global solutions and time decay of the non cutoff vlasov maxwell boltzmann system in the whole space the global in time classical solutions to the cauchy problem of the vlasov maxwell boltzmann system near maxwellians are obtained under the lower regularity index assumption and the weaker smallness condition on the initial perturbation in comparison with the work duan et al in kinet relat models 6 1 159 204 2013 in particular we show the relation between time decay rates and spatial derivatives of solutions our analysis relies on a refined energy estimates and the interpolation techniques between negative sobolev norms and positive sobolev norms without linear decay analysis

lp stability of solutions of boltzmann equation with external force in soft potentials in this paper we present a uniform lp stability theory for the solutions to boltzmann equation with small or large external forces in vacuum in soft potentials we assume that the initial data are sufficiently small and a fast decay in space and velocity our stability analysis is divided into two parts for small external forces we first consider the time derivative of the pth power of lp norm of the distance between two mild solutions f and g then define nonlinear functionals and apply the gronwall s inequality to obtain the lp stability for large external forces we make use of the constructive assumptions on external force minkowski s inequality and gronwall s inequality to derive the lp stability estimate

decay of the two species vlasov poisson boltzmann system specialized research fund for the doctoral program of higher education 20120121120023 natural science foundation of fujian province of china 2012j05011 national natural science foundation of china 11201389

uniform stability of solutions of boltzmann equation for soft potential with external force abstract the study of cauchy problem of the boltzmann equation is important in both theory and applications existence of global solutions to the equation and uniform stability of solutions in the absence of external force were introduced in the previous work on the boltzmann equation in this paper we will investigate the uniform stability of solutions in l 1 for the cauchy problem of the boltzmann equation when there is an external force for the case of soft potentials

the vlasov poisson boltzmann system with angular cutoff for soft potentials abstract this paper is concerned with the cauchy problem of the vlasov poisson boltzmann system near a given global maxwellian with angular cutoff for a class of soft potentials in three space dimensions and the main purpose here is to derive the global solvability of such a problem without the neutral condition which was imposed in duan et al 2013 11 on the initial perturbation our analysis is based on a time weighted energy method and some delicate estimates on its solutions which show that the l 2 r x 3 r 3 norms of the solutions with higher order x derivatives enjoy sharper time decay rates

the same class of stationary solutions to some multidimensional kinetic systems with extensive background density abstract in this study we investigate the existence and uniqueness of the same class of stationary solutions for some kinetic systems comprising the vlasov poisson fokker planck system vlasov poisson boltzmann system and vlasov maxwell boltzmann system in arbitrary space dimensions n n 3 essentially the problem can be reduced to solving the problem of a second order elliptic equation with exponential nonlinearity this result had been proved in three spatial dimensions but the extension to a higher dimensional setting makes the existence proof nontrivial in particular it necessary to highlight that the requirement regarding the background density function is relaxed compared with previous studies in the case of three spatial dimensions

energy method for boltzmann equation abstract a basic simple energy method for the boltzmann equation is presented here it is based on a new macro micro decomposition of the boltzmann equation as well as the h theorem this allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method as an illustration we apply the method for the study of the time asymptotic nonlinear stability of the global maxwellian states previous energy method starting with grad and finishing with ukai involves the spectral analysis and regularity of collision operator through sophisticated weighted norms

classical solutions to the boltzmann equation for molecules with an angular cutoff an important class of collision kernels in the boltzmann theory are governed by the inverse power law in which the intermolecular potential between two particles is an inverse power of their distance under the grad angular cutoff assumption global in time classical solutions near maxwellians are constructed in a periodic box for all soft potentials with 3 0

relation between volatility correlations in financial markets and omori processes occurring on all scales we analyze the memory in volatility by studying volatility return intervals defined as the time between two consecutive fluctuations larger than a given threshold in time periods following stock market crashes such an aftercrash period is characterized by the omori law which describes the decay in the rate of aftershocks of a given size with time t by a power law with exponent close to 1 a shock followed by such a power law decay in the rate is here called omori process we find self similar features in the volatility specifically within the aftercrash period there are smaller shocks that themselves constitute omori processes on smaller scales similar to the omori process after the large crash we call these smaller shocks subcrashes which are followed by their own aftershocks we also show that the omori law holds not only after significant market crashes as shown by lillo and mantegna phys rev e 68 016119 2003 but also after intermediate shocks by appropriate detrending we remove the influence of the crashes and subcrashes from the data and find that this procedure significantly reduces the memory in the records moreover when studying long term correlated fractional brownian motion and autoregressive fractionally integrated moving average artificial models for volatilities we find omori type behavior after high volatilities thus our results support the hypothesis that the memory in the volatility is related to the omori processes present on different time scales

automatic algorithm for monotone trend removal the available numerical algorithms for trend removal require a direct subjective intervention in choosing critical parameters in this paper an algorithm is presented which needs no initial subjective assumptions monotone trends are approximated by piecewise linear curves obtained by dividing into subintervals the signal values interval not the time interval the slope of each linear segment of the estimated trend is proportional to the average one step displacement of the signal values included into the corresponding subinterval the evaluation of the trend removal is performed on statistical ensembles of artificial time series with the random component given by realizations of autoregressive of order one stochastic processes or by fractional brownian motions the accuracy of the algorithm is compared with that of two well tested methods polynomial fitting and a nonparametric method based on moving average for stationary noise the results of the algorithm are slightly better but for nonstationary noise the preliminary results indicate that the polynomial fitting has the best accuracy as a verification on a real time series the time periods with monotone variation of global average temperature over the last 1800 phantom rule 0 3em 0ex text years are established the removal of a nonmonotone trend is also briefly discussed

