global weak solution to the inhomogeneous navier stokes vlasov equations abstract the inhomogeneous navier stokes vlasov equations for fluid particle flows are considered in the three dimensional space the coupling in the fluid particle system arises from the drag force in the fluid equations and the acceleration in the vlasov equation an initial boundary value problem is studied in a bounded domain with large initial data the existence of global weak solution is established through an approximation scheme a fixed point argument energy estimates and a weak convergence method

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

diffusion approximation for linear transport with multiplying boundary conditions we consider the diffusion limit of a suitably rescaled model transport equation in a slab with multiplying boundary conditions as the scaling parameter varepsilon tends to zero we show that for sufficiently smooth data the solution converges in the l2 norm for each t 0 to the solution of a diffusion equation with robin boundary conditions corresponding to an incoming flux the derivation of the diffusive limit is based on an asymptotic expansion which is rigorously justified

the elusive asymptotic behavior of the high temperature expansion of the hierarchical ising model we present a differential formulation of the recursion formula of the hierarchical model which provides a recursive method of calculation for the high temperature expansion we calculate the first 30 coefficients of the high temperature expansion of the magnetic susceptibility of the ising hierarchical model with 12 significant digits we study the departure from the approximation which consists in identifying the coefficients with the values they would take if a 0 1 pade approximant were exact we show that when the order in the high temperature expansion increases the departure from this approximation grows more slowly than for nearest neighbor models as a consequence the value of the critical exponent estimated using pade approximants converges very slowly and the estimations using 30 coefficients have errors larger than 0 05 a presumably much larger number of coefficients is necessary to obtain the critical exponents with a precision comparable to the precision obtained for nearest neighbor models with fewer coefficients we also discuss the possibility of constructing models where a 0 1 pade approximant would be exact

iterations of the non classical symmetries method and conditional lie backlund symmetries iterations of the non classical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the original equation invariant solutions of these heir equations supply new solutions of the original equation we show that particular cases of such invariant solutions correspond to zhdanov s conditional lie backlund symmetries

on the application of substochastic semigroup theory to fragmentation models with mass loss a linear integro differential equation modelling multiple fragmentation with inherent mass loss is investigated by means of substochastic semigroup theory the existence of a semigroup is established and under natural conditions on certain coefficients the generator of this semigroup is identified this yields in particular a validation of the formal mass loss rate equation for the model

asymptotic spectrum of the linear boltzmann equation with general boundary conditions in finite bodies abstract the purpose of this paper is the study of the spectral properties of both streaming operator and transport operator with general boundary conditions in multidimensional bounded geometry we discuss the asymptotic spectrum existence and nonexistence results of eigenvalues in the half plane c re s t h where s t h stands for the spectral bound of the streaming operator t h next we discuss the irreducibility of the transport semigroup in particular we establish that the transport semigroup is irreducible if the boundary operator is strictly positive afterwards we discuss the strict monotonicity of the leading eigenvalue when it exists of the transport operator with respect to different parameters of the equation our analysis is based essentially on results from the theory of positive linear operators

strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss abstract in the paper we shall present a survey of recent results on substochastic semigroups and provide new methods for determining their honesty these methods are applied to the fragmentation equation with mass loss yielding sufficient conditions for the existence of conservative and shattering solutions our results provide a mathematical framework that clarifies the discussion of phys rev a 43 1991 656 phys rev a 41 1990 5755 j phys a 24 1991 3967 on shattering fragmentation in such models showing among others that there occurs an unexpected mass loss associated with shattering which is not accounted for by the discrete and continuous mass loss contrary to the conjecture of phys rev a 43 1991 656 phys rev a 41 1990 5755

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

conditional symmetry of a porous medium equation abstract we obtain an exhaustive description of conditional symmetries admitted by a nonlinear porous medium equation in one spatial dimension it is proved that solving the determining equations for the conditional symmetries either yields classical lie symmetries or is equivalent to solving the porous medium equation itself the second possibility is shown to be typical for an arbitrary evolution type partial differential equation in one spatial variable we show that this very property is responsible for the well known phenomenon when using non classical symmetries yield exact solutions which prove to be invariant ones

