A classification of orbits admitting a unique invariant measure

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_{\infty }$-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique $S_{\infty }$-invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order $(\mathbb{Q},<)$.     

A non-ergodic probabilistic cellular automaton with a unique invariant measure

We exhibit a Probabilistic Cellular Automaton (PCA) on {0,1}^Z with a neighborhood of size 2 which is non-ergodic although it has a unique invariant measure. This answers by the negative an old open question on whether uniqueness of the invariant measure implies ergodicity for a PCA.     

Ergodicity of the 2D Navier-Stokes equations with degenerate multiplicative noise

Consider the two-dimensional, incompressible Navier-Stokes equations on torus T² = [?π, π]² driven by a degenerate multiplicative noise in the vorticity formulation (abbreviated as SNS): dw _t = νΔw _t dt + B(Kw _t ,w _t )dt + Q(w _t )dW _t . We prove that the solution to SNS is continuous differentiable in initial value. We use the Malliavin calculus to prove that the semigroup {P _t }_t≥0 generated by the SNS is asymptotically strong Feller. Moreover, we use the coupling method to prove that the solution to SNS has a weak form of irreducibility. Under almost the same Hypotheses as that given by Odasso, Prob. Theory Related Fields, 140: 41–82 (2005) with a different method, we get an exponential ergodicity under a stronger norm.     

Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations

Summary. The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection of the pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference between these two vectors (pressure gradient and projection) is introduced in the continuity equation. The resulting formulation is shown to be stable and optimally convergent, both in a norm associated to the problem and in the norm for both velocities and pressure. This is proved first for the Stokes problem, and then it is extended to the nonlinear case. All the analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in spite of the fact that the present method is only a minor modification of this. Received May 25, 1998 / Revised version received August 31, 1999 / Published online July 12, 2000     

Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations

The combined quasi-neutral and non-relativistic limit of compressible Navier-Stokes-Maxwell equations for plasmas is studied. For well-prepared initial data, it is shown that the smooth solution of compressible Navier-Stokes-Maxwell equations converges to the smooth solution of incompressible Navier-Stokes equations by introducing new modulated energy functional.     

From the BGK model to the Navier-Stokes equations

We give here a complete derivation of the Navier-Stokes-Fourier equations from a model collisional kinetic equation, the BGK model. Though physically unrealistic, this model shares some common features with more classical models such as the Boltzmann equation.Then the program developed by Bardos, Golse and Levermore [Fluid dynamic limits of kinetic equations II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46 (5) (1993) 667-753] to study hydrodynamic limits of the steady Boltzmann equation, and extended by Lions and Masmoudi [From Boltzmann equations to Navier-Stokes equations I, Archive Rat. Mech. Anal. 158 (2001) 173-193] in the time-dependent case, can be adapted here, and gives the expected convergence result provided that the particle density f satisfies some integrability assumption.The originality of the present work is to remove this assumption by establishing refined a priori estimates. The crucial idea is to decompose f as (f−M_f)+M_f where M_f is the local Maxwellian associated with f. The first term is then estimated by means of the entropy dissipation, while the other is smooth in v. A mixing property of the operator (ε∂_t+v.∇_x) allows to transfer some of this extra-integrability on the variable x.     

Stochastic uniform observability of linear differential equations with multiplicative noise

The aim of this paper is to give a deterministic characterization of the uniform observability property of linear differential equations with multiplicative white noise in infinite dimensions. We also investigate the properties of a class of perturbed evolution operators and we used these properties to give a new representation of the covariance operators associated to the mild solutions of the investigated stochastic differential equations. The obtained results play an important role in obtaining necessary and sufficient conditions for the stochastic uniform observability property.     

Splitting algorithms as applied to the Navier-Stokes equations

Implicit finite-difference schemes of approximate factorization and predictor-corrector schemes based on a special splitting of operators are proposed for the numerical solution of the Navier-Stokes equations governing a viscous compressible heat-conducting gas. The schemes are based on scalar tridiagonal Gaussian elimination and are unconditionally stable. The accuracy and efficiency of the algorithms are confirmed by computing two-dimensional flows of complex geometry.     

Self-similar solutions to the isothermal compressible Navier-Stokes equations

^** Email: guo_zhenhua{at}iapcm.ac.cn^*** Email: jiang{at}iapcm.ac.cn We investigate the self-similar solutions to the isothermalcompressible Navier–Stokes equations. The aim of thispaper is to show that there exist neither forward nor backwardself-similar solutions with finite total energy. This generalizesthe results for the incompressible case in Neas, J., Rika, M.& verák, V. (1996, On Leray's self-similar solutionsof the Navier-Stokes equations. Acta. Math., 176, 283–294),and is consistent with the (unproved) existence of regular solutionsglobally in time for the compressible Navier–Stokes equations.     

Special global regular solutions to the Navier-Stokes equations

Ezistence results for global regular solutions to the Navier-Stokes equations, which are close either to two-dimensional or to axially symmetric solutions are presented. Slip boundary conditions are assumed. Moreover, the domains considered are either cylindrical or axially symmetric. Problems with and without inflow-outflow are examined. All proofs can be divided into two steps; (1) long time existence established either by the Leroy Schauder fixed point theorem or by the method of successive approximations; (2) global existence proved by prolongation of a local solution with respcct to time, Bibliography: 32 titles. I dedicate this paper to Vsevolod Alekseevich Solonnikov, the great mathematician and my teacher Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 120–152.     

On the existence of a finite invariant measure

A necessary and sufficient condition is given for a Borel automorphism on a standard Borel space to admit an invariant probability measure.