If you want to receive e-mail announcements of talks please contact
Dionisios Margetis (dio@math.umd.edu).
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September 4
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NO SEMINAR
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September 11
JOINT PDE/CSCAMM SEMINAR
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Stability of Strong Viscous Shock Layers in an Ideal Gas
Kevin Zumbrun
Department of Mathematics,
Indiana University
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Abstract:
By a combination of asymptotic ODE estimates and numerical
Evans function computations, we examine the spectral stability of shock-wave solutions of the compressible Navier--Stokes equations with ideal gas equation of state, for arbitrary strength waves.
Our main results are that, in appropriately rescaled coordinates, the Evans function associated with the linearized operator about the wave, an analytic function analogous to the characteristic polynomial whose zeros correspond to eigenvalues of L, (i) converges in the strong shock limit to the Evans function for a limiting shock profile of the same equations, for which internal energy vanishes at one endstate; and (ii) has no unstable (positive real part) zeros outside a uniform ball. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with the (nonphysical) limiting case, form a compact family of boundary-value problems that may be conveniently studied numerically. An intensive numerical study then yields unconditional stability, independent of amplitude, for a range of parameter values including all common gases.
Besides its physical interest, we believe that this analysis has interest as an example where it is possible to carry out a rigorous globl stability analysis by numerical techniques, the obvious obstace being the need to treat an unbounded parameter range using finitely many operations.
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September 18
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Global existence for the defocusing cubic wave equation in dimension 3
Tristan Roy
Department of Mathematics, UCLA
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Abstract:
In this talk I will discuss recent developments regarding the global existence of solutions to the defocusing cubic wave equation in dimension 3 below the energy
norm. This initial value problem is known to be globally well-posed in $H^{s} \times H^{s-1}$, $s \geq \frac{3}{4}$. We design the $I$ method (originally invented
by the "I team" for the semilinear Schrodinger equations) for this wave equation and give another proof of global existence for $s>\frac{3}{4}$. Then we prove
global well-posedness for $s>\frac{7}{10}$ under the additional assumption of spherical data by adding new components to the method, such as Morawetz-type
estimates, radial Sobolev inequalities and a "greedy" algorithm. Finally we get back to the general problem and we show that this equation is globally well-posed
for $s>\frac{13}{18}$.
The proof is based upon an adapted linear-nonlinear decomposition of the solution.
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September 25
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RESERVED FOR AMSC EVENT
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Abstract:
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October 2
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Nonlinear dynamical phenomena in mesoscale modeling of polycrystals
Maria Emelianenko
Department of Mathematical Sciences,
George Mason University
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Abstract:
Polycrystalline materials are important in many technological applications, yet there are still many challenges they present for
mathematical modeling and analysis. One such challenge lies in understanding how statistical distributions develop in the process of coarsening
of materials microstructure and how these distributions in turn relate to materials properties. In this talk, we will discuss and compare
several recent continuum level models resulting in nonlinear evolution equations. Special focus will be placed on newly discovered features of
interface dynamics that connect this problem to the theory of nonhomogeneous Poisson processes in industrial applications and Boltzmann
equations in statistical physics. Numerical and analytical characteristics of the solutions will be discussed and compared against the results
produced by experiments and large-scale simulations.
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October 9
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Inertial manifolds for nonlinear Fokker-Planck
(Smoluchowski) equations arising in the modeling of nematic
polymers
Jesenko Vukadinovic
Department of Mathematics,
College of Staten Island and
Graduate Center of the City Univ. of New York (CUNY)
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Abstract:
Although intrinsically infinite-dimensional, many
dissipative parabolic systems exhibit long-term dynamics with
properties typical of finite-dimensional dynamical systems. The
global attractor, often considered the central object in the
study of long-term behavior of dynamical systems, appears to be
inadequate in capturing this finite-dimensionality. If a very
restrictive spectral gap condition is satisfied, an
exponentially attracting invariant finite-dimensional manifold
- termed inertial manifold - appears much more appropriate. On
it, the PDE reduces to an ODE termed inertial form.
Unfortunately, many physically relevant systems fail to satisfy
the spectral-gap condition, amongst them also the Fokker-Planck
(Smoluchowski) equations arising in the theory of nematic
polymers. It turns out that via a gauge transform, the
equations from this class can be transformed into nonlinear
Schroedinger-like equations for which the spectral gap
condition holds, and the existence of inertial manifolds is
proven.
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October 16
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Antoine Mellet
Department of Mathematics, UMCP
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Abstract:
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October 23
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Matei Machedon
Department of Mathematics, UMCP
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Abstract:
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October 30
Different Room:
MTH 3206 reserved for Statistics Seminar
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OPEN
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Abstract:
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November 6
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OPEN
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Abstract:
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November 13
AZIZ LECTURER
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Diffractive behavior of the wave equation in periodic media
Gregoire Allaire
Centre de Mathematiques Appliques
L'Ecole Polytechnique
Paris, France
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Abstract:
We study the homogenization and singular perturbation of the
wave equation in a periodic media for long times of the order
of the inverse of the period. We consider inital data that are
Bloch wave packets, i.e., that are the product of a fast
oscillating Bloch wave and of a smooth envelope function.
We prove that the solution is approximately equal to two waves
propagating in opposite directions at a high group velocity with
envelope functions which obey a Schrödinger
type equation.
Our analysis extends the usual WKB approximation by adding a
dispersive, or diffractive, effect due to the non uniformity
of the group velocity which yields the dispersion tensor of
the homogenized
Schrödinger
equation. This is a joint work
with M. Palombaro and J. Rauch.
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November 20
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Konstantina Trivisa
Department of Mathematics, UMCP
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Abstract:
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November 27
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THANKSGIVING HOLIDAY -- NO SEMINAR
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December 4
Joint Norbert Wiener Center/PDE Seminar
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Solving Certain Fokker-Planck Equations
Using Noncommutative Harmonic Analysis
Gregory Chirikjian
Mechanical Engineering, Johns Hopkins University
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Abstract:
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December 11
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Pair excitation in
Bose-Einstein condensation:
Beyond the nonlinear Schrödinger equation
Dionisios Margetis
Dept. of Mathematics and IPST, UMCP
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Abstract:
Quantum mechanical treatments of
atomic gases at extremely low temperatures often
invoke a nonlinear Schrödinger equation (NLSE)
for the wave function of a
macroscopic state
(``Bose-Einstein condensate'').
In this talk, I will
address recent progress and challenges in analyzing
effects that cannot be described solely by the NLSE.
A primary effect is ``pair excitation'', according to which
pairs of particles are scattered off the macroscopic state.
For non-translationally invariant settings,
this mechanism leads to
to a system of
nonlocal dispersive PDEs.
A particular
asymptotic solution
of a minimal model
for pair excitation will be discussed.
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Special accomodations for individuals with disabilities can be made
by calling in advance (301) 405-5048. It would be appreciated if we are
notified at least one week in advance.
