3.8 Partial Differential Equations

Organizers S. Mueller (Leipzig), I.M. Sigal (Toronto)

3.8.1 Weak Turbulence for Periodic NSL

James Colliander

Univ. of Toronto colliand@math.toronto.edu

I will describe recent work with G. Staffilani, M. Keel, H. Takaoka and T. Tao. We construct a global-in-time solution of periodic cubic NLS on the 2-torus which transfers the conserved L2 mass from low frequencies to arbitrarily high frequencies. In particularly, norms measuring smoothness of the solution can grow from their initial size to arbitrarily large size in a finite time. The solution is built using a combinatorial construction on resonant frequencies and the existence of a travelling wave solution in a system of ordinary differential equations which moves from low to high frequencies.

3.8.2 Ginzburg-Landau Dynamics

Stephen Gustafson

University of British Columbia gustaf@math.ubc.ca

Ginbzburg-Landau type equations have attracted a great deal of interest from mathematicians in recent years. This is a family of nonlinear partial differential equations which describe statics and dynamics in superconductors and superfluids, and which admit interesting localized solutions such as vortices. I will describe some of the recent results on Ginzburg-Landau dynamics.

3.8.3 On the Derivation of Furier’s Law

Antti Kupiainen

Helsinki Univ. ajkupiai@mappi.helsinki.fi

We study the Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice and subjected to a stochastic forcing on the boundary. We introduce a truncation of the Hopf equations describing the stationary state of the system which leads to a nonlinear generalized Boltzman equation for the two-point stationary correlation functions. We prove that these equations have a unique solution which, for N large, is approximately a local equilibrium state

satisfying Fourier law that relates the heat current to a local temperature gradient. The temperature exhibits a nonlinear profile. We discuss also implications for the study of Boltzman equations.

3.8.4 TBA

Stefan Mueller

Max Planck Institute for Mathematics in the Sciences, Leipzig sm@mis.mpg.de

3.8.5 A Criterion for the Logarithmic Sobolev Inequality

Felix Otto

Univ. Bonn otto@iam.uni-bonn.de

This is joint work with Maria Reznikoff. We present a criterion for the logarithmic Sobolev inequality (LSI) on the product space X1 × · · · × XN . We have in mind an N -site lattice, unbounded continuous spin variables, and Glauber dynamics. The interactions are described by the Hamiltonian H of the Gibbs measure. The criterion for LSI is formulated in terms of the LSI constants of the single-site conditional measures and the size of the off-diagonal entries of the Hessian of H . It is optimal for Gaussians with positive covariance matrix.

3.8.6 Singular Behaviour in Chemotaxis Models

Juan J. L. Velazquez

Universidad Complutense

JJ_Velazquez@mat.ucm.es

In this talk I will describe some of the singular behaviours that arise in the partial differential equations that describe the phenomenon of chemotactic aggregation. In particular I will describe solutions that generate Dirac masses in finite time. The continuation beyond the blow-up will be also considered as well as the study of the long time asymptotics for some models of chemotactic aggregation where the interaction takes place by means of nondiffusive chemicals.