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

functional separable solutions of nonlinear reaction diffusion equations with variable coefficients abstract the paper presents a number of new functional separable solutions to nonlinear reaction diffusion equations of the form c x u t a x u x x b x u x p x f u where f u is an arbitrary function it is shown that any three of the four variable coefficients a x b x c x p x of such equations can be chosen arbitrarily and the remaining coefficient can be expressed through the others examples of specific equations and their exact solutions are given the results obtained are generalized to more complex multidimensional nonlinear reaction diffusion equations with variable coefficients also some functional separable solutions to nonlinear reaction diffusion equations with delay u t u x x a x f u w w u x t where 0 is the delay time and f u w is an arbitrary function of two arguments are obtained it is important to note that the exact solutions of nonlinear pdes and delay pdes that contain arbitrary functions and therefore have sufficient generality are of the greatest practical interest for testing and evaluating the accuracy of various numerical and approximate analytical methods for solving corresponding initial boundary value problems

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

nonclassical symmetry reductions of a porous medium equation with convection in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with convection those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method the asymptotic behaviour of some of the new solutions is analysed

global weak solutions of the vlasov maxwell system with boundary conditions boundaries occur naturally in physucal systems which satisfy the vlasov maxwell system assume perfect conductor boundary conditions for maxwell and either specular reflection or partial absorption for vlasov then weak solutions with finite energy exist for all time

asymptotic stability of the boltzmann equation with maxwell boundary conditions abstract in a general c 1 domain we study the perturbative cauchy theory for the boltzmann equation with maxwell boundary conditions with an accommodation coefficient in 2 3 1 and discuss this threshold we consider polynomial or stretched exponential weights m v and prove existence uniqueness and exponential trend to equilibrium around a global maxwellian in l x v m of important note is the fact that the methods do not involve contradiction arguments

on the weak symmetry groups of partial differential equations abstract we define an algorithm to characterize all weak symmetry groups of a partial differential equation new invariant solutions to the fokker planck equation are found

the oscillatory behavior of the high temperature expansion of dyson s hierarchical model a renormalization group analysis we calculate 800 coefficients of the high temperature expansion of the magnetic susceptibility of dyson s hierarchical model with a landau ginzburg measure log periodic corrections to the scaling laws appear as in the case of an ising measure the period of oscillation appears to be a universal quantity given in good approximation by the logarithm of the largest eigenvalue of the linearized rg transformation in agreement with a possibility suggested by wilson and developed by niemeijer and van leeuwen we estimate to be 1 300 with a systematic error of the order of 0 002 in good agreement with the results obtained with other methods such as the e expansion we briefly discuss the relationship between the oscillations and the zeros of the partition function near the critical point in the complex temperature plane

indefinite sturm liouville problems and half range completeness where the function h changes sign on i such problems occur in certain physical models particularly in transport theory and statistical physics see 2 9 11 131 and references therein as for the standard eigenvalue problem it is true under quite general conditions and not difficult to prove that there is a set of eigenfunctions un which is complete in an appropriate hilbert space of functions this is known as full range completeness for the physical problems the question of interest is generally that of half range completeness to formulate this we set

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

spectrum analysis of some kinetic equations we analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory the models include the classical boltzmann equation for hard potentials with or without angular cutoff and the landau equation with gamma geqq 2 as an application we show that the solutions to these two fundamental equations are asymptotically equivalent mod time decay rate t 5 4 as t to infty to that of the compressible navier stokes equations for initial data around an equilibrium state

nonclassical symmetry solutions and the methods of bluman cole and clarkson kruskal nonclassical symmetry solutions of physically relevant partial differential equations are considered via the reduction methods of bluman and cole and clarkson and kruskal consistency conditions will be provided to show that if satisfied these two methods are equivalent in the sense that they lead to the same symmetry solutions the boussinesq equation and burgers equation are used as illustrative examples exact solutions one of which is new will be presented for burgers equation obtained from the bluman and cole method yet not obtainable by clarkson and kruskal s method

densities for piecewise deterministic markov processes with boundary abstract we study the existence of densities for distributions of piecewise deterministic markov processes we also obtain relationships between invariant densities of the continuous time process and that of the process observed at jump times in our approach we use functional analytic methods and the theory of linear operator semigroups by imposing general conditions on the characteristics of a given markov process we show the existence of a substochastic semigroup describing the evolution of densities for the process and we identify its generator our main tool is a new perturbation theorem for substochastic semigroups where we perturb both the action of the generator and of its domain allowing to treat general transport type equations with non local boundary conditions a couple of particular examples illustrate our general results

