entropy dissipation and long range interactions we study boltzmann s collision operator for long range interactions i e without grad s angular cut off assumption we establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function in a precise sense our estimate is optimal and gives a unified treatment of both the linear and the nonlinear cases we also give simple and self contained proofs of several useful results that were scattered in previous works as an application we obtain several helpful estimates for the cauchy problem and for the landau approximation in plasma physics

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

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

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

rate of convergence to equilibrium for the spatially homogeneous boltzmann equation with hard potentials for the spatially homogeneous boltzmann equation with hard potentials and grad s cutoff e g hard spheres we give quantitative estimates of exponential convergence to equilibrium and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation on which we provide a lower bound our approach is based on establishing spectral gap like estimates valid near the equilibrium and then connecting the latter to the quantitative nonlinear theory this leads us to an explicit study of the linearized boltzmann collision operator in functional spaces larger than the usual linearization setting

a diagonalized multilevel fast multipole method with spherical harmonics expansion of the k space integrals diagonalization of the fast multipole method fmm for the helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves the resulting k space integral over the ewald sphere is numerically evaluated storing the k space quadrature samples of the method of moments mom basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems in this paper it is proposed to expand the k space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules aggregations plane wave translations and disaggregations in the realized multilevel fast multipole method mlfmm are carried out using the k space samples of a numerical quadrature rule however the incoming plane waves on the finest mlfmm level are expanded in spherical harmonics again thus due to the orthonormality of spherical harmonics the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients overall the resulting mlfmm can save a considerable amount of memory without compromising accuracy and numerical speed

on the optimality of simple schedules for networks with multiple half duplex relays this paper studies networks that consist of n half duplex relays assisting the communication between a source and a destination in isit 12 brahma et al conjectured that in gaussian half duplex diamond networks i e without a direct link between the source and the destination and with n non interfering relays an approximately optimal relay scheduling policy i e achieving the cut set upper bound to within a constant gap uniformly over all channel gains has at most n 1 active states i e at most n 1 out of the 2 n possible relay listen transmit configurations have a strictly positive probability such relay scheduling policies were referred to as simple in itw 13 we conjectured that simple approximately optimal relay scheduling policies exist for any gaussian half duplex multi relay network irrespectively of the topology this paper formally proves this more general version of the conjecture and shows it holds beyond gaussian noise networks in particular for any class of memoryless half duplex n relay networks with independent noises and for which independent inputs are approximately optimal in the cut set upper bound an approximately optimal simple relay scheduling policy exists the key step of the proof is to write the minimum of the submodular cut set function by means of its lovasz extension and use the greedy algorithm for submodular polyhedra to highlight structural properties of the optimal solution this together with the saddle point property of min max problems and the existence of optimal basic feasible solutions for linear programs proves the conjecture as an example for n relay gaussian networks with independent noises where each node is equipped with multiple antennas and where each antenna can be configured to listen or transmit irrespectively of the others the existence of an approximately optimal simple relay scheduling policy with at most n 1 active states irrespectively of the total number of antennas in the system is proved

smoothing estimates for boltzmann equation with full range interactions spatially homogeneous case in this work we are concerned with the regularities of the solutions to the boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials r p 1 with p 2 we give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional sobolev norms as an application we get the global entropy dissipation estimate which is a little stronger than that described by alexandre et al arch rational mech anal 152 4 327 355 2000 as another application we prove the smoothing effects for the strong solutions constructed by desvillettes and mouhot arch rational mech anal 193 2 227 253 2009 of the spatially homogeneous boltzmann equation with true hard potential and true moderately soft potential

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

regularizing effect and local existence for the non cutoff boltzmann equation the boltzmann equation without grad s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non integrable angular singularity of the cross section however even though this has been justified satisfactorily for the spatially homogeneous boltzmann equation it is still basically unsolved for the spatially inhomogeneous boltzmann equation in this paper by sharpening the coercivity and upper bound estimates for the collision operator establishing the hypo ellipticity of the boltzmann operator based on a generalized version of the uncertainty principle and analyzing the commutators between the collision operator and some weighted pseudo differential operators we prove the regularizing effect in all time space and velocity variables on the solutions when some mild regularity is imposed on these solutions for completeness we also show that when the initial data has this mild regularity and a maxwellian type decay in the velocity variable there exists a unique local solution with the same regularity so that this solution acquires the c regularity for any positive time

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

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

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

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

ultra analytic effect of cauchy problem for a class of kinetic equations abstract the smoothing effect of the cauchy problem for a class of kinetic equations is studied we firstly consider the spatially homogeneous nonlinear landau equation with maxwellian molecules and inhomogeneous linear fokker planck equation to show the ultra analytic effects of the cauchy problem those smoothing effect results are optimal and similar to heat equation in the second part we study a model of spatially inhomogeneous linear landau equation with maxwellian molecules and show the analytic effect of the cauchy problem

