells of the energy will have an interface, moving with the speed of the wave. We show that for suitable fixed subsonic velocities, there is a family of ``heteroclinic" travelling waves (heteroclinic means here that they connect both wells of the energy). Though the microscopic picture is Hamiltonian, we derive non-trivial so-called kinetic relations on the continuum scale; they can be related to the dissipation generated by a moving phase boundary. We then investigate the question of when the kinetic relation does not vanish (dissipation is generated). It turns out that a microscopic asymmetry determines here the macroscopic dissipation. This is joint work with Hartmut Schwetlick (Bath). If time permits, the talk will finish with with a glimpse on a similar model of dislocation dynamics, proposed by Frenkel and Kontorova in 1939. We will sketch a rigorous existence result for travelling waves, obtained in collaboration with Carl-Friedrich Kreiner (Munich).

Tuesday, September 8, 2009

Tuesday, September 22, 2009

Tuesday, September 29, 2009

Tuesday, October 13, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, October 13, 2009

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Submitted by verahur.

Francisco Gancebo (University of Chicago)
Contour dynamics for 2D active scalars
Abstract: In this talk we discuss two free boundary problems given by fluid domains which are weak solutions of incompressible equations. We consider the contour dynamics Muskat problem and the evolution of a sharp front by the 2D surface Quasi-geostrophic equation. Both systems are described by means of a transport equation for the active scalar \rho(x,t) which takes constant values on complementary domains. The velocity field is determined by \rho(x,t) by singular integral operators. However the solutions of these two physical scenarios have completely different outcomes regarding well-posedness and regularity issues.

Tuesday, October 20, 2009

Tuesday, October 27, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, October 27, 2009

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Submitted by ekirr.

Boaz Ilan [email] (U. of California, Merced)
Solitons at the interface between bands and gaps
Abstract: Solitons or localized bound states arise in nonlinear wave systems including nonlinear optics, ultra-cold atomic systems, and water waves. In some cases solitons can be very stable while in others highly unstable and can undergo collapse (singularity formation). We study solitons in focusing Nonlinear Schr�dinger (NLS) equations with periodic potentials. Rigorous asymptotic analysis reveals that when a soliton bifurcates from a band edge into a gap, the soliton profile is constructed from a linear Bloch wave that is slowly modulated by a bound state solution of a homogenized NLS equation. A consequence of the analysis is that in the L2-critical case, the soliton power (L2 norm) is below the threshold for collapse. Direct computations of soliton dynamics in L2-critical NLS equations elucidate these results.

Tuesday, November 3, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, November 3, 2009

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Submitted by laugesen.

Joe Lakey [email] (New Mexico State University)
Time- and band-limiting
Abstract: This talk will survey some of the classical and recent results concerning operators composed of a projection onto a compact set in time, followed by a projection onto a compact set in frequency. Such "time- and band-limiting" operators were studied by Landau, Slepian, and Pollak in a series of papers published in the Bell Systems Tech. Journal in the early 1960s. Among other important results, Landau and Pollak gave an initial precise statement of the "folklore" observation that the dimension of the space of signals that are essentially timelimited to a given duration and bandlimited to a given frequency bandwidth is the time-bandwidth product. Other useful versions were proved by Slepian in the early 1970s and by Landau and Widom in 1980.
Further progress on time- and bandlimiting has been intermittent, but genuine recent progress has been made in terms of numerical analysis, sampling theory, and extensions to multiband signals, all driven to some extent by potential applications in wireless communications.
After providing a brief outline of the historical developments in the mathematical theory of time- and bandlimiting, some details of the sampling theory and multiband setting will be given.