487 research outputs found

### Singularities and nonhyperbolic manifolds do not coincide

We consider the billiard flow of elastically colliding hard balls on the flat
$\nu$-torus ($\nu\ge 2$), and prove that no singularity manifold can even
locally coincide with a manifold describing future non-hyperbolicity of the
trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli
mixing property) of all such systems, i.e. the verification of the
Boltzmann-Sinai Ergodic Hypothesis.Comment: Final version, to appear in Nonlinearit

### 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

### Limiting Distribution of Frobenius Numbers for $n=3$

The purpose of this paper is to give a complete derivation of the limiting
distribution of large Frobenius numbers outlined in earlier work of J. Bourgain
and Ya. Sinai and fill some gaps formulated there as hypotheses.Comment: 13 page

### Non-ergodicity of the motion in three dimensional steep repelling dispersing potentials

It is demonstrated numerically that smooth three degrees of freedom
Hamiltonian systems which are arbitrarily close to three dimensional strictly
dispersing billiards (Sinai billiards) have islands of effective stability, and
hence are non-ergodic. The mechanism for creating the islands are corners of
the billiard domain.Comment: 6 pages, 8 figures, submitted to Chao

### One-particle and few-particle billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles

### Evolution of collision numbers for a chaotic gas dynamics

We put forward a conjecture of recurrence for a gas of hard spheres that
collide elastically in a finite volume. The dynamics consists of a sequence of
instantaneous binary collisions. We study how the numbers of collisions of
different pairs of particles grow as functions of time. We observe that these
numbers can be represented as a time-integral of a function on the phase space.
Assuming the results of the ergodic theory apply, we describe the evolution of
the numbers by an effective Langevin dynamics. We use the facts that hold for
these dynamics with probability one, in order to establish properties of a
single trajectory of the system. We find that for any triplet of particles
there will be an infinite sequence of moments of time, when the numbers of
collisions of all three different pairs of the triplet will be equal. Moreover,
any value of difference of collision numbers of pairs in the triplet will
repeat indefinitely. On the other hand, for larger number of pairs there is but
a finite number of repetitions. Thus the ergodic theory produces a limitation
on the dynamics.Comment: 4 pages, published versio

### 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

### Turbulence without pressure

We develop exact field theoretic methods to treat turbulence when the effect
of pressure is negligible. We find explicit forms of certain probability
distributions, demonstrate that the breakdown of Galilean invariance is
responsible for intermittency and establish the operator product expansion. We
also indicate how the effects of pressure can be turned on perturbatively.Comment: 12 page

### Invariant measures for Burgers equation with stochastic forcing

In this paper we study the following Burgers equation
du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t)
where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x
and white noise in t. We prove the existence and uniqueness of an invariant
measure by establishing a ``one force, one solution'' principle, namely that
for almost every realization of the force, there is a unique distinguished
solution that exists for the time interval (-infty, +infty) and this solution
attracts all other solutions with the same forcing. This is done by studying
the so-called one-sided minimizers. We also give a detailed description of the
structure and regularity properties for the stationary solutions. In
particular, we prove, under some non-degeneracy conditions on the forcing, that
almost surely there is a unique main shock and a unique global minimizer for
the stationary solutions. Furthermore the global minimizer is a hyperbolic
trajectory of the underlying system of characteristics.Comment: 84 pages, published version, abstract added in migratio

- …