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

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

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

on measure solutions of the boltzmann equation part ii rate of convergence to equilibrium abstract the paper considers the convergence to equilibrium for measure solutions of the spatially homogeneous boltzmann equation for hard potentials with angular cutoff we prove the exponential sharp rate of strong convergence to equilibrium for conservative measure solutions having finite mass and energy the proof is based on the regularizing property of the iterated collision operators exponential moment production estimates and some previous results on the exponential rate of strong convergence to equilibrium for square integrable initial data we also obtain a lower bound of the convergence rate and deduce that no eternal solutions exist apart from the trivial stationary solutions given by the maxwellian equilibrium the constants in these convergence rates depend only on the collision kernel and conserved quantities mass momentum and energy we finally use these convergence rates in order to deduce global in time strong stability of measure solutions

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

hybrid fe bi modeling of 3 d doubly periodic structures utilizing triangular prismatic elements and an mpie formulation accelerated by the ewald transformation we present the formulation of a finite element boundary integral method for the analysis of three dimensional doubly periodic structures based on arbitrary nonorthogonal lattice configurations the method starts from a functional description of the field problem where only a single unit cell of the array is considered this unit cell is meshed with triangular prismatic volume elements and the electric field intensity is discretized with edge based expansion functions on the sidewalls of the unit cell phase boundary conditions are employed to relate the fields on opposing walls of the unit cell on the top and or bottom unit cell planar surfaces the mesh is terminated using a mixed potential integral equation the required space domain periodic green s function is calculated after applying the ewald 1921 transformation to convert the slowly converging series representation into two rapidly converging series the method is validated for simple slot and strip frequency selective surfaces as well as microstrip dipole arrays more complex geometries investigated are slot coupled microstrip patches photonic bandgap materials and the so called artificial puck plate frequency selective surface bandpass structure

network simplification in half duplex building on submodularity this paper explores the network simplification problem in the context of gaussian half duplex diamond networks specifically given an n relay diamond network this problem seeks to derive fundamental guarantees on the capacity of the best k relay subnetwork as a function of the full network capacity simplification guarantees are presented in terms of a particular approximate capacity termed independent gaussian ig approximate capacity that characterizes the network capacity to within an additive gap which is independent of the channel coefficients and operating snr the main focus of this work is when k n 1 relays are selected out of n relays in a diamond network first a simple algorithm is proposed which selects all relays except the one with the minimum ig approximate half duplex capacity it is shown that the selected n 1 relay subnetwork has an ig approximate half duplex capacity that is at least 1 2 of the ig approximate half duplex capacity of the full network and that for the proposed algorithm this guarantee is tight furthermore this work proves the following tight fundamental guarantee there always exists a subnetwork of k n 1 relays that have an ig approximate half duplex capacity that is at least equal to n 1 n of the ig approximate half duplex capacity of the full network finally these results are extended to derive lower bounds on the fraction guarantee when k in 1 n relays are selected the key steps in the proofs lie in the derivation of properties of submodular functions which provide a combinatorial handle on the network simplification problem for gaussian half duplex diamond networks

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

a fast fourier transform accelerated ewald summation technique for the vector electromagnetic rectangular cavity green s function a fast fourier transform accelerated ewald method for the computation of the vector electromagnetic rectangular cavity green s function in terms of the electric field due to electric currents is presented and used in a boundary integral formulation the ewald summation technique suffers from the high frequency breakdown when it is applied to green s functions of wave problems in the case of the rectangular cavity green s function the number of necessary terms in the spectral series grows therefore cubically with frequency for a given accuracy to counteract the high frequency breakdown the evaluation of the spectral series is accelerated with an inverse fast fourier transform in this work at high frequencies a speed up of up to four orders of magnitude is achieved as an application example a reverberation chamber containing a metallic enclosure and a mode stirrer is modeled

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

a stochastic galerkin method for the boltzmann equation with uncertainty we develop a stochastic galerkin method for the boltzmann equation with uncertainty the method is based on the generalized polynomial chaos gpc approximation in the stochastic galerkin framework and can handle random inputs from collision kernel initial data or boundary data we show that a simple singular value decomposition of gpc related coefficients combined with the fast fourier spectral method in velocity space allows one to compute the high dimensional collision operator very efficiently in the spatially homogeneous case we first prove that the analytical solution preserves the regularity of the initial data in the random space and then use it to establish the spectral accuracy of the proposed stochastic galerkin method several numerical examples are presented to illustrate the validity of the proposed scheme

global in time estimates for the spatially homogeneous landau equation with soft potentials abstract this paper deals with some global in time a priori estimates of the spatially homogeneous landau equation for soft potentials 2 0 for the first result we obtain the estimate of weak solutions in l t l v 3 e for 2 3 e 3 2 e and 0 e 1 which is an improvement over estimates by fournier and guerin 10 for the second result we have the estimate of weak solutions in l t l v p p 1 which extends part of results by fournier and guerin 10 and alexandre liao and lin 1 as an application we deduce some global well posedness results for 2 0 our estimates include the case 2 which is the key point in this paper

