about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

estimation of sojourn time distributions for cyclic semi markov processes in equilibrium summary the em algorithm is used to develop procedures for estimating the interoccurrence distributions when n independent and identically distributed cyclic semi markov processes each being ergodic irreducible and in equilibrium are observed over finite windows this work is an extension of vardi 1982a b who considers a similar problem for renewal processes and develops the rt algorithm for estimation of the common interoccurrence distribution the results are illustrated using simulations for alternating renewal processes and compared with an extension of an approximate approach of denby vardi 1985

the bgk model with external confining potential existence long time behaviour and time periodic maxwellian equilibria we study global existence and long time behaviour for the inhomogeneous nonlinear bgk model for the boltzmann equation with an external confining potential for an initial datum f0 0 with bounded mass entropy and total energy we prove existence and strong convergence in l1 to a maxwellian equilibrium state by compactness arguments and multipliers techniques of particular interest is the case with an isotropic harmonic potential in which boltzmann himself found infinitely many time periodic maxwellian steady states this behaviour is shared with the boltzmann equation and other kinetic models for all these systems we study the multistability of the time periodic maxwellians and provide necessary conditions on f0 to identify the equilibrium state both in l1 and in lyapunov sense under further assumptions on f these conditions become also sufficient for the identification of the equilibrium in l1

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

maximum likelihood estimation in the mover stayer model abstract the discrete time mover stayer model a special mixture of two independent markov chains has been widely used in modeling the dynamics of social processes the problem of maximum likelihood estimation of its parameters from the data however which consist of a sample of independent realizations of this process has not been considered in the literature i present a maximum likelihood procedure for the estimation of the parameters of the mover stayer model and develop a recursive method of computation of maximum likelihood estimators that is very simple to implement i also verify that obtained maximum likelihood estimators are strongly consistent i show that the two estimators of the parameters of the mover stayer model previously proposed in the literature are special cases of the maximum likelihood estimator derived in this article that is they coincide with the maximum likelihood estimator under special conditions i thus explain the interconnection between existing estimators i also pre

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

the boltzmann equation with large amplitude initial data in bounded domains abstract the paper is devoted to constructing the global solutions around global maxwellians to the initial boundary value problem on the boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries we allow a class of non negative initial data which have arbitrary large amplitude and even contain vacuum the result shows that the oscillation of solutions away from global maxwellians becomes small after some positive time provided that they are initially close to each other in l 2 this yields the disappearance of any initial vacuum and the exponential convergence of large amplitude solutions to equilibrium in large time the isothermal diffuse reflection boundary condition plays a vital role in the analysis the most key ingredients in our strategy of the proof include i l x v 2 l x l v 1 l x v estimates along a bootstrap argument ii pointwise estimates on the upper bound of the gain term by the product of l norm and l 2 norm iii an iterative procedure on the nonlinear term

the design of a panel study under an alternating poisson process assumption we discuss the design of a panel study for the estimation of the average durations in the two states of an alternating poisson process two types of designs are examined the first fixes the number of follow up waves but permits the total length of follow up to vary and the second varies the number of follow up waves with fixed length simple expressions for nearly optimal designs are presented and we compare these with designs allowing for continuous observation both equilibrium and nonequilibrium cases are examined an example is given using morbidity data to illustrate how these results can be applied

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

on equations of benjamin bona mahony type abstract the objective of this paper is to establish existence uniqueness and regularity of solutions for a mixed problem associated with equations of benjamin bona mahony type in a domain q with moving boundary the technique to show the existence of solutions consists in transforming q into a cylinder q by using a diffeomorphism t t and to apply in q the faedo galerkin method in the transformed mixed problem the uniqueness of solutions is proved by using the energy method in the non cylindrical domain q

the analysis of panel data under a markov assumption abstract methods for the analysis of panel data under a continuous time markov model are proposed we present procedures for obtaining maximum likelihood estimates and associated asymptotic covariance matrices for transition intensity parameters in time homogeneous models and for other process characteristics such as mean sojourn times and equilibrium distributions generalizations to handle covariance analysis and to the fitting of certain nonhomogeneous models are presented and an example based on a longitudinal study of the smoking habits of school children is discussed questions of embeddability and estimation are examined

abstract time dependent transport equations abstract we consider the abstract time dependent linear transport equation as an initial boundary value evolution problem in the banach spaces lp 1 p

coupled system of korteweg de vries equations type in domains with moving boundaries we consider the initial boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for the coupled system of equations of korteweg de vries kdv type modelling strong interactions between internal solitary waves finite domains of wave propagation changing in time arise naturally in certain practical situations when the equations are used as a model for waves and a numerical scheme is needed we prove a global existence and uniqueness for strong solutions for the coupled system of equations of kdv type as well as the exponential decay of small solutions in asymptotically cylindrical domains finally we present a numerical scheme based on semi implicit finite differences and we give some examples to show the numerical effect of the moving boundaries for this kind of systems

the boltzmann equation with weakly inhomogeneous data in bounded domain this paper is concerned with the boltzmann equation with specular reflection boundary condition we construct a unique global solution and obtain its large time asymptotic behavior in the case that the initial data is close enough to a radially symmetric homogeneous datum the result extends the case of cauchy problem considered by arkeryd esposito pulvirenti 4 to the specular reflection boundary value problem in bounded domain

on weak and strong convergence to equilibrium for solutions to the linear boltzmann equation this paper considers the linear space inhomogeneous boltzmann equation for a distribution function in a bounded domain with general boundary conditions together with an external potential force the paper gives results on strong convergence to equilibrium whent for general initial data first in the cutoff case and then for infinite range collision forces the proofs are based on the properties of translation continuity and weak convergence to equilibrium to handle these problems generalh theorems concerning monotonicity in time of convex entropy functionals are presented furthermore the paper gives general results on collision invariants i e on functions satisfying detailed balance relations in a binary collision

