218 research outputs found
Infinite dimensional SRB measures
We review the basic steps leading to the construction of a Sinai-Ruelle-Bowen
(SRB) measure for an infinite lattice of weakly coupled expanding circle maps,
and we show that this measure has exponential decay of space-time correlations.
First, using the Perron-Frobenius operator, one connects the dynamical system
of coupled maps on a -dimensional lattice to an equilibrium statistical
mechanical model on a lattice of dimension . This lattice model is, for
weakly coupled maps, in a high-temperature phase, and we use a general, but
very elementary, method to prove exponential decay of correlations at high
temperatures.Comment: 19 page
Probabilistic estimates for the Two Dimensional Stochastic Navier-Stokes Equations
We consider the Navier-Stokes equation on a two dimensional torus with a
random force, white noise in time and analytic in space, for arbitrary Reynolds
number . We prove probabilistic estimates for the long time behaviour of the
solutions that imply bounds for the dissipation scale and energy spectrum as
.Comment: 10 page
Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations
We present a general method for studying long time asymptotics of nonlinear
parabolic partial differential equations. The method does not rely on a priori
estimates such as the maximum principle. It applies to systems of coupled
equations, to boundary conditions at infinity creating a front, and to higher
(possibly fractional) differential linear terms. We present in detail the
analysis for nonlinear diffusion-type equations with initial data falling off
at infinity and also for data interpolating between two different stationary
solutions at infinity.Comment: 29 page
KAM Theorem and Quantum Field Theory
We give a new proof of the KAM theorem for analytic Hamiltonians. The proof
is inspired by a quantum field theory formulation of the problem and is based
on a renormalization group argument treating the small denominators inductively
scale by scale. The crucial cancellations of resonances are shown to follow
from the Ward identities expressing the translation invariance of the
corresponding field theory.Comment: 32 page
Global Large Time Self-similarity of a Thermal-Diffusive Combustion System with Critical Nonlinearity
We study the initial value problem of the thermal-diffusive combustion
system: , for non-negative spatially decaying initial data of arbitrary size
and for any positive constant . We show that if the initial data decays to
zero sufficiently fast at infinity, then the solution converges to
a self-similar solution of the reduced system: , in the large time limit. In particular, decays to
zero like , where is an
anomalous exponent depending on the initial data, and decays to zero with
normal rate . The idea of the proof is to combine
the a priori estimates for the decay of global solutions with the
renormalization group (RG) method for establishing the self-similarity of the
solutions in the large time limit.Comment: 22pages, Latex, [email protected],[email protected],
[email protected]
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