ICMP 2003: Fluid dynamics and nonlinear PDEs [Sessions]

Dario Bambusi
Dario.Bambusi@mat.unimi.it
Dipartimento di Matematica,
Via Saldini, 50
20 133 Milano
Italy

Some results on Birkhoff Normal Form in Hamiltonian PDEs

Consider a Hamiltonian system with a linearly stable equilibrium point. Assume that the frequencies of small oscillation are nonresonant then there exist coordinates (p,q) in which the Hamiltonian is the sum of a part depending only the actions (p_j² + q_j²) / 2 and a small remainder. From this one can deduce that all small amplitude solutions remain close to a torus up to very long times. I will show how this result can be extended to a quite general class of Hamiltonian semilinear PDEs including the Nonlinear Wave Equation in one space dimensions and the Nonlinear Schr\"odinger Equation in arbitrary space dimensions. An extension to quasilinear equations will also be discussed.

Martin Hairer
Martin.Hairer@math.unige.ch
Mathematics Research Centre
University of Warwick

Coupling stochastic PDEs

We consider a class of parabolic stochastic PDEs driven by white noise in time, and we are interested in showing ergodicity for some cases where the noise is very degenerate, i.e. acts only on a ``small'' part of the equation. In some cases where the standard Strong Feller - Irreducibility argument fails, one can nevertheless implement a coupling construction that ensures uniqueness of the invariant measure. Several examples will be investigated.

Vadim Kaloshin
kaloshin@ias.edu
IAS
Princeton

Dynamics of oil spill

We consider an oil spill Ω on the surface of ocean R². Ocean evolves in time and the ultimate goal is to remove the spill Ω_t after time t>0 from the surface. The problem gives rise to various questions about the size and shape of the spill Ω_t.

We model motion of an oil spot by a stochastic differential equation driven by a finite-dimensional Brownian motion. Let Ω⊂R² be an open set and let Ω_t evolve according to that equation. Then:

Central Limit Theorem (CLT) for distribution of the spill holds true. This, in particular, says that for any t>0 and some large C>0 with probability 99% the ball of radius C√t contains 99% of the spill Ω_t.
(Shape Theorem) Define the set of poison points by time t by W_t = {x∈ R²: d(x, Ω_s)<1 for 0<s<t}. Then there is a nonrandom convex set B⊂R² such that for any ε>0 almost surely we have (1-ε) t B ⊂ W_t ⊂ (1+ε) t B.

(joint work with D.Dolgopyat, L. Koralov.)

Guido Schneider
guido.schneider@math.uni-karlsruhe.de
Universtät Karlsruhe
Mathematisches Institut I
D-76128 Karlsruhe
Germany

Diffusive behavior in hydrodynamical stability problems

We consider pattern forming systems, like the Taylor-Couette problem, exhibiting spatially periodic traveling wave solutions. First we are interested in the asymptotic behavior of solutions converging initially towards spatially periodic traveling wave solutions with the same wavenumbers at infinity, but different phases. Our result is that there occurs a universal behavior, namely a local convergence towards a spatially periodic traveling wave solution, where a self-similar mixing of the phases can be observed. In case of non-varying group velocity and different wave numbers at infinity we also consider the self-similar mixing of wavenumbers.

Armen Shirikyan
Armen.Shirikyan@math.u-psud.fr
Laboratoire de Mathématiques
Université de Paris-Sud XI
Bâtiment 425
91405 Orsay Cedex
France

Some mathematical problems of statistical hydrodynamics

We give a survey of recent advances in the problem of ergodicity for a class of dissipative PDE's perturbed by an external random force. One of the main results in this direction asserts that if the external force is sufficiently non-degenerate, then there is a unique stationary measure, which is exponentially mixing. Moreover, a strong law of large numbers and a central limit theorem hold for solutions of the problem in question. The results obtained apply, for instance, to the 2D Navier-Stokes system and the complex Ginzburg-Landau equation. Some open problems will also be discussed.

D. Bambusi	Milano
M. Hairer	Warwick
V. Kaloshin	Princeton
G. Schneider	Kalsruhe
A. Shirikyan	Orsay

Fluid dynamics and nonlinear PDEs

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