fluid dynamic limit to rarefaction wave for the boltzmann equation abstract the hydrodynamic limit for the boltzmann equation to the compressible euler equation as knudsen number e vanishes is a difficult and challenging problem in the mathematics when the corresponding compressible euler equation has a single rarefaction wave xin and zeng 2010 23 recently verified the hydrodynamic limit as e tends to zero with a convergence rate e 1 5 ln e in this paper the convergence rate of xin and zeng 2010 23 is improved to e 1 3 ln e 2 by different scaling arguments

global existence of classical solutions to the vlasov poisson boltzmann system with given magnetic field abstract this paper considers the vlasov poisson boltzmann system with given magnetic field the global existence of classical solutions was obtained when the initial data is a small perturbation around a global maxwellian the proof is based on the theory of compressible navier stokes poisson equations with forcing and the macro microdecomposition of the solution to the boltzmann equation with respect to the local maxwellian introduced in t p liu t yang s h yu energy method for the boltzmann equation physica d 188 3 4 2004 178 192 and elaborated in t yang h j zhao a new energy method for the boltzmann equation j math phys 47 2006 the result shows that the existence of solutions is independent of the magnetic field

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 cauchy problem for the bgk equation with an external force abstract in this paper we study the cauchy problem for the bgk equation with an external force firstly we establish an l existence result for this equation and obtain some weighted l estimates then by means of the regularizing effects to the initial datum we construct the approximate solutions and obtain some uniform estimates of the approximate solutions finally by using compactness method and passing to the limit we prove the existence theorems of the l 1 and l p solutions and establish the propagation properties of the l p moments

hydrodynamic limit of the boltzmann equation with contact discontinuities the hydrodynamic limit for the boltzmann equation is studied in the case when the limit system that is the system of euler equations contains contact discontinuities when suitable initial data is chosen to avoid the initial layer we prove that there exist a family of solutions to the boltzmann equation globally in time for small knudsen number and this family of solutions converge to the local maxwellian defined by the contact discontinuity of the euler equations uniformly away from the discontinuity as the knudsen number e tends to zero the proof is obtained by an appropriately chosen scaling and the energy method through the micro macro decomposition

cercignani s conjecture is sometimes true and always almost true we establish several new functional inequalities comparing boltzmann s entropy production functional with the relative h functional first we prove a longstanding conjecture by cercignani under the nonphysical assumption that the boltzmann collision kernel is superquadratic at infinity the proof rests on the method introduced in 39 combined with a novel use of the blachman stam inequality if the superquadraticity assumption is not satisfied then it is known that cercignani s conjecture is not true however we establish a slightly weakened version of it for all physically relevant collision kernels thus extending previous results from 39 finally we consider the entropy entropy production version of kac s spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension the first two results are sharp in some sense and the third one is likely to be too they contain all previously known entropy estimates as particular cases this gives a first coherent picture of the study of entropy production according to a program started by carlen and carvalho 12 ten years ago these entropy inequalities are one step in our study of the trend to equilibrium for the boltzmann equation in both its spatially homogeneous and spatially inhomogeneous versions

the landau equation in a periodic box the landau equation which was proposed by landau in 1936 is a fundamental equation to describe collisions among charged particles interacting with their coulombic force in this article global in time classical solutions near maxwellians are constructed for the landau equation in a periodic box our result also covers a class of generalized landau equations which describes grazing collisions in a dilute gas

interaction between deterministic trend and autoregressive process 1 when both an autoregressive ar process stochastic trend and a deterministic trend systematic changes over mean exist within a time series there may be some interactions between them this study investigates whether these two components interact with each other only time series consisting of a linear trend and a lag 1 autoregressive ar 1 process are explored which are commonly used in hydrology results indicate that 1 the presence of a deterministic trend will overestimate positive serial correlation and underestimate negative serial correlation while 2 the existence of an ar 1 process does not affect the estimate of the magnitude of the deterministic trend however a positive ar 1 will inflate the variance of the trend and a negative ar 1 will shrink the variance of the trend for a time series with a deterministic trend the trend has to be removed in order to correctly model the stochastic properties of the time series in the literature both differencing and detrending have been proposed to fulfill such a task their applicability is based on the assumption that they cannot distort the residual series however no evidence has been provided to certify whether this assumption is appropriate this study examines this issue and it indicates that differencing can remove a deterministic trend from a time series but it can seriously damage the existing ar 1 process in contrast to differencing detrending can remove a deterministic trend without distorting the existing ar process

