Distribution of energy and convergence to equilibria in extended dissipative systems

Friday, September 28, 2012 - 10:15am - 11:05am
Keller 3-180
Thierry Gallay (Université de Grenoble I (Joseph Fourier))
We study the local energy dissipation in gradient-like nonlinear
partial differential equations on unbounded domains. Our basic
assumption, which happens to be satisfied in many classical
examples, is a pointwise upper bound on the energy flux in terms
of the energy dissipation rate. Under this hypothesis, we derive
a simple and general bound on the integrated energy flux which
implies that, in low space dimensions, our extended dissipative
system has a gradient-like dynamics in a suitable averaged sense.
In particular, we can estimate the time spent by any trajectory
outside a neighborhood of the set of equilibria. As an application,
we study the long-time behavior of solutions to the two-dimensional
vorticity equation in an infinite cylinder. This talk is based
on a joint work with S. Slijepcevic (Zagreb)
MSC Code: