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

optimal gradient estimates and asymptotic behaviour for the vlasov poisson system with small initial data the vlasov poisson system describes interacting systems of collisionless particles for solutions with small initial data in three dimensions it is known that the spatial density of particles decays as t 3 at late times in this paper this statement is refined to show that each derivative of the density which is taken leads to an extra power of decay so that in n dimensions for n geqq 3 the derivative of the density of order k decays as t n k an asymptotic formula for the solution at late times is also obtained

finite group extensions and the atiyah conjecture the atiyah conjecture for a discrete group g states that the l 2 betti numbers of a finite cw complex with fundamental group g are integers if g is torsion free and in general that they are rational numbers with denominators determined by the finite subgroups of g here we establish conditions under which the atiyah conjecture for a torsion free group g implies the atiyah conjecture for every finite extension of g the most important requirement is that h g z p is isomorphic to the cohomology of the p adic completion of g for every prime p an additional assumption is necessary e g that the quotients of the lower central series or of the derived series are torsion free we prove that these conditions are fulfilled for a certain class of groups which contains in particular artin s pure braid groups and more generally fundamental groups of fiber type arrangements free groups fundamental groups of orientable compact surfaces certain knot and link groups primitive positive one relator groups and products of these therefore every finite in fact every elementary amenable extension of these groups satisfies the atiyah conjecture as a consequence if such an extension h is torsion free then the group ring ch contains no non trivial zero divisors i e h fulfills the zero divisor conjecture in the course of the proof we prove that if these extensions are torsionfree then they have plenty of non trivial torsion free quotients which are virtually nilpotent all of this applies in particular to artin s full braid group therefore answering question b6 on www grouptheory org our methods also apply to the baum connes conjecture this is discussed in 47 where for example the baum connes conjecture is proved for the full braid groups msc 55n25 homology with local coecients 16s34 group rings laurent rings 57m25 knots and links

velocity averaging kinetic formulations and regularizing effects in quasi linear pdes we prove in this paper new velocity averaging results for second order multidimensional equations of the general form l x v f x v g x v where l x v a v x b v x these results quantify the sobolev regularity of the averages v f x v v dv in terms of the nondegeneracy of the set

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

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

a boundary value problem for the stationary vlasov poisson equations the plane diode the stationary vlasov poisson boundary value problem in a spatially one dimensional domain is studied the equations describe the flow of electrons in a plane diode existence is proved when the boundary condition the cathode emission distribution is a bounded function which decays super linearly or a dirac mass uniqueness is proved for physically realistic boundary conditions which are decreasing functions of the velocity variable it is shown that uniqueness does not always hold for the dirac mass boundary conditions

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

propagation of moments and regularity for the 3 dimensional vlasov poisson system we prove that if the initial data has moments inv higher than three then the solution of vlasov poisson has also moments inv higher than three we deduce from this different regularity results on the local density the force field or the solution itself also we give a new uniqueness result and new regularity results for solutions satisfying only the energy andl bounds our proofs are based on a new representation formula and logarithmic estimates for the force field

initial layers and zero relaxation limits of euler maxwell equations in this paper we consider zero relaxation limits for periodic smooth solutions of euler maxwell systems for well prepared initial data we propose an approximate solution based on a new asymptotic expansion up to any order for ill prepared initial data we construct initial layer corrections in an explicit way in both cases the asymptotic expansions are valid in time intervals independent of the relaxation time and their convergence is justified by establishing uniform energy estimates

on the controllability of the vlasov poisson system in this paper we study the controllability of the vlasov poisson system in a periodic domain by means of an interior control located in an spatial subdomain the first result proves the local exact zero controllability in the two dimensional torus between two small acceptable distribution functions with an arbitrary control zone a second result establishes the global exact controllability in arbitrary dimension provided the control zone satisfies the condition that it contains a hyperplane of the torus

the classical limit of the relativistic vlasov maxwell system solutions of the relativistic vlasov maxwell system of partial differential equations are considered in three space dimensions the speed of light c appears as a parameter in this system for smooth cauchy data classical solutions are shown to exist on a time interval that is independent ofc then using an integral representation for the electric and magnetic fields due to glassey and strauss 6 conditions are given under which solutions of the relativistic vlasov maxwell system converge in pointwise sense to solutions of the non relativistic vlasov poisson system at the asymptotic rate of 1 c asc tends to infinity

stable self similar blow up dynamics for the three dimensional relativistic gravitational vlasov poisson system r f x v dv describes the mechanical state of a stellar system subject to its own gravity smooth initial data yield unique global in time solutions from a celebrated result by pfaffelmoser 37 there exists a hierarchy of physical models which aim at taking into account further relativistic effects the simplest one is the three dimensional relativistic gravitational vlasov poisson system tf v 1 v xf ef vf 0 which we study here a striking feature as observed by glassey schaeffer 18 is that this system now admits finite blow up solutions nevertheless the existence argument is purely obstructive and provides no insight into the description of the singularity formation we exhibit in this paper a family of finite time blow up self similar solutions and prove that their blow up dynamic is stable with respect to radially symmetric perturbations our analysis applies to the four dimensional gravitational vlasov poisson system as well whose study began in 26

higher regularity of the tangential fields in the relativistic vlasov maxwell system abstract it is shown that the tangential electric and magnetic fields in the glassey strauss representation formulas are in fact bounded in l loc t l x 2 for some 0

averaging regularity results for pdes under transversality assumptions let u u x y and f f x y be two functions related by a pde p x y dx dy u f the regularity of the y average u x dy as a function of x is investigated knowing that of u and f our method consists in reducing p to a microlocal normal form under a natural transversality assumption the 2 microlocal regularity of u is also determined knowing that of f these results are then applied to a homogenization problem this article generalizes the results of 16 and 12 on velocity averaging and homogenization for kinetic equations

the perturbed korteweg de vries equation considered anew the perturbed korteweg de vries equation is studied in a new way by a green s function formalism without use of inverse scattering methods the green s function is determined by employing the backlund transformation and green s theorem after a thorough analysis of the exact first order solution with regard to secular terms a two time scale expansion leads to the adiabatic approximation and the first order correction in accordance with the results of karpman and maslov contrary to statements in the literature the term tanh2 z in the expression for the modified phase of the perturbed soliton arises as a consequence of the systematically conducted first order perturbation theory

classical solutions and the glassey strauss theorem for the 3d vlasov maxwell system r glassey and w strauss have proved in arch rational mech anal 92 1986 59 90 that c1 solutions to the relativistic vlasov maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded the present paper simplifies their proof

the semigeostrophic equations discretized in reference and dual variables we study the evolution of a system of n particles x i v i i 1 n in mathbb r 2d that system is a conservative system with a hamiltonian of the form h mu w 2 2 mu nu n where w2 is the wasserstein distance and is a discrete measure concentrated on the set x i v i i 1 n typically 0 is a discrete measure approximating an initial l density and can be chosen randomly when d 1 our results prove convergence of the discrete system to a variant of the semigeostrophic equations we obtain that the limiting densities are absolutely continuous with respect to the lebesgue measure when nu n n 1 infty converges to a measure concentrated on a special d dimensional set we obtain the vlasov monge ampere vma system when d 1 the vma system coincides with the standard vlasov poisson system

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

conservation laws and the perturbed kdv equation examines the integral for the correction to the soliton position in the adiabatic approximation using the modified conservation laws for the mass and the energy the general correction to the soliton position is obtained it is easily seen that this correction is due to the formation of the shelf

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

quasi stationary perturbations of the kdv soliton in this paper we show how the results of quasi stationary perturbation theory relate to those based upon squared eigenfunction expansions we show that the two give the same results for a quasi stationary approximation and also show how standard adiabatic perturbation theory connects to the quasi stationary theory

on the lagrangian structure of transport equations the vlasov poisson system the vlasov poisson system is an important nonlinear transport equation used to describe the evolution of particles under their self consistent electric or gravitational field the existence of classical solutions is limited to dimensions d 3 under strong assumptions on the initial data whereas weak solutions are known to exist under milder conditions however in the setting of weak solutions it is unclear whether the eulerian description provided by the equation physically corresponds to a lagrangian evolution of the particles in this article we develop several general tools concerning the lagrangian structure of transport equations with nonsmooth vector fields and we apply these results to show that weak renormalized solutions of vlasov poisson are lagrangian and actually that the concepts of renormalized and lagrangian solutions are equivalent as a corollary we prove that finite energy solutions in dimension d 4 are transported by a global flow in particular they preserve all the natural casimir invariants and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data

a new continuation criterion for the relativistic vlasov maxwell system the global existence of solutions to the relativistic vlasov maxwell system given sufficiently regular finite energy initial data is a longstanding open problem the main result of glassey and strauss arch ration mech anal 92 59 90 1986 shows that a solution f e b remains c 1 as long as the momentum support of f remains bounded alternate proofs were later given by bouchut et al arch ration mech anal 170 1 15 2003 and klainerman and staffilani commun pure appl anal 1 103 125 2002 we show that only the boundedness of the momentum support of f after projecting to any two dimensional plane is needed for f e b to remain c 1

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

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

stability of periodic steady state solutions to a non isentropic euler poisson system abstract we study the stability of periodic smooth solutions near non constant steady states for a non isentropic euler poisson system without temperature damping term the system arises in the theory of semiconductors for which the doping profile is a given smooth function in this stability problem there are no special restrictions on the size of the doping profile but only on the size of the perturbation we prove that small perturbations of periodic steady states are exponentially stable for large time for this purpose we introduce new variables and choose a non diagonal symmetrizer of the full euler equations to recover dissipation estimates this also allows to make the proof of the stability result very simple and concise

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 the 3 d vlasov poisson system with point charges global solutions with unbounded supports and propagation of velocity spatial moments abstract a new energy functional which is shown to decay to zero as time tends to infinite is introduced for the three dimensional vlasov poisson plasma in the presence of heavy point charges with repulsive interaction moreover a new kind of pointwise energy of a plasma particle relative to each point charge is also constructed and the variation of its root is shown to be controlled by the electrostatic fields based on those results we prove global existence and uniqueness of a classical solution possibly having infinite kinetic energy but with compact velocity spatial support specified in theorem 1 1 furthermore for the single point charge vlasov poisson plasma global existence and polynomial propagation of velocity spatial moments of weak solutions are also established

discontinuous galerkin methods for the vlasov maxwell equations discontinuous galerkin methods are developed for solving the vlasov maxwell system methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy such properties in general are hard to achieve within other numerical method frameworks for simulating the vlasov maxwell system the proposed scheme employs discontinuous galerkin discretizations for both the vlasov and the maxwell equations resulting in a consistent description of the distribution function and electromagnetic fields it is proven up to some boundary effects that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the maxwell equations and the underlying approximation spaces error estimates are established for several flux choices the scheme is tested on the streaming weibel instability the order of accuracy and conservation properties of the proposed method are verified

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

the vlasov poisson system with infinite kinetic energy and initial data in lp r6 abstract in this paper the global existence of weak solutions to the three dimensional vlasov poisson system with infinite kinetic energy and initial data in l p r 6 is studied the propagation of low moments in the velocity variables is also proved

mean dimension mean rank and von neumann luck rank we introduce an invariant called mean rank for any module m of the integral group ring of a discrete amenable group as an analogue of the rank of an abelian group it is shown that the mean dimension of the induced action on the pontryagin dual of m the mean rank of m and the von neumann luck rank of m all coincide as applications we establish an addition formula for mean dimension of algebraic actions prove the analogue of the pontryagin schnirelmnn theorem for algebraic actions and show that for elementary amenable groups with an upper bound on the orders of finite subgroups algebraic actions with zero mean dimension are inverse limits of finite entropy actions

on the spectrum of lamplighter groups and percolation clusters let mathfrak g be a finitely generated group and x its cayley graph with respect to a finite symmetric generating set s furthermore let mathfrak h be a finite group and mathfrak h wr mathfrak g the lamplighter group wreath product over mathfrak g with group of lamps mathfrak h we show that the spectral measure plancherel measure of any symmetric switch walk switch random walk on mathfrak h wr mathfrak g coincides with the expected spectral measure integrated density of states of the random walk with absorbing boundary on the cluster of the group identity for bernoulli site percolation on x with parameter mathsf p 1 mathfrak h the return probabilities of the lamplighter random walk coincide with the expected annealed return probabilities on the percolation cluster in particular if the clusters of percolation with parameter mathsf p are almost surely finite then the spectrum of the lamplighter group is pure point this generalizes results of grigorchuk and uk resp dicks and schick regarding the case when mathfrak g is infinite cyclic analogous results relate bond percolation with another lamplighter random walk in general the integrated density of states of site or bond percolation with arbitrary parameter mathsf p is always related with the plancherel measure of a convolution operator by a signed measure on mathfrak h wr mathfrak g where mathfrak h mathbb z or another suitable group

time asymptotics for solutions of vlasov poisson equation in a circle we prove that there exists a class of solutions of the nonlinear vlasov poisson equation vpe on a circle that converges weakly as t to a stationary homogeneous solution of vpe this behavior is called in the linear case landau damping the result is obtained by constructing a suitable scattering problem for the solutions of the vlasov poisson problem a consequence of this result is that a class of stationary solutions of the vlasov poisson equation is unstable in a weak topology

averaging lemmas without time fourier transform and application to discretized kinetic equations we prove classical averaging lemmas in the l2 framework with the help of the fourier transform in variables x and v but not t this method is then used to study discretized problems arising out of the numerical analysis of kinetic equations

regularizing effects for the classical solutions to the landau equation in the whole space abstract the landau equation describes the binary collisional effects through long range coulombian interaction in a plasma in this paper we prove that the known classical solutions to the landau equation near maxwellian in the whole space have a regularizing effect in all time space and velocity variables that is become immediately smooth with respect to all variables

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

group ring elements with large spectral density given delta 0 we construct a group g and a group ring element s in mathbb z g such that the spectral measure mu of s fulfils mu 0 varepsilon frac c log varepsilon 1 delta for small varepsilon in particular the novikov shubin invariant of any such s is 0 the constructed examples show that the best known upper bounds on mu 0 varepsilon are not far from being optimal

bounds on the growth of the velocity support for the solutions of the vlasov poisson equation in a torus a bound on the growth of the velocity for the vlasov poisson equation in a torus is given in one and two dimensions the main tool used in the proof is a partition into fast and slow particles and the ergodic property of the free motion in a torus

numerical approximation of the vlasov poisson fokker planck system in two dimensions a numerical method is developed for approximating the solution to the vlasov poisson fokker planck system in two spatial dimensions the method generalizes the approximation for the system in one dimension given in s wollman e ozizmir numerical approximation of the vlasov poisson fokker planck system in one dimension j comput phys 202 2005 602 644 the numerical procedure is based on a change of variables that puts the convection diffusion equation into a form so that finite difference methods for parabolic type partial differential equations can be applied the computational cycle combines a type of deterministic particle method with a periodic interpolation of the solution along particle trajectories onto a fixed grid computational work is done to demonstrate the accuracy and effectiveness of the approximation method parts of the numerical procedure are adapted to run on a parallel computer

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

a direct approach to studying soliton perturbations starting with an integrable nonlinear evolution equation the author investigates perturbations about a one soliton solution through the inversion of a linear equation for the first order correction to the soliton solution this inversion differs from past methods as the proposed method takes place in coordinate space not spectral space while it employs some of the tools of inverse scattering theory the method is applied to the korteweg de vries nonlinear schrodinger and sine gordon equations the first order corrections are then obtained

existence of global weak solutions to a hybrid vlasov mhd model for magnetized plasmas we prove the global in time existence of large data nite energy weak solutions to an incompressible hybrid vlasov magnetohydrodynamic model in three space dimensions the model couples three essential ingredients of magnetized plasmas a transport equation for the probability density function which models energetic rare ed particles of one species the incompressible navier stokes system for the bulk uid and a parabolic evolution equation involving magnetic di usivity for the magnetic eld the physical derivation of our model is given it is also shown that the weak solution whose existence is established has nonincreasing total energy and that it satis es a number of physically relevant properties including conservation of the total momentum conservation of the total mass and nonnegativity of the probability density function for the energetic particles the proof is based on a one level approximation scheme which is carefully devised to avoid increase of the total energy for the sequence of approximating solutions in conjunction with a weak compactness argument for the sequence of approximating solutions the key technical challenges in the analysis of the mathematical model are the nondissipative nature of the vlasov type particle equation and passage to the weak limits in the multilinear coupling terms

permanent regimes for the vlasov maxwell equations with specular boundary conditions the subject matter of this paper concerns the existence of permanent regimes i e stationary or time periodic solutions for the vlasov maxwell system in a bounded domain we are looking for equilibrium configurations by imposing specular boundary conditions the main difficulty is the treatment of such boundary conditions our analysis relies on perturbative techniques based on uniform a priori estimates

asymptotic behavior of weak solutions for the relativistic vlasov maxwell equations with large light speed abstract we study here the behavior of time periodic weak solutions for the relativistic vlasov maxwell boundary value problem in a three dimensional bounded domain with strictly star shaped boundary when the light speed becomes infinite we prove the convergence toward a time periodic weak solution for the classical vlasov poisson equations

strichartz estimates and moment bounds for the relativistic vlasov maxwell system we consider the relativistic vlasov maxwell system with data of unrestricted size and without compact support in momentum space in the two dimensional and the two and a half dimensional cases glassey schaeffer proved commun math phys 185 257 284 1997 arch ration mech anal 141 331 354 1998 arch ration mech anal 141 355 374 1998 that for regular initial data with compact momentum support this system has unique global in time classical solutions in this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space in the two dimensional and the two and a half dimensional cases we prove the global existence uniqueness and regularity for solutions arising from this class of initial data to this end we use strichartz estimates and prove that suitable moments of the solution remain bounded moreover we obtain a slight improvement of the temporal growth of the l infty x norms of the electromagnetic fields compared to glassey and schaeffer commun math phys 185 257 284 1997 arch ration mech anal 141 355 374 1998 in the three dimensional case we apply strichartz estimates and moment bounds to show that a regular solution can be extended as long as p 0 theta f l q x l 1 p remains bounded for theta frac 2 q 2 q leqq infty this improves previous results of pallard indiana univ math j 54 5 1395 1409 2005 commun math sci 13 2 347 354 2015