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

a half space problem for the boltzmann equation with specular reflection boundary condition there are many open problems on the stability of nonlinear wave patterns to the boltzmann equation even though the corresponding stability theory has been comparatively well established for the gas dynamical systems in this paper we study the nonlinear stability of a rarefaction wave profile to the boltzmann equation with the boundary effect imposed by specular reflection for both the hard sphere model and the hard potential model with angular cut off the analysis is based on the property of the solution and its derivatives which are either odd or even functions at the boundary coming from specular reflection and the decomposition on both the solution and the boltzmann equation introduced in 24 26 for energy method

global existence to boltzmann equation with external force in infinite vacuum in this paper we give a condition on the bicharacteristic which guarantees the global existence of the mild solution to the boltzmann equation with an external force for the hard sphere model and potentials with angular cutoff in infinite vacuum this generalizes the previous results to the case when the force can have arbitrary strength the constructive condition on the bicharacteristic is used to obtain the pointwise estimates on the collision operator so that the global existence comes from the contraction mapping theorem

effect of nonlinear filters on detrended fluctuation analysis when investigating the dynamical properties of complex multiple component physical and physiological systems it is often the case that the measurable system s output does not directly represent the quantity we want to probe in order to understand the underlying mechanisms instead the output signal is often a linear or nonlinear function of the quantity of interest here we investigate how various linear and nonlinear transformations affect the correlation and scaling properties of a signal using the detrended fluctuation analysis dfa which has been shown to accurately quantify power law correlations in nonstationary signals specifically we study the effect of three types of transforms i linear y i a x i b ii nonlinear polynomial y i a x k i and iii nonlinear logarithmic y i log x i delta filters we compare the correlation and scaling properties of signals before and after the transform we find that linear filters do not change the correlation properties while the effect of nonlinear polynomial and logarithmic filters strongly depends on a the strength of correlations in the original signal b the power k of the polynomial filter and c the offset delta in the logarithmic filter we further apply the dfa method to investigate the apparent scaling of three analytic functions i exponential exp x a ii logarithmic log x a and iii power law x a lambda which are often encountered as trends in physical and biological processes while these three functions have different characteristics we find that there is a broad range of values for parameter a common for all three functions where the slope of the dfa curves is identical we further note that the dfa results obtained for a class of other analytic functions can be reduced to these three typical cases we systematically test the performance of the dfa method when estimating long range power law correlations in the output signals for different parameter values in the three types of filters and the three analytic functions we consider

stability of the front under a vlasov fokker planck dynamics we consider a kinetic model for a system of two species of particles interacting through a long range repulsive potential and a reservoir at given temperature the model is described by a set of two coupled vlasov fokker plank equations the important front solution which represents the phase boundary is a stationary solution on the real line with given asymptotic values at infinity we prove the asymptotic stability of the front for small symmetric perturbations

exponential decay of the landau equation with potential forces abstract the cauchy problem of the landau equation with potential forces on torus is investigated the global existence of solutions with the symmetry of origin and the exponential convergence rate in time to the steady state are obtained for any initially smooth periodic origin symmetric small perturbation

on the cauchy problem of the vlasov poisson bgk system global existence of weak solutions global in time existence of weak solutions to the cauchy problem of the three dimensional vlasov poisson bgk system is shown for initial data belonging to the space lp 3 3 with p 9 and having finite second order velocity moments this result solves partially the well posed problem for the vlasov poisson bgk system proposed by b perthame higher moments for kinetic equations the vlasov poisson and fokker planck cases math meth appl sci 13 441 452 1990

diffusive expansion for solutions of the boltzmann equation in the whole space this paper is concerned with the diffusive expansion for solutions of the rescaled boltzmann equation in the whole space 0 1 tf v xf 1 q f f x rn v rn t 0 with prescribed initial data 0 2 f t x v t 0 f0 x v x rn v rn our main purpose is to justify the global validity of the diffusive expansion 0 3 f t x v f1 t x v 2f2 t x v n 1fn 1 t x v nfn t x v for a solution f t x v of the rescaled boltzmann equation 0 1 in the whole space rn for all t 0 with initial data f0 x v satisfying the initial expansion 0 4 f0 x v f1 0 x v 2f2 0 x v n 1fn 1 0 x v nfn 0 x v x v r2n here v 2 n2exp v 22 is a normalized global maxwellian under the assumption that the fluid components of the coefficients fm 0 x v 1 m n of the initial expansion f0 x v have divergence free velocity fields um0 x as well as temperature fields m0 x if we assume further that the velocity temperature fields u10 x 10 x of f1 0 x v have small amplitude in hs rn s 2 n n 2 we can determine these coefficients fm t x v 1 m n in the diffusive expansion 0 3 uniquely by an iteration method and energy method the hydrodynamic component of these coefficients fm t x v 1 m n satisfies the incompressible condition the boussinesq relations and or the navier stokes fourier system respectively while the microscopic component of these coefficients is determined by a recursive formula compared with the corresponding problem inside a periodic box studied in y guo 2006 18 the main difficulty here is due to the fact that poincare s inequality is not valid in the whole space rn and this difficulty is overcome by using the lp lq estimate on the riesz potential moreover by exploiting the energy method we can also deduce certain the space time energy estimates on these coefficients fm t x v 1 m n once the coefficients fm t x v 1 m n in the diffusive expansion 0 3 are uniquely determined and some delicate estimates have been obtained the uniform estimates with respect to on the remainders fn t x v are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion 0 3 in the large provided that 0 5 n 2n 2 notice that for m 2 um t x is no longer a divergence free vector and it is worth to pointing out that for m 3 it was in deducing certain estimates on pm t x by the lp lq estimate on the riesz potential that we need to require that n 2n 2

