bounds on coarsening rates for the lifschitz slyozov wagner equation this paper is concerned with the large time behavior of solutions to the lifschitz slyozov wagner lsw system of equations point wise in time upper and lower bounds on the rate of coarsening are obtained for solutions with fairly general initial data these bounds complement the time averaged upper bounds obtained by dai and pego and the point wise in time upper and lower bounds obtained by niethammer and velasquez for solutions with initial data close to a self similar solution

on the rate of convergence to equilibrium in the becker doring equations abstract we provide an explicit rate of convergence to equilibrium for solutions of the becker doring equations using the energy energy dissipation relation the main difficulty is the structure of equilibria of the becker doring equations which do not correspond to a gaussian measure such that a logarithmic sobolev inequality is not available we prove a weaker inequality which still implies for fast decaying data that the solution converges to equilibrium as e ct1 3

graded contractions of representations of orthogonal and symplectic lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras o n c n or 5 and sp 2n c n or 2 with complex parameters are described for all maximal parabolic subalgebras all contractions which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras are found contractions of the irreducible representations for each parabolic contraction of the lie algebra are the main results of the article

batalin vilkovisky formalism in the functional approach to classical field theory we develop the batalin vilkovisky formalism for classical field theory on generic globally hyperbolic spacetimes a crucial aspect of our treatment is the incorporation of the principle of local covariance which amounts to formulate the theory without reference to a distinguished spacetime in particular this allows a homological construction of the poisson algebra of observables in classical gravity our methods heavily rely on the differential geometry of configuration spaces of classical fields

graded contractions of casimir operators we describe graded contractions of casimir operators of lie algebras the formalism applies to any casimir operator given as a symmetric form we deal with the quadratic casimir operator in exhaustive detail and indicate the modifications for operators of higher degree

the becker doring equations at large times and their connection with the lsw theory of coarsening a systematic procedure is given for obtaining the asymptotic late time behavior of the becker doring equations describing the time evolution of a population of clusters of particles in lowest order of approximation the distribution of the sizes of the largest clusters satisfies the equations of the lifshitz slyozov wagner theory of coarsening

a functional central limit theorem for the becker doring model we investigate the fluctuations of the stochastic becker d oring model of polymerization when the initial size of the system converges to infinity a functional central limit problem is proved for the vector of the number of polymers of a given size it is shown that the stochastic process associated to fluctuations is converging to the strong solution of an infinite dimensional stochastic differential equation sde in a hilbert space we also prove that at equilibrium the solution of this sde is a gaussian process the proofs are based on a specific representation of the evolution equations the introduction of a convenient hilbert space and several technical estimates to control the fluctuations especially of the first coordinate which interacts with all components of the infinite dimensional vector representing the state of the process

from the becker doring to the lifshitz slyozov wagner equations connections between two classical models of phase transitions the becker doring bd equations and the lifshitz slyozov wagner lsw equations are investigated homogeneous coefficients are considered and a scaling of the bd equations is introduced in the spirit of the previous works by penrose and collet goudon poupaud and vasseur convergence of the solutions to these rescaled bd equations towards a solution to the lsw equations is shown for general coefficients an approach in the spirit of numerical analysis allows to approximate the lsw equations by a sequence of bd equations a new uniqueness result for the bd equations is also provided

the becker doring process pathwise convergence and phase transition phenomena in this note we study an infinite reaction network called the stochastic becker doring process a sub class of the general coagulation fragmentation models we prove pathwise convergence of the process towards the deterministic becker doring equations which improves classical tightness based results also we show by studying the asymptotic behavior of the stationary distribution that the phase transition property of the deterministic model is also present in the finite stochastic model such results might be interpreted closed to the so called gelling phenomena in coagulation models we end with few numerical illustrations that support our results

the becker doring equations and the lifshitz slyozov theory of coarsening in this paper the relation between the kinetic set of becker doring bd equations and the classical lifshitz slyozov ls theory of coarsening is studied a model that resembles the ls theory but keeps some of the nucleation effects is derived for this model a solution is described that shows how the kinetic effects explain the particular solution selected in the ls theory by means of a renormalization procedure a discrete group of transformations is shown to play an important role in describing the structure of the solution near the critical size of the ls theory