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

uncertainty principle and kinetic equations abstract a large number of mathematical studies on the boltzmann equation are based on the grad s angular cutoff assumption however for particle interaction with inverse power law potentials the associated cross sections have a non integrable singularity corresponding to the grazing collisions smoothing properties of solutions are then expected on the other hand the uncertainty principle established by heisenberg in 1927 has been developed so far in various situations and it has been applied to the study of the existence and smoothness of solutions to partial differential equations this paper is the first one to apply this celebrated principle to the study of the singularity in the cross sections for kinetic equations precisely we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations in particular we give some estimates on the regularity of solutions in sobolev spaces w r t all variables for both linearized and nonlinear space inhomogeneous boltzmann equations without angular cutoff as well as the linearized space inhomogeneous landau equation

a hybrid ewald spectral cavity green s function boundary element method with spectral domain acceleration for modeling of over moded cavities a hybrid ewald spectral cavity green s function cgf boundary element method bem with spectral domain acceleration is presented in contrast to all previous work on the cgf bem the presented method relieves the two major drawbacks of the cgf approach 1 the high frequency complexity is decreased using a hybrid ewald spectral representation of the cgf and 2 the costly evaluation of the interaction integrals is accelerated with a fast group spectral domain approach results demonstrate that cavities containing small medium sized and large objects are efficiently modeled over a wide frequency range

the landau equation with the specular reflection boundary condition the existence and stability of the landau equation 1936 in a general bounded domain with a physical boundary condition is a long outstanding open problem this work proves the global stability of the landau equation with the coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the maxwellian equilibrium states the highlight of this work also comes from the low regularity assumptions made for the initial distribution this work generalizes the recent global stability result for the landau equation in a periodic box kim et al in peking math j 2020 our methods consist of the generalization of the wellposedness theory for the fokker planck equation hwang et al siam j math anal 50 2 2194 2232 2018 hwang et al arch ration mech anal 214 1 183 233 2014 and the extension of the boundary value problem to a whole space problem as well as the use of a recent extension of de giorgi nash moser theory for the kinetic fokker planck equations golse et al ann sc norm super pisa cl sci 19 1 253 295 2019 and the morrey estimates bramanti et al j math anal appl 200 2 332 354 1996 to further control the velocity derivatives which ensures the uniqueness our methods provide a new understanding of the grazing collisions in the landau theory for an initial boundary value problem

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

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

analytic smoothness effect of solutions for spatially homogeneous landau equation in this paper we study the smoothness e ect of cauchy problem for the spatially homogeneous landau equation in the hard potential case and themaxwellian molecules case we obtain the analytic smoothing e ect for the solutions under rather weak assumptions on the initial datum

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

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

the approximate capacity of half duplex line networks this paper investigates the problem of characterizing the capacity of half duplex hd line networks where a source node communicates to a destination node through a multi hop path of n relays if the relays operate in full duplex fd it is well known that the capacity of the line network equals the minimum among the point to point link capacities in the path in contrast this paper considers a different case where the relays operate in hd in the first part of the paper it is shown that the approximate capacity optimal up to a constant additive gap that only depends on the number of nodes in the network of an hd n relay line network equals half the minimum of the harmonic means of the point to point link capacities of each two consecutive links in the path it is then proved that the n 1 listen transmit states out of the 2 n possible ones sufficient to characterize the approximate capacity can be found in linear time in the second part of the paper it is shown that the problem of finding the path that has the largest hd approximate capacity in a network that can be represented as a graph is np hard however if the number of cycles in the network is polynomial in the number of nodes then a polynomial time algorithm can indeed be designed

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

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

global existence and full regularity of the boltzmann equation without angular cutoff we prove the global existence and uniqueness of classical solutions around an equilibrium to the boltzmann equation without angular cutoff in some sobolev spaces in addition the solutions thus obtained are shown to be non negative and c in all variables for any positive time in this paper we study the maxwellian molecule type collision operator with mild singularity one of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator this norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated h theorem

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

a direct method for the regularity of the gain term in the boltzmann equation this paper presents an essentially simple and direct method to prove the regularizing properties of the collision gain term in the first equation of the boltzmann hierarchy some corollaries of the main results of this paper are given to improve the previous results on the regularity of the gain term in the boltzmann equation an extension of a new fundamental result concerning separated variable kernels is also given to some general kernels