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

nonlinear stability of rarefaction waves for the boltzmann equation it is well known that the boltzmann equation is related to the euler and navier stokes equations in the field of gas dynamics the relation is either for small knudsen number or for dissipative waves in the time asymptotic sense in this paper we show that rarefaction waves for the boltzmann equation are time asymptotic stable and tend to the rarefaction waves for the euler and navier stokes equations our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro macro decomposition of the boltzmann equation the expansion nature of the rarefaction waves and the suitable microscopic version of the h theorem are essential elements of our analysis

on the cauchy problem for the boltzmann equation in the whole space global existence and uniform stability in lξ2 hxn abstract based on a refined energy method in this paper we prove the global existence and uniform in time stability of solutions in the space l 2 h x n to the cauchy problem for the boltzmann equation around a global maxwellian in the whole space r 3 compared with the solution space used by the spectral analysis and the classical energy method the velocity weight functions or time derivatives need not be included in the norms of l 2 h x n which is realized by introducing some temporal interactive energy functionals to estimate the macroscopic dissipation rate the key proof is carried out in terms of the macroscopic equations together with the local conservation laws it is also found that the perturbed macroscopic variables actually satisfy the linearized compressible navier stokes equations with remaining terms only related to the microscopic part

global weak solutions to the cometary flow equation with a self generated electric field abstract we study the cauchy problem of a cometary flow equation with a self generated electric field this kinetic model originates from the theory of astrophysical plasmas and can be viewed as a perturbation by a wave particle collision operator of the classical vlasov poisson system by asymptotic methods in kinetic theory global existence of nonnegative weak solutions to the cauchy problem in three space variables is established for bounded initial data having finite second order velocity moments

exponential decay for soft potentials near maxwellian we consider both soft potentials with angular cutoff and landau collision kernels in the boltzmann theory inside a periodic box we prove that any smooth perturbation near a given maxwellian approaches zero at the rate of e lambda t p for some 0 and 0 p 1 our method is based on an unified energy estimate with appropriate exponential velocity weight our results extend the classical result caflisch of 2 to the case of very soft potential and coulomb interactions and also improve the recent almost exponential decay results by 5 14

on the trend to equilibrium for the vlasov poisson boltzmann equation abstract the dynamics of dilute electrons and plasma can be modeled by vlasov poisson boltzmann equation for which the equilibrium state can be a global maxwellian in this paper we show that the rate of convergence to equilibrium is o t by using a method developed for the boltzmann equation without external force in l desvillettes c villani on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation invent math 159 2005 245 316 in particular the idea of this method is to show that the solution f cannot stay near any local maxwellians for long the improvement in this paper is to handle the effect from the external force governed by the poisson equation moreover by using the macro micro decomposition we simplify the estimation on the time derivatives of the deviation of the solution from the local maxwellian with same macroscopic components

scaling properties and entropy of long range correlated time series we calculate the shannon entropy of a time series by using the probability density functions of the characteristic sizes of the long range correlated clusters introduced in a carbone g castelli h e stanley phys rev e 69 2004 026105 we define three different measures of the entropy related respectively to the length the duration and the area of the clusters for all the three cases the entropy increases as the logarithm of a power of the size with exponents equal to the fractal dimension of the cluster length duration and area respectively r 2007 elsevier b v all rights reserved

cauchy problem for the vlasov poisson boltzmann system the dynamics of dilute electrons can be modelled by the fundamental vlasov poisson boltzmann system which describes mutual interactions of the electrons through collisions in the self consistent electric field in this paper it is shown that any smooth perturbation of a given global maxwellian leads to a unique global in time classical solution when either the mean free path is small or the background charge density is large moreover the solution converges to the global maxwellian when time tends to infinity the analysis combines the techniques used in the study of conservation laws with the decomposition of the boltzmann equation introduced in 17 19 by obtaining new entropy estimates for this physical model

on the global existence of mild solutions to the boltzmann equation for small data in ld we develop a new theory of existence of global solutions to the boltzmann equation for small initial data these new mild solutions are analogous to the mild solutions for the navier stokes equations the existence comes as a result of the study of the competing phenomena of dispersion due to the transport operator and of singularity formation due to the nonlinear boltzmann collision operator it is the joint use of the so called dispersive estimates with new convolution inequalities on the gain term of the collision operator that allows to obtain uniform bounds on the solutions and thus demonstrate the existence of solutions

existence of stationary solutions to the vlasov poisson boltzmann system abstract in this paper we study the existence of stationary solutions to the vlasov poisson boltzmann system when the background density function tends to a positive constant with a very mild decay rate as x in fact the stationary vlasov poisson boltzmann system can be written into an elliptic equation with exponential nonlinearity under the assumption on the decay rate being ln e x for some 0 it is shown that this elliptic equation has a unique solution this result generalizes the previous work r glassey j schaeffer y zheng steady states of the vlasov poisson fokker planck system j math anal appl 202 1996 1058 1075 where the decay rate 1 x 1 2 is assumed

negative sobolev spaces and the two species vlasov maxwell landau system in the whole space abstract a global solvability result of the cauchy problem of the two species vlasov maxwell landau system near a given global maxwellian is established by employing an approach different than that of 2 compared with that of 2 the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker our analysis does not rely on the decay of the corresponding linearized system and the duhamel principle and thus it can be used to treat the one species vlasov maxwell landau system for the case of 3 and the one species vlasov maxwell boltzmann system for the case of 1 1 to deduce the global existence results together with the corresponding temporal decay estimates

contact discontinuity with general perturbations for gas motions the contact discontinuity is one of the basic wave patterns in gas motions the stability of contact discontinuities with general perturbations for the navier stokes equations and the boltzmann equation is a long standing open problem general perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties in this paper we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the navier stokes equations and the boltzmann equation in a uniform way one of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of thus this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the l8 norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

boundary conditions for the microscopic fene models we consider the microscopic equation of finite extensible nonlinear elasticity fene models for polymeric fluids under a steady flow field it is shown that for the underlying fokker planck type of equations any preassigned distribution on the boundary will become redundant once the nondimensional number text it li frac hb k bt geq 2 where h is the elasticity constant sqrt b is the maximum dumbbell extension t is the temperature and k b is the usual boltzmann constant moreover if the probability density function is regular enough for its trace to be defined on the sphere m sqrt b then the trace is necessarily zero when text it li 2 these results are consistent with our numerical simulations as well as some recent well posedness results by preassuming a zero boundary distribution

the vlasov maxwell boltzmann system near maxwellians perhaps the most fundamental model for dynamics of dilute charged particles is described by the vlasov maxwell boltzmann system in which particles interact with themselves through collisions and with their self consistent electromagnetic field despite its importance no global in time solutions weak or strong have been constructed so far it is shown in this article that any initially smooth periodic small perturbation of a given global maxwellian which preserves the same mass total momentum and reduced total energy 22 leads to a unique global in time classical solution for such a master system the construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation the linear collision operator l not its time integration is positive definite for any solutionf t x v with small amplitude to the vlasov maxwell boltzmann system 8 and 12 as a by product such an estimate also yields an exponential decay for the simpler vlasov poisson boltzmann system 24

the vlasov poisson boltzmann system without angular cutoff this paper is concerned with the vlasov poisson boltzmann system for plasma particles of two species in three space dimensions the boltzmann collision kernel is assumed to be angular non cutoff with 3 2s and 1 2 s 1 where s are two parameters describing the kinetic and angular singularities respectively we establish the global existence and convergence rates of classical solutions to the cauchy problem when initial data is near maxwellians this extends the results in duan et al j diff eqs 252 12 6356 6386 2012 math models methods appl sci 23 6 927 2013 for the cutoff kernel with 2 1 to the case 3 2 as long as the angular singularity exists instead and is strong enough i e s is close to 1 the proof is based on the time weighted energy method building also upon the recent studies of the non cutoff boltzmann equation in gressman and strain j amer math soc 24 3 771 847 2011 and the vlasov poisson landau system in guo j amer math soc 25 759 812 2012

hard frequency domain model error bounds from least squares like identification techniques the problem of deriving so called hard error bounds for estimated transfer functions is addressed a hard bound is one that is sure to be satisfied i e the true system nyquist plot will be confined with certainty to a given region provided that the underlying assumptions are satisfied by blending a priori knowledge and information obtained from measured data it is shown how the uncertainty of transfer function estimates can be quantified the emphasis is on errors due to model mismatch the effects of unmodeled dynamics can be considered as bounded disturbances hence techniques from set membership identification can be applied to this problem the approach taken corresponds to weighted least squares estimation and provides hard frequency domain transfer function error bounds the main assumptions used in the current contribution are that the measurement errors are bounded that the true system is indeed linear with a certain degree of stability and that there is some knowledge about the shape of the true frequency response

stability of the boltzmann equation with potential forces on torus abstract in this paper we are concerned with the stability of solutions to the cauchy problem of the boltzmann equation with potential forces on torus it is shown that the natural steady state with the symmetry of origin is asymptotically stable in the sobolev space with exponential rate in time for any initially smooth periodic origin symmetric small perturbation which preserves the same total mass momentum and mechanical energy for the non symmetric steady state it is also shown that it is stable in l 1 norm for any initial data with the finite total mass mechanical energy and entropy

on a boundary layer problem for the nonlinear boltzmann equation this article deals with a boundary layer problem arising in the kinetic theory of gases when the mean free path of molecules tends to zero the model considered here is the stationary nonlinear boltzmann equation in one dimension with a slightly perturbed reflection boundary condition we restrict our attention to the case of hard spheres collisions with grad s cutoff assumption existence uniqueness and asymptotic behavior are derived by means of energy estimates

optimal large time behavior of the vlasov maxwell boltzmann system in the whole space in this paper we study the large time behavior of classical solutions to the two species vlasov maxwell boltzmann system in the whole space input amssym bbb r 3 the existence of global in time nearby maxwellian solutions is known from strain in 2006 however the asymptotic behavior of these solutions has been a challenging open problem building on our previous work on time decay for the simpler vlasov poisson boltzmann system we prove that these solutions converge to the global maxwellian with the optimal decay rate of o t 3 2 3 2r in the l l norm for any 2 r if initial perturbation is smooth enough and decays in space velocity fast enough at infinity moreover some explicit rates for the electromagnetic field tending to 0 are also provided 2011 wiley periodicals inc

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

closed loop system identification using the dual youla control parametrization identification of an unknown plant operated in a closed loop environment in the presence of external disturbance is considered under closed loop experiment the objectives of system identification are not only to obtain a best fit of the unknown plant but also to make close prediction or estimation of closed loop system responses by using the a priori information of a known stabilizing controller and the dual youla control parametrization identification of an unknown plant in a closed loop system can be simplified to that of an open loop reduced system it is shown that the output prediction errors of the estimates for both the original plant and the reduced system are equal furthermore if the feedback controller is of normalized coprime factorization then to find a best fit of the reduced system amounts to establishing a nominal closed loop system with which the signals of the true closed loop system can be predicted in a least squares sense in addition the optimal experimental designs for reduci