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions

holder continuous homomorphisms between infinite dimensional lie groups are smooth let f g h be a homomorphism between smooth lie groups modelled on mackey complete locally convex real topological vector spaces we show that if f is holder continuous at 1 then f is smooth

graded contractions of representations of special linear lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras sl n c n or 3 are described for all maximal parabolic subalgebras all contractions are found which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras contractions of the irreducible representations of sl n c for each parabolic contraction of the lie algebra are the main results of the article

direct limits of infinite dimensional lie groups compared to direct limits in related categories let g be a lie group which is the union of an ascending sequence g1 g2 of lie groups all of which may be infinite dimensional we study the question when g limgn in the category of lie groups topological groups smooth manifolds respectively topological spaces full answers are obtained for g the group diffc m of compactly supported c diffeomorphisms of a compact smooth manifold m and for test function groups cc m h of compactly supported smooth maps with values in a finite dimensional lie group h we also discuss the cases where g is a direct limit of unit groups of banach algebras a lie group of germs of lie group valued analytic maps or a weak direct product of lie groups

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

some remarks on large time asymptotic of the lifshitz slyozov equations we obtain an entropy functional for the lifshitz slyozov system this can be used to investigate the time asymptotics of the system in particular we describe situations in which the monomers concentration either tend to 0 or saturate as time becomes large the latter situation can be excluded under assumptions on the support of the initial data

the modified becker doring system with aggregation dominated rate coefficients we consider a system of becker doring equations modified in the manner of dreyer duderstadt 2006 j stat phys 123 55 we analyse the case of size independent rate coefficients in the case of weak fragmentation aggregation dominated we use matched asymptotic expansions to construct approximations to the cluster size distribution as it evolves from purely monomeric initial data to the equilibrium solution in which the typical cluster size is extremely large we compare the results with earlier calculations of a similar limit in the standard becker doring system and find a similar sequence of timescales with similar profile shapes however there are notable differences in the kinetic behaviour within timescales due to the modified system having a fragmentation rate which depends on the total number of clusters present in the system

contractions of invariants of lie algebras with applications to classical inhomogeneous lie algebras we study in full generality the contraction of invariants of finite dimensional lie algebras and give some illustrative examples our applications show that the contraction method is not only quite flexible but also has the advantage over other methods that independent invariants can be contracted into independent ones and that they appear in compact tensorial form for our applications of physical interest we can restrict ourselves to polynomial invariants casimir operators and generalized inonu wigner contractions we show how to get all invariants of the inhomogeneous lie algebras iso p q iu p q isl n r and isp 2n r from those of the homogeneous lie algebras these invariants are all known but our results for isl n r and isp 2n r appear in a much more desirable form

well posedness of smoluchowski s coagulation equation for a class of homogeneous kernels the uniqueness and existence of measure valued solutions to smoluchowski s coagulation equation are considered for a class of homogeneous kernels denoting by 2 0 the degree of homogeneity of the coagulation kernel a measure valued solutions are shown to be unique under the sole assumption that the moment of order of the initial datum is finite a similar result was already available for the kernels a x y 2 x y and xy and is extended here to a much wider class of kernels by a different approach the uniqueness result presented herein also seems to improve previous results for several explicit kernels furthermore a comparison principle and a contraction property are obtained for the constant kernel

graded contractions of the lie algebra e 2 1 the graded contractions of pseudo euclidean lie algebra are studied the non equivalent gradings of of type and a are extended to the entire lie algebra using the action of on the abelian ideal the translations the graded contractions embed into a large family of six dimensional lie algebras the family includes solvable nilpotent and nonsolvable lie algebras both decomposable and indecomposable ones the distinction between graded and inonu wigner contractions is analysed the physically most interesting lie algebras obtained by the contractions are the inhomogeneous galilei and pseudo galilei

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

from the discrete to the continuous coagulation fragmentation equations the connection between the discrete and the continuous coagulation fragmentation models is investigated a weak stability principle relying on a priori estimates and weak compactness in l 1 is developed for the continuous model we approximate the continuous model by a sequence of discrete models and writing the discrete models as modified continuous ones we prove the convergence of the latter towards the former with the help of the above mentioned stability principle another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation fragmentation model

metastable states for the becker doring cluster equations the becker doring equations in whichc l t can represent the concentration ofl particle clusters or droplets in say a condensing vapour at timet are begin array 20 c dc l t mathord left vphantom dc l t dt right kern nulldelimiterspace dt j l 1 t j l t l 2 3 end array with j l t a l c 1 t c l t b l 1 c l 1 t and eitherc1 const case a or rho sum limits 1 infty lc l const case b the equilibrium solutions arec l q l z l where q l prod limits 2 l a r 1 mathord left vphantom a r 1 b r right kern nulldelimiterspace b r the density of the saturated vapour defined as rho s sum limits 1 infty lq l z s l wherez s is the radius of convergence of the series is assumed finite it is proved here that subject to some further plausible conditions on the kinetic coefficientsa l andb l there is a class of metastable solutions of the equations withc1 z s small and positive which take an exponentially long time to decay to their asymptotic steady states an exponentially long time means one that increases more rapidly than any negative power of the given value ofc1 z s or in caseb s as the latter tends to zero the main ingredients in the proof are i a time independent upper bound on the solution of the kinetic equations this upper bound is a steady state solution of case a of the equations of the type used in the becker doring theory of nucleation and ii an upper bound on the total concentration of particles in clusters greater than a certain critical size which with suitable initial conditions remains exponentially small until the time becomes exponentially large