well posedness of weak and strong solutions to the kinetic cucker smale model abstract in this paper we develop a unified framework that can be used to establish the well posedness of kinetic cucker smale model with or without noise for general initial data regardless of the supports meanwhile we rigorously justify the vanishing noise limit our proof is based on weighted energy estimates and the velocity averaging lemma in kinetic theory

generalized solutions of the vlasov poisson system with singular data abstract we study spherically symmetric solutions of the vlasov poisson system in the context of algebras of generalized functions this allows to model highly concentrated initial configurations and provides a consistent setting for studying singular limits of the system the proof of unique solvability in our approach depends on new stability properties of the system with respect to perturbations

free lamplighter groups and a question of atiyah we compute the von neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational thus providing an elementary constructive answer to a question of atiyah

global weak solutions to the nordstrom vlasov system abstract the nordstrom vlasov system is a lorentz invariant model for a self gravitating collisionless gas we establish suitable a priori bounds on the solutions of this system which together with energy estimates and the smoothing effect of momentum averaging yield the existence of global weak solutions to the corresponding initial value problem in the process we improve the continuation criterion for classical solutions which was derived recently the weak solutions are shown to preserve mass

three new results on continuation criteria for the 3d relativistic vlasov maxwell system abstract we consider continuation criteria for the three dimensional relativistic vlasov maxwell system when the particle density f t x p is compactly supported at t 0 we prove p 0 18 5 r 1 f l t l x r l p 1 1 where 1 r 2 and 0 is arbitrarily small is a continuation criteria our continuation criteria is an improvement in the 1 r 2 range to the previously best known criteria p 0 4 r 1 f l t l x r l 1 p 1 due to kunze 7 we also consider continuation criteria when f 0 x p has noncompact support in this regime luk strain 9 proved that p 0 f l x 1 l p 1 1 is a continuation criteria for 5 we improve this result to 3 finally we build on another result by luk strain 8 the authors proved boundedness of momentum support on a fixed two dimensional plane is a sufficient continuation criteria we prove the same result even if the plane varies continuously in time

finite speed propagation of the solutions for the relativistic vlasov maxwell system in this paper we investigate the continuous dependence with respect to the initial data of the solutions for the 1d and 1 5d relativistic vlasov maxwell system more precisely we prove that these solutions propagate with finite speed we formulate our results in the framework of mild solutions i e the particle densities are solutions by characteristics and the electro magnetic fields are lipschitz continuous functions

yet another criterion for global existence in the 3d relativistic vlasov maxwell system abstract we prove that solutions of the 3d relativistic vlasov maxwell system can be extended as long as the quantity 1 t x max 1 r 3 d p 1 p 2 1 1 v f t x p is bounded in l x 2

global classical solutions to the 3d nordstrom vlasov system the nordstrom vlasov system describes the kinetic evolution of self gravitating collisionless matter in the framework of a relativistic scalar theory of gravitation we prove global existence and uniqueness of classical solutions for the corresponding initial value problem in three dimensions when the initial data for the scalar field are smooth and the initial particle density is smooth with compact support

global weak solutions of vlasov maxwell systems we study here the vlasov maxwell system in its classical and relativistic form we prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data the main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work

smoothing effects for classical solutions of the full landau equation in this work we consider the smoothness of the solutions to the full landau equation in particular we prove that any classical solutions such as the ones obtained by guo in a close to equilibrium setting become immediately smooth with respect to all variables this shows that the landau equation is a nonlinear and nonlocal analog of an hypoelliptic equation

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

global existence for the vlasov poisson system in bounded domains in this paper we prove global existence for solutions of the vlasov poisson system in convex bounded domains with specular boundary conditions and with a prescribed outward electrical field at the boundary

a uniqueness criterion for unbounded solutions to the vlasov poisson system we prove uniqueness for the vlasov poisson system in two and three dimensions under the condition that the l p norms of the macroscopic density grow at most linearly with respect to p this allows for solutions with logarithmic singularities we provide explicit examples of initial data that fulfill the uniqueness condition and that exhibit a logarithmic blow up in the gravitational two dimensional case such states are intimately related to radially symmetric steady solutions of the system our method relies on the lagrangian formulation for the solutions exploiting the second order structure of the corresponding ode