the vlasov poisson boltzmann system for the whole range of cutoff soft potentials abstract the dynamics of dilute electrons can be modeled by the fundamental one species vlasov poisson boltzmann system which describes mutual interactions of the electrons through collisions in the self consistent electrostatic field for cutoff intermolecular interactions although there is some progress on the construction of global smooth solutions to its cauchy problem near maxwellians recently the problem for the case of very soft potentials remains unsolved by introducing a new time velocity weighted energy method and based on some new optimal temporal decay estimates on the solution itself and some of its derivatives with respect to both the spatial and the velocity variables it is shown in this manuscript that the cauchy problem of the one species vlasov poisson boltzmann system for all cutoff soft potentials does exist a unique global smooth solution for general initial perturbation which is unnecessary to satisfy the neutral condition imposed in 13 for the case of cutoff moderately soft potentials but is assumed to be small in certain weighted sobolev spaces our approach applies also to the case of cutoff hard potentials and thus provides a satisfactory global well posedness theory to the one species vlasov poisson boltzmann system near maxwellians for the whole range of cutoff intermolecular interactions in the perturbative framework

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

critical scale invariance in a healthy human heart rate we demonstrate the robust scale invariance in the probability density function pdf of detrended healthy human heart rate increments which is preserved not only in a quiescent condition but also in a dynamic state where the mean level of the heart rate is dramatically changing this scale independent and fractal structure is markedly different from the scale dependent pdf evolution observed in a turbulentlike cascade heart rate model these results strongly support the view that a healthy human heart rate is controlled to converge continually to a critical state

the vlasov poisson landau system in a periodic box the classical vlasov poisson landau system describes dynamics of a collisional plasma interacting with its own electrostatic eld as well as its grazing collisions such grazing collisions are modeled by the famous landau fokker planck collision kernel proposed by landau in 1936 we construct global unique solutions to such a system for initial data which have small weighted h norms but can have large h k 3 norms with high velocity moments our construction is based on accumulative study on the landau kernel in the past decade g1 sg1 3 with four extra ingredients to overcome the speci c mathematical di culties present in the vlasov poisson landau system a new exponential weight of electric potential to cancel the growth of the velocity a new velocity weight to capture the weak velocity di usion in the landau kernel a decay of the electric eld to close the energy estimate and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude

stability of rarefaction waves for the bipolar vlasov poisson boltzmann system abstract in this paper we construct the global solutions near a local maxwellian for the one dimensional bipolar vlasov poisson boltzmann system the macroscopic components of this local maxwellian are the approximate rarefaction wave solutions to the associated one dimensional compressible euler equations thus we prove the stability of the rarefaction waves for the bipolar vlasov poisson boltzmann system in the weighted function space moreover we obtain some time decay rates of the disparity between two species and the electric field which decay at an exponential rate

effect of trends on detrended fluctuation analysis scaling behavior we find that crossovers result from the competition between the scaling of the noise and the apparent scaling of the trend we study how the characteristics of these crossovers depend on i the slope of the linear trend ii the amplitude and period of the periodic trend iii the amplitude and power of the power law trend and iv the length as well as the correlation properties of the noise surprisingly we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws i e long range power law dependence of the position of the crossover on the parameters of the trends we show that the dfa result of noise with a trend can be exactly determined by the superposition of the separate results of the dfa on the noise and on the trend assuming that the noise and the trend are not correlated if this superposition rule is not followed this is an indication that the noise and the superposed trend are not independent so that removing the trend could lead to changes in the correlation properties of the noise in addition we show how to use dfa appropriately to minimize the effects of trends how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation or if the crossover is due to a trend without any transition in the dynamical properties of the noise

asymptotic stability of traveling wave solutions of systems for one dimensional gas motion the asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics 1 the solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet for some 0 ast provided that the initial disturbance is small and of integral zero and in addition decays at an algebraic rate for x 2 the traveling wave solution with nishida and smoller s condition of the system of a viscous heat conductive ideal gas is asymptotically stable provided the initial disturbance is small and of integral zero 3 the traveling wave solution with weak shock profile of the broadwell model system of the boltzmann equation is asymptotically stable provided the initial disturbance is small and its hydrodynamical moments are of integral zero each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one the property of integral zero of the initial disturbance plays a crucial role in this procedure

quantifying the error in estimated transfer functions with application to model order selection previous results on estimating errors or error bounds on identified transfer functions have relied on prior assumptions about the noise and the unmodeled dynamics this prior information took the form of parameterized bounding functions or parameterized probability density functions in the time or frequency domain with known parameters it is shown that the parameters that quantify this prior information can themselves be estimated from the data using a maximum likelihood technique this significantly reduces the prior information required to estimate transfer function error bounds the authors illustrate the usefulness of the method with a number of simulation examples how the obtained error bounds can be used for intelligent model order selection that takes into account both measurement noise and under modeling is shown another simulation study compares the method to akaike s well known fpe and aic criteria

the vlasov poisson boltzmann system in the whole space the hard potential case abstract this paper is concerned with the cauchy problem on the vlasov poisson boltzmann system for hard potentials in the whole space when the initial data is a small perturbation of a global maxwellian a satisfactory global existence theory of classical solutions to this problem together with the corresponding temporal decay estimates on the global solutions is established our analysis is based on time decay properties of solutions and a new time velocity weight function which is designed to control the large velocity growth in the nonlinear term for the case of non hard sphere interactions