analytic mappings between lb spaces and applications in infinite dimensional lie theory we give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces this tool is then used to construct new classes of real and complex infinite dimensional lie groups the group diffgerm k x of germs of analytic diffeomorphisms around a compact set k in a banach space x and the group s n n gn where the gn are banach lie groups

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

asymptotic behaviour of solutions to the generalized becker doring equations for general initial data we prove the following asymptotic behaviour for solutions to the generalized becker doring system for general initial data under a detailed balance assumption and in situations where density is conserved in time there is a critical density s such that solutions with an initial density s converge strongly to the equilibrium with density and solutions with initial density s converge in a weak sense to the equilibrium with density s this extends the previous knowledge that this behaviour happens under more restrictive conditions on the initial data the main tool is a new estimate on the tail of solutions with density below the critical density

lie group structures on quotient groups and universal complexifications for infinite dimensional lie groups we characterize the existence of lie group structures on quotient groups and the existence of universal complexifications for the class of baker campbell hausdorff bch lie groups which subsumes all banach lie groups and linear direct limit lie groups as well as the mapping groups ckr m g cr m g m k 1 for every bch lie group g second countable finite dimensional smooth manifold m compact subset k of m and 0 r also the corresponding test function groups dr m g kckr m g are bch lie groups

graded contractions of bilinear invariant forms of lie algebra we introduce a new construction for bilinear invariant forms on lie algebras based on the method of graded contractions the general method is described and the z2 z3 and z2 x z2 contractions are found the results can be applied to all lie algebras and superalgebras finite or infinite dimensional which admit the chosen gradings we consider some examples contractions of the killing form toroidal contractions of sl 3 c and we briefly discuss the limit to new wess zumino witten actions

continuity of bilinear maps on direct sums of topological vector spaces abstract we prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces as a first application we get a new proof for the fact due to hirai et al 2001 that the map f c c r n c c r n c c r n taking a pair of test functions to their convolution is continuous the criterion also allows an open problem by k h neeb to be solved if e is a locally convex space regard the tensor algebra t e j n 0 t j e as the locally convex direct sum of projective tensor powers of e we show that t e is a topological algebra if and only if every sequence of continuous seminorms on e has an upper bound in particular if e is metrizable then t e is a topological algebra if and only if e is normable also t e is a topological algebra if e is dfs or k

long time behavior of a modified becker doring system a modification based on asymptotic behavior of the becker doring system is introduced in which the concentration of monomers is slaved to the concentrations of the other clusters this modified system has the same continuum limit as the usual becker doring system for one member of the modified systems it is proved for compact initial data that all solutions will converge to the same self similar form as time tends to infinity

on a thermodynamically consistent modification of the becker doring equations abstract recently dreyer and duderstadt have proposed a modification of the becker doring cluster equations which now have a different lyapunov function we start with existence and uniqueness results for the modified equations next we derive an explicit criterion for the existence of equilibrium states and solve the minimization problem for the lyapunov function finally we discuss the long time behavior in the case that equilibrium solutions do exist

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

casimir invariants for the complete family of quasisimple orthogonal algebras a complete choice of generators of the centre of the enveloping algebras of real quasisimple lie algebras of orthogonal type for arbitrary dimension is obtained in a unified setting the results simultaneously include the well known polynomial invariants of the pseudo orthogonal algebras as well as the casimirs for many non simple algebras such as the inhomogeneous the newton hooke and galilei type etc which are obtained by contraction s starting from the simple algebras the dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction the structure of the higher order invariants is given in a convenient pyramidal manner in terms of certain sets of pauli lubanski elements in the enveloping algebras as an example showing this approach at work the scheme is applied to recovering the casimirs for the 3 1 kinematical algebras some prospects on the relevance of these results for the study of expansions are also given

asymptotic behaviour of solutions to the becker doring equations for arbitrary initial data on determine le comportement asymptotique des solutions des equations d amas de becker doring sous certaines hypotheses de decroissances sur les donnees initiales faites dans un article anterieur