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

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

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

on the boltzmann equation for long range interactions we study the boltzmann equation without grad s angular cutoff assumption we introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables angular singularities occur as soon as one is interested in long range interactions while singularities in the relative velocity variable occur in the study of soft potentials in particular coulomb interaction together with several new estimates this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by lions appearance of strong compactness under general fully realistic assumptions 2001 john wiley sons inc

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

fast and slow convergence to equilibrium for maxwellian molecules via wild sums we consider the spatially homogeneous boltzmann equation for maxwellian molecules and general finite energy initial data positive borel measures with finite moments up to order 2 we show that the coefficients in the wild sum converge strongly to the equilibrium and quantitatively estimate the rate we show that this depends on the initial data f essentially only through on the behavior near r 0 of the function j f r v 1 r v 2 df v these estimates on the terms in the wild sum yield a quantitative estimate in the strongest physical norm on the rate at which the solution converges to equilibrium as well as a global stability estimate we show that our upper bounds are qualitatively sharp by producing examples of solutions for which the convergence is as slow as permitted by our bounds these are the first examples of solutions of the boltzmann equation that converge to equilibrium more slowly than exponentially

two dimensional method of moments modelling of lossless overmoded transverse magnetic cavities we present a method of moment mom analysis technique allowing to model overmoded transverse magnetic tm cavities in two dimensions this technique uses the single series form of the green s function of the empty cavity the convergence behaviour of this series is analysed in detail

local existence lower mass bounds and a new continuation criterion for the landau equation abstract we consider the spatially inhomogeneous landau equation with soft potentials first we establish the short time existence of solutions assuming the initial data has sufficient decay in the velocity variable and regularity no decay assumptions are made in the spatial variable next we show that the evolution instantaneously spreads mass throughout the domain the resulting lower bounds are sub gaussian which we show is optimal the proof of mass spreading is based on a stochastic process and makes essential use of nonlocality by combining this theorem with prior results we derive two important applications c smoothing even for initial data with vacuum regions and a continuation criterion the solution can be extended as long as the mass and energy densities stay bounded from above this is the weakest condition known to prevent blow up in particular it does not require a lower bound on the mass density or an upper bound on the entropy density

regularity theory for the spatially homogeneous boltzmann equation with cut off we develop the regularity theory of the spatially homogeneous boltzmann equation with cut off and hard potentials for instance hard spheres by i revisiting the lp theory to obtain constructive bounds ii establishing propagation of smoothness and singularities iii obtaining estimates on the decay of the singularities of the initial datum our proofs are based on a detailed study of the regularity of the gain operator an application to the long time behavior is presented

gaussian half duplex relay networks improved constant gap and connections with the assignment problem this paper considers a gaussian relay network where a source transmits a message to a destination with the help of n half duplex relays the information theoretic cut set upper bound to the capacity is shown to be achieved to within 1 96 n 2 bits by noisy network coding thereby reducing the previously known gap this gap is obtained as a special case of a more general constant gap result for gaussian half duplex multicast networks it is then shown that the generalized degrees of freedom of this network is the solution of a linear program where the coefficients of the linear inequality constraints are proved to be the solution of several linear programs referred as the assignment problem in graph theory for which efficient numerical algorithms exist the optimal schedule that is the optimal value of the 2 n possible transmit receive configuration states for the relays is investigated and known results for diamond networks are extended to general relay networks it is shown for the case of n 2 relays that only n 1 3 out of the 2 n 4 possible states have a strictly positive probability and suffice to characterize the capacity to within a constant gap extensive experimental results show that for a general n relay network with n leq 8 the optimal schedule has at most n 1 states with a strictly positive probability as an extension of a conjecture presented for diamond networks it is conjectured that this result holds for any half duplex relay network and any number of relays finally a network with n 2 relays is studied in detail to illustrate the channel conditions under which selecting the best relay is not optimal and to highlight the nature of the rate gain due to multiple relays

approximately achieving gaussian relay network capacity with lattice based qmf codes recently a new relaying strategy quantize map and forward qmf scheme has been demonstrated to approximately achieve within an additive constant number of bits the gaussian relay network capacity universally i e for arbitrary topologies channel gains and snrs this was established using gaussian codebooks for transmission and random mappings at the relays in this paper we develop structured lattice codes that implement the qmf strategy the main result of this paper is that such structured lattice codes can approximately achieve the gaussian relay network capacity universally again within an additive constant in addition we establish a similar result for half duplex networks where we demonstrate that one can approximately achieve the capacity using fixed transmit receive tx rx schedules for the relays with no transmit power optimization across the different tx rx states of the network

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 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

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

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