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

a numerical study of the hierarchical ising model high temperature versus epsilon expansion we study numerically the magnetic susceptibility of the hierarchical model with ising spins 1 above the critical temperature and for two values of the epsilon parameter the integrations are performed exactly using recursive methods which exploit the symmetries of the model lattices with up to 218 sites have been used surprisingly the numerical data can be fitted very well with a simple power law of the form 1 0 g for thewhole temperature range considered this approximate law implies a simple approximate formula for the coefficients of the high temperature expansion and more importantly approximate relations among the coefficients themselves we found that some of these approximate relations hold with errors less then 2 on the other hand g differs significantly from the critical exponent calculated with the epsilon expansion even when the fit is restricted to intervals closer to c we discuss this discrepancy in the context the renormalization group analysis of the hierarchical model

global well posedness in spatially critical besov space for the boltzmann equation the unique global strong solution in the chemin lerner type space to the cauchy problem on the boltzmann equation for hard potentials is constructed in a perturbation framework such a solution space is of critical regularity with respect to the spatial variable and it can capture the intrinsic properties of the boltzmann equation for the proof of global well posedness we develop some new estimates on the nonlinear collision term through the littlewood paley theory

on the classical solutions of the boltzmann equation on etudie le probleme aux valeurs initiales pour l equation de boltzmann on montre que des solutions au sens classique existent pour tout temps si la donnee initiale est assez proche de l equilibre on se restreint a un domaine cubique et on considere seulement la condition aux limites periodique

the high temperature expansion of the hierarchical ising model from poincare symmetry to an algebraic algorithm it is shown that the hierarchical model at finite volume has a symmetry group which can be decomposed into rotations and translations as the familiar poincare groups using these symmetries it is shown that the intricate sums appearing in the calculation of the high temperature expansion of the magnetic susceptibility can be performed at least up to the fourth order using elementary algebraic manipulations which can be implemented with a computer these symmetries appear more clearly if we use the two adic fractions to label the sites then the new algebraic methods are applied to the calculation of quantities having a random walk interpretation in particular it is shown that the probability of returning to the starting point after m steps has poles at d 2 4 2m where d is a free parameter playing a role similar to the dimensionality in nearest neighbor models

a non gaussian renormalization group fixed point for hierarchical scalar lattice field theories a rigorous method is developed to handle the large field problems in the wilson kadanoff renormalization group approach to critical lattice systems of unbounded spins we use this method to study in a hierarchical approximation the non gaussian renormalization group fixed point which governs the infrared behaviour of critical lattice field theories in three dimensions the method is an improvement of the analyticity techniques of gawedzki and kupiainen using borel summation techniques we are able to incorporate the large field region into the perturbative region so that the theory is completely described in terms of convergent expansions

existence and convergence to equilibrium of a kinetic model for cometary flows a kinetic equation with a relaxation time model for wave particle collisions is considered similarly to the bgk model of gas dynamics it involves a projection onto the set of equilibrium distributions nonlinearly dependent on moments of the distribution function an earlier existence result is extended to bounded domains with reflecting boundaries and to initial conditions permitting vacuum regions the long time behaviour is investigated convergence on compact time intervals shifted to infinity to the set of equilibrium solutions is proven the set of smooth equilibrium solutions is computed

stability and exponential convergence for the boltzmann equation we prove existence uniqueness and stability for solutions of the nonlinear boltzmann equation in a periodic box in the case when the initial data are sufficiently close to a spatially homogeneous function the results are given for a range of spaces including l1 and extend previous results in l for the non homogeneous equation as well as the more developed lp theory for the spatially homogeneous boltzmann equation

evolution equations invariant surface conditions and functional separation of variables abstract this paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables the relationship of these methods with nonclassical and weak point symmetries is stressed applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided

on the determining equations for the nonclassical reductions of the heat and burgers equation abstract the determining equations for the nonclassical reductions of the heat and burgers equations are considered it is shown that both systems belong to a burgers equation hierarchy each system is written in terms of the same matrix burgers equation that is linearized via a matrix hopf cole transformation in essence it is shown that both systems can be solved simultaneously their respective solutions are then presented in a very compact form

new conditional symmetries and exact solutions of reaction diffusion convection equations with exponential nonlinearities abstract a complete description of q conditional symmetries for a class of reaction diffusion convection equations with exponential diffusivities is derived it is shown that all the known results for reaction diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa the symmetries obtained for constructing exact solutions of the relevant equations are successfully applied the exact solutions are compared with those found earlier by means of different approaches an application of the solutions for solving a boundary value problem arising in population dynamics is presented

