Global Existence for a Kinetic Model of Chemotaxis via Dispersion and Strichartz Estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of ‘run and tumble’. The existence of solutions has been obtained in several papers Chalub et al. (2004 Chalub , F. A. C. C. , Markowich , P. A. , Perthame , B. , Schmeiser , C. ( 2004 ). Kinetic models for chemotaxis and their drift-diffusion limits . Monatsh. Math. 142 : 123 – 141 .[Crossref], [Web of Science ®] , [Google Scholar]), Hwang et al. (2005a Hwang , H. J. , Kang , K. , Stevens , A. ( 2005a ). Global solutions of nonlinear transport equations for chemosensitive movement . SIAM J. Math. Anal. 36 ( 4 ): 1177 – 1199 . [Google Scholar] b Hwang , H. J. , Kang , K. , Stevens , A. ( 2005b ). Drift-diffusion limits of kinetic models for chemotaxis: a generalization . Discrete Contin. Dyn. Syst. Ser. B 5 ( 2 ): 319 – 334 . [Google Scholar]) using direct and strong dispersive effects.

Here, we use the weak dispersion estimates of Castella and Perthame (1996 Castella , F. , Perthame , B. ( 1996 ). Estimations de Strichartz pour les équations de transport cinétique. [Strichartz’ estimates for kinetic transport equations.] C. R. Acad. Sci. Paris Sér. I 322 ( 6 ): 535 – 540 . [Google Scholar]) to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.     

Kinetic salt effects in the reaction of permanganate ions with iodide ions in concentrated electrolyte solutions

The oxidation of iodide ion by permanganate ion has been studied in several electrolyte solutions over the concentration range (0–6M). The observed kinetic salt effect may be interpreted as arising as a result of ionsolvent interactions. This salt effect is additive for different salts. Support for these conclusions comes from an examination of the activity coefficients of species estimated from the treatment given by Pitzer. It has also been shown that the species relevant in terms of kinetic control are unassociated.     

Growth Estimates for the Solutions of the Vlasov-Poisson System in the Plasma Physics Case

In [3] new a—priori—estimates for solutions of the Vlasov-Poisson system in the plasma physics case were obtained. We show that these lead to improved estimates on the possible growth of the solutions.     

Non-commutative geometry and irreversibility

A kinetics built upon q-calculus, the calculus of discrete dilatations, is shown to describe diffusion on a hierarchical lattice. The only observable on this ultrametric space is the “quasi-position” whose eigenvalues are the levels of the hierarchy, corresponding to the volume of phase space available to the system at any given time. Motion along the lattice of quasi-positions is irreversible. Received: 24 June 1997 / Revised: 15 September 1997 / Accepted: 6 October 1997     

带电粒子在高温等离子体中的减速

本文利用考虑三体效应的量子等离子体动力学方程,推导了带电粒子在高温等离子体中减速公式。对氚核在LiD等离子体和氦核在DT等离子体中的减速作了数值计算,并与其它方法计算结果进行了比较。     

Collision-induced absorption in dense mixtures of simple fluids

Summary Collision-induced absorption (CIA) spectroscopy is a probe of the dynamical properties of dense systems. Two simple cases of CIA spectra are here discussed: the translational band observed in solutions of a noble gas in liquid Ar and the rotational lines of H₂ observed in solutions of H₂ in Ar. In the first case, the analysis of the translational band allows one to derive the characteristic frequency ω₀, associated with the rattling component of the motion of the impurity in the Ar medium. In the second case, the analysis of the rotational lines allows one to derive the parameter δ, that characterizes the density narrowing effect. In the high-density region, the parameter δ is directly related to the diffusive motion of the Ar atoms around the impurity. Paper presented at the workshop ?Highlights on Simple Liquids?, held in Turin at ISI on 1–3 May, 1989.     

Velocity-space contours of collision integrals

Based on the recently found closed-form expressions of the Boltzmann collision integrals in a rigid-sphere gas for multi-Maxwellian distributions, a few typical sets of contour surfaces of the integrals in the space of molecular velocities are presented. These show graphically the tendency toward equilibrium under the influence of collisions. A brief preliminary comparison with Monte Carlo results is also given.     

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R ^d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation models.