An Elementary Proof of the Existence and Uniqueness Theorem for the Navier-Stokes Equations

We give a geometric approach to proving know regularity and existence theorems for the 2D Navier-Stokes Equations. We feel this point of view is instructive in better understanding the dynamics. The technique is inspired by constructions in the Dynamical Systems.Comment: 15 Page

On cross-ratio distortion and Schwarz derivative

We prove asymptotic estimates for the cross-ratio distortion with respect to a smooth or holomorphic function in terms of its Schwarz derivative.Comment: the spelling of the name `Schwarz' correcte

Herman's Theory Revisited

We prove that a $C^{2+\alpha}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\delta$, $0<\delta<\alpha\le1$, is $C^{1+\alpha-\delta}$-smoothly conjugate to a rigid rotation. We also derive the most precise version of Denjoy's inequality for such diffeomorphisms.Comment: 10 page

Thermodynamic formalism for field driven Lorentz gases

We analytically determine the dynamical properties of two dimensional field driven Lorentz gases within the thermodynamic formalism. For dilute gases subjected to an iso-kinetic thermostat, we calculate the topological pressure as a function of a temperature-like parameter \ba up to second order in the strength of the applied field. The Kolmogorov-Sinai entropy and the topological entropy can be extracted from a dynamical entropy defined as a Legendre transform of the topological pressure. Our calculations of the Kolmogorov-Sinai entropy exactly agree with previous calculations based on a Lorentz-Boltzmann equation approach. We give analytic results for the topological entropy and calculate the dimension spectrum from the dynamical entropy function.Comment: 9 pages, 5 figure

Proving The Ergodic Hypothesis for Billiards With Disjoint Cylindric Scatterers

In this paper we study the ergodic properties of mathematical billiards describing the uniform motion of a point in a flat torus from which finitely many, pairwise disjoint, tubular neighborhoods of translated subtori (the so called cylindric scatterers) have been removed. We prove that every such system is ergodic (actually, a Bernoulli flow), unless a simple geometric obstacle for the ergodicity is present.Comment: 24 pages, AMS-TeX fil

Random matrices: Universality of local eigenvalue statistics up to the edge

This is a continuation of our earlier paper on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in that paper from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.Comment: 24 pages, no figures, to appear, Comm. Math. Phys. One new reference adde

Ensemble averages and nonextensivity at the edge of chaos of one-dimensional maps

Ensemble averages of the sensitivity to initial conditions $\xi(t)$ and the entropy production per unit time of a {\it new} family of one-dimensional dissipative maps, $x_{t+1}=1-ae^{-1/|x_t|^z}(z>0)$, and of the known logistic-like maps, $x_{t+1}=1-a|x_t|^z(z>1)$, are numerically studied, both for {\it strong} (Lyapunov exponent $\lambda_1>0$) and {\it weak} (chaos threshold, i.e., $\lambda_1=0$) chaotic cases. In all cases we verify that (i) both $[\ln_q x \equiv (x^{1-q}-1)/(1-q); \ln_1 x=\ln x]$ and $<S_q > [S_q \equiv (1-\sum_i p_i^q)/(q-1); S_1=-\sum_i p_i \ln p_i]$ {\it linearly} increase with time for (and only for) a special value of $q$, $q_{sen}^{av}$, and (ii) the {\it slope} of $$and that of$$ {\it coincide}, thus interestingly extending the well known Pesin theorem. For strong chaos, $q_{sen}^{av}=1$, whereas at the edge of chaos, $q_{sen}^{av}(z)<1$.Comment: 5 pages, 5 figure

Anomalous diffusion in disordered multi-channel systems

We study diffusion of a particle in a system composed of K parallel channels, where the transition rates within the channels are quenched random variables whereas the inter-channel transition rate v is homogeneous. A variant of the strong disorder renormalization group method and Monte Carlo simulations are used. Generally, we observe anomalous diffusion, where the average distance travelled by the particle, []_{av}, has a power-law time-dependence []_{av} ~ t^{\mu_K(v)}, with a diffusion exponent 0 \le \mu_K(v) \le 1. In the presence of left-right symmetry of the distribution of random rates, the recurrent point of the multi-channel system is independent of K, and the diffusion exponent is found to increase with K and decrease with v. In the absence of this symmetry, the recurrent point may be shifted with K and the current can be reversed by varying the lane change rate v.Comment: 16 pages, 7 figure

A limit result for a system of particles in random environment

We consider an infinite system of particles in one dimension, each particle performs independant Sinai's random walk in random environment. Considering an instant $t$, large enough, we prove a result in probability showing that the particles are trapped in the neighborhood of well defined points of the lattice depending on the random environment the time $t$ and the starting point of the particles.Comment: 11 page

Distribution of the Oscillation Period in the Underdamped One Dimensional Sinai Model

We consider the Newtonian dynamics of a massive particle in a one dimemsional random potential which is a Brownian motion in space. This is the zero temperature nondamped Sinai model. As there is no dissipation the particle oscillates between two turning points where its kinetic energy becomes zero. The period of oscillation is a random variable fluctuating from sample to sample of the random potential. We compute the probability distribution of this period exactly and show that it has a power law tail for large period, P(T)\sim T^{-5/3} and an essential singluarity P(T)\sim \exp(-1/T) as T\to 0. Our exact results are confirmed by numerical simulations and also via a simple scaling argument.Comment: 9 pages LateX, 2 .eps figure