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

global existence and asymptotic behavior for the nonlinear enskog equation the existence of a classical solution to the nonlinear enskog equation in all mathbb r 3 is proved the solution which is global in time exists if the initial data go to zero fast enough at infinity and the mean path is sufficiently large the solution is smooth in space variable if the initial value is smooth the asymptoticbehavior of solutions is also given it is shown that as t to infty the solution to the enskog equation can be approximated by the solution to the free motion problem key words partial differential equations nonlinear integral equation asymptotic behavior kinetic theory boltzmann equation

on non linear partial differential equations with an infinite dimensional conditional symmetry abstract the invariance of non linear partial differential equations under a certain infinite dimensional lie algebra a n z in n spatial dimensions is studied the special case a 1 2 was introduced in j stat phys 75 1994 1023 and contains the schrodinger lie algebra sch 1 as a lie subalgebra it is shown that there is no second order equation which is invariant under the massless realizations of a n z however a large class of strongly non linear partial differential equations is found which are conditionally invariant with respect to the massless realization of a n z such that the well known monge ampere equation is the required additional condition new exact solutions are found for some representatives of this class

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

some non local transformations between nonlinear diffusion equations generalized backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations including many which are of physical significance some new exactly linearizable forms are determined

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

l2 solution of linear non cutoff boltzmann equation with boundary conditions abstract in this paper we are concerned about the existence of solution to linear non cutoff boltzmann equation with boundary conditions in 9 and 18 the existence of the solutions to the kinetic equation and linear boltzmann equation were obtained by characteristic line and iteration methods however their methods are not applicable in the context of non cutoff linear operators to overcome this difficulty we cut off the non cutoff linear operator l to l r the latter is compatible with the method in 9 via this process we are able to use the sequence of solutions associated to l r to approximate the solution associated to l which is corresponding to the original equation in turn obtain the weak solutions of linear non cutoff boltzmann equation with specular and diffusion reflection boundary conditions

new q conditional symmetries and exact solutions of some reaction diffusion convection equations arising in mathematical biology abstract a theorem giving a complete description of q conditional symmetries of a class of nonlinear reaction diffusion convection equations is proved furthermore the q conditional symmetries obtained and the method of additional generating conditions are applied for finding exact solutions of the generalized fisher fitzhugh nagumo and kolmogorov petrovskii piskunov equations the symmetries and solutions constructed are compared with those obtained by other authors in particular it was established that the known travelling wave solutions of these equations are particular cases of more general non lie solutions the relation between q conditional symmetries and generalized conditional symmetries is also shown

relatively strictly singular perturbations essential spectra and application to transport operators p strictly singular operator was discussed in a previous paper by k latrach and a jeribi 1998 j math anal appl 225 461485 in the present paper we prove the invariance of the gustafsonweidmann wolf schechter and browder essential spectra of a under relatively strictly singular not necessarily bounded perturba tions on these spaces further a precise characterization of the schechter essential spectrum is given we show that these results are also valid on c where is a compact hausdorff space the results are applied to the one dimensional transport equations with anisotropic scattering and abstract boundary conditions 2000 academic press

limit sets for multidimensional nonlinear transport equations abstract we propose a hyperbolic generalisation of the well known reaction diffusion equation and study the structure of its limit sets under a dissipativity condition on the nonlinearity we show quasi convergence of the flow saying that the limit set of any orbit is non empty compact connected and contained in the set of equilibrium solutions

exact interaction of solitary waves for certain nonintegrable equations physically interesting exact solutions are constructed for a large class of nonlinear nonintegrable evolution equations these solutions describe the interaction of traveling waves they exhibit rich phenomenology including breaking of solitary waves generalized conditional symmetries and bilinearity are used to derive these exact results