ICMP 2003 > Sessions > Fluid dynamics and nonlinear PDEs | |
Session organized by S. Kuksin (Edinburgh), J.P. Eckmann (Genève).
D. Bambusi | Milano |
M. Hairer | Warwick |
V. Kaloshin | Princeton |
G. Schneider | Kalsruhe |
A. Shirikyan | Orsay |
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_{j}^{2} + q_{j}^{2}) / 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.
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.
Dynamics of oil spill
We consider an oil spill Ω on the surface of ocean R^{2}. 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⊂R^{2} be an open set and let Ω_{t} evolve according to that equation. Then:
(joint work with D.Dolgopyat, L. Koralov.)
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.
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.
Zero-energy flows and vortex patterns in quantum mechanics
Phase turbulence in the complex Ginzburg-Landau equation via Kuramoto-Sivashinsky phase dynamics
Generalized nonlinear equation and solutions for fluid contour/surface deformations
New symmetries for some diffusion equations
A comparison analysis of Sivashinsky's type evolution equations describing flame propagation in channels
Solution of some classes of mathematical physics inverse problems
Bounds on the bifurcation branches of an equation that models long-wave instabilities
Two-loop calculation of expansion improved by summation of nearest dimensional singularities for randomly forced incompressible fluid
Ideal turbulence and symbolic dynamics
Zero-energy flows and vortex patterns in quantum mechanics
The nonlocal Burgers equation
Gas dynamics applications of a characteristic Cauchy problem
Generalized solutions for the one-dimensional Fokker-Planck equation: partner potentials
Korteweg-deVries and Boussinesq equations and nonlinear transformation solitary waves at a bottom step
Euler equations of Riemannian manifolds: classes of smooth solutions and their properties