optimal decay estimates on the linearized boltzmann equation with time dependent force and their applications although the decay in time estimates of the semi group generated by the linearized boltzmann operator without forcing have been well established there is no corresponding result for the case with general external force this paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted sobolev spaces for the linearized boltzmann equation with a time dependent external force no time decay assumption is made on the force the proof is based on both the energy method through the macro micro decomposition and the lp lq estimates from the spectral analysis the decay estimates thus obtained are applied to the study on the global existence of the cauchy problem to the nonlinear boltzmann equation with time dependent external force and source precisely for space dimension n 3 the global existence and decay rates of solutions to the cauchy problem are obtained under the condition that the force and source decay in time with some rates this time decay restriction can be removed for space dimension n 5 moreover the existence and asymptotic stability of the time periodic solution are given for space dimension n 5 when the force and source are time periodic with the same period

global hilbert expansion for the vlasov poisson boltzmann system the dynamics of an electron gas in a constant ion background can be decribed by the vlasov poisson boltzmann system at the kinetic level or by the compressible euler poisson system at the fluid level we prove that any solution of the vlasov poisson boltzmann system near a smooth local maxwellian with a small irrotational velocity converges global in time to the corresponding solution to the euler poisson system as the mean free path e goes to zero we use a recent l2 l framework in the boltzmann theory to control the higher order remainder in the hilbert expansion uniformly in e and globally in time

steady states of the vlasov poisson fokker planck system abstract the form of steady state solutions to the vlasov poisson fokker planck system is known from the works of dressler and others in these papers an external potential is present which tends to infinity as x it is shown here that this assumption is needed to obtain nontrivial steady states this is achieved by showing that for a given nonnegative background density satisfying certain integrability conditions only the trivial solution is possible this result is sharp and exactly matches the known existence criteria of f bouchut and j dolbeault differential integral equations 8 1995 487 514 and others these steady states are solutions to a nonlinear elliptic equation with an exponential nonlinearity for a given background density which is asymptotically constant it is directly shown by elementary means that this nonlinear elliptic equation possesses a smooth and uniquely determined global solution

detecting long range correlations with detrended fluctuation analysis abstract we examine the detrended fluctuation analysis dfa which is a well established method for the detection of long range correlations in time series we show that deviations from scaling which appear at small time scales become stronger in higher orders of dfa and suggest a modified dfa method to remove them the improvement is necessary especially for short records that are affected by non stationarities furthermore we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of dfa

iterative algorithm for robust performance optimization this paper proposes an iterative algorithm for achieving robust performance optimization for uncertain linear systems the objective is to obtain a controller which minimizes the weighted infinity norm of sensitivity for the worst uncertainty scenario the algorithm seeks the solution to this mini max problem by solving the associated maximal and minimal problems in an iterative sequence the paper describes the conceptual basis of the algorithm supported by simulation examples

on the bgk equation with some force fields in lp space abstract the cauchy problem for the bgk equation with two kinds of force fields is studied in this paper the first one considers a given force field f t x v which is assumed to be divergence free with respect to the velocity variable and has an l p integrability it is shown that there exists a nonnegative distributional solution which extends a previous result obtained by the authors the second one deals with a lorentz field e t x v b t x with e t x and b t x belonging to l p space which gives a similar existence result to the cauchy problem

stability of the vlasov poisson boltzmann system in r3 abstract we construct global in time classical solutions to the cauchy problem for the 2 species vlasov poisson boltzmann system near maxwellian in the whole space r 3 moreover the convergence rate in time to the global maxwellian and the uniform in time stability of solutions are also obtained through energy method

a new energy method for the boltzmann equation an energy method for the boltzmann equation was proposed by liu yang and yu physica d 188 178 192 2004 based on the decomposition of the boltzmann equation and its solution around the local maxwellian the main idea is to rewrite the boltzmann equation as a fluid type dynamics system with the nonfluid component appearing in the source terms coupled with an equation for the time evolution of the nonfluid component in this paper we will elaborate this method and our main observation is that the microscopic projection of the local maxwellian with respect to a given global maxwellian is not linear but quadratic based on this and by analyzing the fluid type system using the analytic techniques for the system of conservation laws we can indeed control the conserved quantities u and 12u2 e of the boltzmann equation by the microscopic projection of the solution of the boltzmann equation with respect to the global maxwellian which is sufficient to deduce the energy estimates for the solution of

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

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

global existence of classical solutions to the vlasov poisson boltzmann system the time evolution of the distribution function for the charged particles in a dilute gas is governed by the vlasov poisson boltzmann system when the force is self induced and its potential function satisfies the poisson equation in this paper we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global maxwellian moreover the convergence rate in time to the global maxwellian is also obtained through the energy method the proof is based on the theory of compressible navier stokes equations with forcing and the decomposition of the solutions to the boltzmann equation with respect to the local maxwellian introduced in 23 and elaborated in 31

analysis of clusters formed by the moving average of a long range correlated time series we analyze the stochastic function c n i ensuremath equiv y i ensuremath y n i where y i is a long range correlated time series of length n mathrm max and y n i ensuremath equiv 1 n ensuremath sum k 0 n ensuremath 1 y i ensuremath k is the moving average with window n we argue that c n i generates a stationary sequence of self affine clusters mathcal c with length mathcal l lifetime ensuremath tau and area s the length and the area are related to the lifetime by the relationships mathcal l ensuremath sim ensuremath tau ensuremath psi mathcal l and s ensuremath sim ensuremath tau ensuremath psi s where ensuremath psi mathcal l 1 and ensuremath psi s 1 h we also find that mathcal l ensuremath tau and s are power law distributed with exponents depending on h p mathcal l ensuremath sim mathcal l ensuremath ensuremath alpha p ensuremath tau ensuremath sim ensuremath tau ensuremath ensuremath beta and p s ensuremath sim s ensuremath ensuremath gamma with ensuremath alpha ensuremath beta 2 ensuremath h and ensuremath gamma 2 1 h these predictions are tested by extensive simulations on series generated by the midpoint displacement algorithm of assigned hurst exponent h ranging from 0 05 to 0 95 of length up to n mathrm max 2 21 and n up to 2 13

