On the blow-up of some complex solutions of the 3D Navier–Stokes equations: theoretical predictions and computer simulations

We consider some complex-valued solutions of the Navier–Stokes equations in R^3 for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the ‘fluid’ remains quiet

On the blow-up of some complex solutions of the 3D Navier–Stokes equations: theoretical predictions and computer simulations

We consider some complex-valued solutions of the Navier–Stokes equations in R^3 for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the ‘fluid’ remains quiet

One-Dimensional Hard-Rod Caricature of Hydrodynamics: Navier-Stokes Correction

One-dimensional system of hard-rod particles of length a is studied in the hydrodynamical limit. The Navier-Stokes correction to Euler's equation is found for an initial locally-equilibrium family of states of constant density ρ ϵ [0,a^(-1)). The correction is given, at t~0, by the non-linear second-order differential operator (Bf)(q,v) = (a^2/2)(∂/∂q)[∫dw|v-w|f(q,w)(∂/∂q)f(q,v) - f(q,v)∫dw|v-w|(∂/∂q)f(q,w)](1-ρa)^(-1) where f(q,v) is the (hydrodynamical) density at a point q ϵ R^1 of the species of particles with velocity v ϵ R^1

Continuity and anomalous fluctuations in random walks in dynamic random environments: numerics, phase diagrams and conjectures

We perform simulations for one dimensional continuous-time random walks in two dynamic random environments with fast (independent spin-flips) and slow (simple symmetric exclusion) decay of space-time correlations, respectively. We focus on the asymptotic speeds and the scaling limits of such random walks. We observe different behaviors depending on the dynamics of the underlying random environment and the ratio between the jump rate of the random walk and the one of the environment. We compare our data with well known results for static random environment. We observe that the non-diffusive regime known so far only for the static case can occur in the dynamic setup too. Such anomalous fluctuations give rise to a new phase diagram. Further we discuss possible consequences for more general static and dynamic random environments.Comment: 33 pages, 23 figure

Random walks in random environment with Markov dependence on time

We consider a simple model of discrete-time random walk on Zν, ν = 1, 2, . . . in a random environment independent in space and with Markov evolution in time. We focus on the application of methods based on the properties of the transfer matrix and on spectral analysis. In section 2 we give a new simple proof of the existence of invariant subspaces, with an explicit condition on the parameters. The remaining part is devoted to a review of the results obtained so far for the quenched random walk and the environment from the point of view of the random walk, with a brief discussion of the methods.Ми розглядаємо просту модель випадкового блукання з дискретним часом у Zν, ν = 1, 2, . . . у випадковому середовищi, що є незалежним у просторi i має маркiвську еволюцiю у часi. Ми зосереджуємось на застосуваннi методiв, що ґрунтуються на властивостях трансфер-матрицi i на спектральному аналiзi. У §2 ми подаємо просте доведення iснування iнварiантних пiдпросторiв, що використовує явну умову для параметрiв. Решта роботи присвячується огляду результатiв одержаних дотепер для замороженого випадкового блукання i оточення з точки зору випадкового блукання, а також короткому обговоренню методiв

Phase transition and correlation decay in Coupled Map Lattices

For a Coupled Map Lattice with a specific strong coupling emulating Stavskaya's probabilistic cellular automata, we prove the existence of a phase transition using a Peierls argument, and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoupling originally developed for weak coupling. This implies the exponential decay, in space and in time, of the correlation functions of the invariant measures

Langevin equation for the extended Rayleigh model with an asymmetric bath

In this paper a one-dimensional model of two infinite gases separated by a movable heavy piston is considered. The non-linear Langevin equation for the motion of the piston is derived from first principles for the case when the thermodynamic parameters and/or the molecular masses of gas particles on left and right sides of the piston are different. Microscopic expressions involving time correlation functions of the force between bath particles and the piston are obtained for all parameters appearing in the non-linear Langevin equation. It is demonstrated that the equation has stationary solutions corresponding to directional fluctuation-induced drift in the absence of systematic forces. In the case of ideal gases interacting with the piston via a quadratic repulsive potential, the model is exactly solvable and explicit expressions for the kinetic coefficients in the non-linear Langevin equation are derived. The transient solution of the non-linear Langevin equation is analyzed perturbatively and it is demonstrated that previously obtained results for systems with the hard-wall interaction are recovered.Comment: 10 pages. To appear in Phys. Rev.

Escape orbits and Ergodicity in Infinite Step Billiards

In a previous paper we defined a class of non-compact polygonal billiards, the infinite step billiards: to a given decreasing sequence of non-negative numbers $\{p_{n}$, there corresponds a table \Bi := \bigcup_{n\in\N} [n,n+1] \times [0,p_{n}]. In this article, first we generalize the main result of the previous paper to a wider class of examples. That is, a.s. there is a unique escape orbit which belongs to the alpha and omega-limit of every other trajectory. Then, following a recent work of Troubetzkoy, we prove that generically these systems are ergodic for almost all initial velocities, and the entropy with respect to a wide class of ergodic measures is zero.Comment: 27 pages, 8 figure

Recent Results on the Periodic Lorentz Gas

The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption -- known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases -- this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) vol. 185 (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg.Comment: 62 pages. Course at the conference "Topics in PDEs and applications 2008" held in Granada, April 7-11 2008; figure 13 and a misprint in Theorem 4.6 corrected in the new versio