Seminar

CIRF seminar on Wednesday, May 23, at 4pm in room ESB 2001

                       Bjorn Birnir
         Center for Complex and Nonlinear Science and
               Department of Mathematics,
           Univ. of California, Santa Barbara


                         Abstract

The Kolmogorov-Obukov statistical theory of turbulence, with intermittency
corrections, is derived from a stochastic Navier-Stokes equation with generic
noise. In this talk we discuss how the laminar solution of the Navier-Stokes
equation becomes unstable for large Reynolds number and the unstable solution
is the solution of the stochastic Navier-Stokes equation. This is the unique
solution that describes fully-developed turbulence. In order to compare with
experiments and simulations, we find the solution of the stochastic Hopf’s
equation for the invariant measure.  Gaussian noise and dissipation
intermittency produce the Kolmogorov-Obukov scaling of the structure
functions of turbulence. The Feynmann-Kac formula produces log-Poisson
processes from the stochastic Navier-Stokes equation. These processes,
first found by She, Leveque, Waymire and Dubrulle in 1995, give the
intermittency corrections to the structure functions of turbulence,
stemming from velocity fluctuations. The probability density function
(PDF) of the two-point statistics that can be compared to experiments
and simulations turn out to be similar to the generalized hyperbolic
distributions first suggested by Barndorff-Nilsen in 1977. We compare
the theoretical PDF with PDFs obtained from DNS simulations and
wind-tunnel experiments.

PDE/Applied Math/Complex Science Seminar

DAY : Friday, June 1

ROOM : SH 4607

Vladislav Panferov
Department of Mathematics
California State University, Northridge

Title: Phase transitions in models of Vlasov-McKean type

Abstract: I will discuss the problem of non-uniqueness and stability
of steady states for equations of Vlasov-McKean type.
These equations provide a mean-field description of a system of
interacting diffusions; particularly we consider the problem with
the spatial variable in a periodic box of size L and particles
interacting through a pairwise potential V. If the Fourier transform
of V has a negative minimum, the system has a critical threshold
for the diffusion constant beyond which the trivial uniform steady
state becomes unstable and the system experiences a phase transition.
We show that for a large class of interactions, when the size of the
domain is sufficiently large, the transition is always discontinuous
and is characterized by coexistence of several stable states in a
certain interval of parameter space.The transition is also shown to
occur at a value of the diffusion constant strictly greater than the
critical threshold.