boltzmann equation micro macro decompositions and positivity of shock profiles we introduce an elementary energy method for the boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components the decomposition is useful for the study of time asymptotic stability of nonlinear waves the wave location is determined by the macroscopic equation the microscopic component has an equilibrating property the coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the chapman enskog expansion our main result is the establishment of the positivity of shock profiles for the boltzmann equation this is shown by the time asymptotic approach and the maximal principle for the collision operator

criticality and phase transition in stock price fluctuations we analyze the behavior of the u s s p 500 index from 1984 to 1995 and characterize the non gaussian probability density functions pdf of the log returns the temporal dependence of fat tails in the pdf of a ten minute log return shows a gradual systematic increase in the probability of the appearance of large increments on approaching black monday in october 1987 reminiscent of parameter tuning towards criticality on the occurrence of the black monday crash this culminates in an abrupt transition of the scale dependence of the non gaussian pdf towards scale invariance characteristic of critical behavior these facts suggest the need for revisiting the turbulent cascade paradigm recently proposed for modeling the underlying dynamics of the financial index to account for time varying phase transitionlike and scale invariant critical like behavior

serial correlation of detrended time series a preliminary essential procedure in time series analysis is the separation of the deterministic component from the random one if the signal is the result of superposing a noise over a deterministic trend then the first one must estimate and remove the trend from the signal to obtain an estimation of the stationary random component the errors accompanying the estimated trend are conveyed as well to the estimated noise taking the form of detrending errors therefore the statistical errors of the estimators of the noise parameters obtained after detrending are larger than the statistical errors characteristic to the noise considered separately in this paper we study the detrending errors by means of a monte carlo method based on automatic numerical algorithms for nonmonotonic trends generation and for construction of estimated polynomial trends alike to those obtained by subjective methods for a first order autoregressive noise we show that in average the detrending errors of the noise parameters evaluated by means of the autocovariance and autocorrelation function are almost uncorrelated to the statistical errors intrinsic to the noise and they have comparable magnitude for a real time series with significant trend we discuss a recursive method for computing the errors of the estimated parameters after detrending and we show that the detrending error is larger than the half of the total error

non isothermal boundary in the boltzmann theory and fourier law in the study of the heat transfer in the boltzmann theory the basic problem is to construct solutions to the following steady problem v cdot nabla x f frac 1 rm k rm n q f f qquad x v in omega times mathbf r 3 quad quad 0 1 f x v n x cdot v 0 f x v prime n x cdot v prime dv prime quad x in partial omega quad quad 0 2 where is a bounded domain in mathbf r d 1 leq d leq 3 kn is the knudsen number and mu theta frac 1 2 pi theta 2 x rm exp frac v 2 2 theta x is a maxwellian with non constant non isothermal wall temperature x based on new constructive coercivity estimates for both steady and dynamic cases for theta theta 0 leq delta ll 1 and any fixed value of kn we construct a unique non negative solution fs to 0 1 and 0 2 continuous away from the grazing set and exponentially asymptotically stable this solution is a genuine non equilibrium stationary solution differing from a local equilibrium maxwellian as an application of our results we establish the expansion f s mu theta 0 delta f 1 o delta 2 and we prove that if the fourier law holds the temperature contribution associated to f1 must be linear in the slab geometry

assessment of long range correlation in time series how to avoid pitfalls due to the ubiquity of time series with long range correlation in many areas of science and engineering analysis and modeling of such data is an important problem while the field seems to be mature three major issues have not been satisfactorily resolved i many methods have been proposed to assess long range correlation in time series under what circumstances do they yield consistent results ii the mathematical theory of long range correlation concerns the behavior of the correlation of the time series for very large times a measured time series is finite however how can we relate the fractal scaling break at a specific time scale to important parameters of the data iii an important technique in assessing long range correlation in a time series is to construct a random walk process from the data under the assumption that the data are like a stationary noise process due to the difficulty in determining whether a time series is stationary or not however one cannot be 100 sure whether the data should be treated as a noise or a random walk process is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process and vice versa in this paper we seek to gain important insights into these issues by examining three model systems the autoregressive process of order 1 on off intermittency and l evy motions and considering an important engineering problem target detection within sea clutter radar returns we also provide a few rules of thumb to safeguard against misinterpretations of long range correlation in a time series and discuss relevance of this study to pattern recognition

hypocoercivity for linear kinetic equations conserving mass we develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law local mass conservation is assumed at the level of the collision kernel while transport involves a confining potential so that the solution relaxes towards a unique equilibrium state our goal is to evaluate in an appropriately weighted l 2 norm the exponential rate of convergence to the equilibrium the method covers various models ranging from diffusive kinetic equations like vlasov fokker planck equations to scattering models like the linear boltzmann equation or models with time relaxation collision kernels corresponding to polytropic gibbs equilibria including the case of the linear boltzmann model in this last case and in the case of vlasov fokker planck equations any linear or superlinear growth of the potential is allowed