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 $d$-dimensional lattice to an equilibrium statistical mechanical model on a lattice of dimension $d+1$. 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 $R$. We prove probabilistic estimates for the long time behaviour of the solutions that imply bounds for the dissipation scale and energy spectrum as $R\to\infty$.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: $u_{1,t} = u_{1,x,x} - u_1 u^2_2, u_{2,t} = d u_{2,xx} + u_1 u^2_2, x \in R^1$, for non-negative spatially decaying initial data of arbitrary size and for any positive constant $d$. We show that if the initial data decays to zero sufficiently fast at infinity, then the solution $(u_1,u_2)$ converges to a self-similar solution of the reduced system: $u_{1,t} = u_{1,xx} - u_1 u^2_2, u_{2,t} = d u_{2,xx}$, in the large time limit. In particular, $u_1$ decays to zero like ${\cal O}(t^{-\frac{1}{2}-\delta})$, where $\delta > 0$ is an anomalous exponent depending on the initial data, and $u_2$ decays to zero with normal rate ${\cal O}(t^{-\frac{1}{2}})$. 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]