asymptotic behavior of an initial boundary value problem for the vlasov poisson fokker planck system the asymptotic behavior for the vlasov poisson fokker planck system in bounded domains is analyzed in this paper boundary conditions defined by a scattering kernel are considered it is proven that the distribution of particles tends for large time to a maxwellian determined by the solution of the poisson boltzmann equation with dirichlet boundary condition in the proof of the main result the conservation law of mass and the balance of energy and entropy identities are rigorously derived an important argument in the proof is to use a lyapunov type functional related to these physical quantities

smooth solutions to the bgk equation and the es bgk equation with infinite energy abstract the bgk model and the es bgk model of the boltzmann equation are of great importance in the kinetic theory of rarefied gases for the cauchy problems with general initial data smooth solutions were obtained only in bounded domains for both models in this paper we establish the existence and uniqueness of smooth solutions in the whole space more important some smooth solutions can permit infiniteness of energy

initial boundary value problems for the boltzmann equation global existence of weak solutions for an open set of 3 bounded or not we consider initial boundary value problems for the boltzmann equation for general gas surface interaction laws and for hard potentials we prove a global existence result for weak solutions the proof uses the regularization of the collision operator and the renormalization method for the regularized problem by using weak compactness in l1 and averaged stability ofq f f we prove the existence of weak solutions of our problem

solutions of the boussinesq equation on a periodic domain abstract the existence and the uniqueness of solutions for a linear feedback controlled boussinesq equation on a periodic domain are studied the continuous dependence of the solution on initial data is also proved the proof is based on conservation laws for the boussinesq equation

global existence of classical solutions to the fokker planck bgk equation a kinetic model of the fokker planck boltzmann equation is introduced by replacing the original boltzmann collision operator with the bhatnagar gross krook collision model bgk collision model this model equation which we call the fokker planck bgk equation has many physical features that the fokker planck boltzmann equation possesses we first establish an l existence result for this equation by which we construct the approximate solutions then by means of the regularizing effects of the linear fokker planck operator and lp estimates of local maxwellians we obtain some uniform estimates of the approximate solutions finally combining those estimates and regularizing effects we prove by a compactness argument that the equation has a global classical solution under rather general initial conditions

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

some applications of the method of moments for the homogeneous boltzmann and kac equations using the method of moments we prove that any polynomial moment of the solution of the homogeneous boltzmann equation with hard potentials or hard spheres is bounded provided that a moment of order strictly higher than 2 exists initially we also give partial results of convergence towards the maxwellian equilibrium in the case of soft potentials finally exponential as well as maxwellian estimates are introduced for the kac equation

trend to equilibrium of weak solutions of the boltzmann equation in a slab with diffusive boundary conditions recently r illner and the author proved that under a physically realistic truncation on the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense here it is proved that when the maxwellians associated with the boundary conditions atx 0 andx 1 are the same maxwellianmw then the solution is uniformly bounded and tends tomw fort

a global existence theorem for the initial boundary value problem for the boltzmann equation when the boundaries are not isothermal we extend the existence theorem recently proved by hamdache for the initial boundary value problem for the nonlinear boltzmann equation in a vessel with isothermal boundaries to more general situations including the case when the boundaries are not isothermal in the latter case a cut off for large speeds is introduced in the collision term of the boltzmann equation

global weak solutions of the boltzmann equation in a slab with diffusive boundary conditions it is proved that under a physically realistic truncation of the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense this solution satisfies the boundary conditions almost everywhere and has at worst exponentially growing total energy

exponential decay towards equilibrium for the inhomogeneous aizenman bak model the aizenman bak model for reacting polymers is considered for spatially inhomogeneous situations in which they diffuse in space with a non degenerate size dependent coefficient both the break up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels we demonstrate that the entropy entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non degenerate diffusion in dimension one we finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial up to our knowledge this is the first result of explicit equilibration rates for spatially inhomogeneous coagulation fragmentation models

on the initial boundary value problem for the boltzmann equation a theorem on the traces of the solutions of initial boundary value problems for the boltzmann equation is proved this result makes it possible to extend a recent theorem of existence proved by hamdache to more realistic situations

lp solutions to the cauchy problem of the bgk equation the bgk model of the boltzmann equation plays an important role in the kinetic theory of rarefied gases some existence and uniqueness results of solutions to its cauchy problem were established for large initial data under various circumstances see for example perthame b global existence to the bgk model of boltzmann equation j differ equations 82 191 205 1989 in this paper by establishing some weighted lp estimates of the hydrodynamical quantities of a gas we prove the existence theorem of the lp solutions to the cauchy problem and establish the propagation properties of the lp moments for this kind of solutions

the quantum liouville bgk equation and the moment problem abstract this work is devoted to the analysis of the quantum liouville bgk equation this equation arises in the work of degond and ringhofer on the derivation of quantum hydrodynamical models from first principles their theory consists in transposing to the quantum setting the closure strategy by entropy minimization used for kinetic equations the starting point is the quantum liouville bgk equation where the collision term is defined via a so called quantum local equilibrium defined as a minimizer of the quantum free energy under a local density constraint we then address three related problems we prove new results about the regularity of these quantum equilibria we prove that the quantum liouville bgk equation admits a classical solution and we investigate the long time behavior of the solutions the core of the proofs is based on a fine analysis of the properties of the minimizers of the free energy

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

on the dissipative boussinesq equation in a non cylindrical domain abstract we investigate the initial boundary value problem for the one dimensional nonlinear boussinesq equation inside domains with moving ends having both small increasing and decreasing displacements global solvability uniqueness of solutions and the exponential decay to the energy are established provided the initial data are bounded in some sense