hierarchical construction of bounded solutions in critical regularity spaces we construct uniformly bounded solutions for the equations divu d f and curlu d f in the critical cases f 2 l d t d r and f2 l 3 t 3 r 3 respectively criticality in this context manifests itself by the lack of a linear solution operator mapping l d 7 l 1 t d thus the intriguing aspect here is that although the problems are linear construction of their solutions is not our constructions are special cases of a general framework for solving linear equations of the form lu d f where l is a linear operator densely defined in banach space b with a closed range in a proper subspace of lebesgue space l p and with an injective dual l the solutions are realized in terms of a multiscale hierarchical representation u d p 1 jd1 uj interesting for its own sake here the uj s are constructed recursively as minimizers of ujc1d arg min u kukbc jc1krj luk p lp

approximation in higher order sobolev spaces and hodge systems abstract let d 2 be an integer 1 l d 1 and be a differential l form on r d with w 1 d coefficients it was proved by bourgain and brezis 5 theorem 5 that there exists a differential l form on r d with coefficients in l w 1 d such that d d in the same work bourgain and brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order space w 2 d 2 or more generally in the fractional sobolev spaces w s p with s p d we give a positive answer to this question provided that d l d 1 where is the largest positive integer such that min p d the proof relies on an approximation result interesting in its own right for functions in w s p by functions in w s p l even though w s p does not embed into l in this critical case the proofs rely on some techniques due to bourgain and brezis but the context of higher order and or fractional sobolev spaces creates various difficulties and requires new ideas and methods

jacobi matrices and transversality the paper deals with smooth nonlinear ode systems in n f x such that the derivative f x has a matrix representation of jacobi type not necessarily symmetric with positive off diagonal entries a discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system f x t y two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system it is also proved that the nonwandering points are critical points a new class of morse smale systems in n is then explicitly constructed

morse theory on spaces of braids and lagrangian dynamics in the first half of the paper we construct a morse type theory on certain spaces of braid diagrams we define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces parabolic flows a type of one dimensional lattice dynamics evolve singular braid diagrams in such a way as to decrease their topological complexity algebraic lengths decrease monotonically this topological invariant is derived from a morse conley homotopy index in the second half of the paper we apply this technology to second order lagrangians via a discrete formulation of the variational problem this culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits

catalysis driven aggregate growth we propose two species catalysis driven aggregation models in which coagulation of one species occurs only in the presence of another species the catalyst by means of generalized smoluchovski rate equations we study the kinetics of the system with the rate kernel ka i j l l at which two a clusters of size i and j bond together under the catalytic action of a b cluster of size l the results show that the cluster mass distribution of species a obeys a conventional scaling law in the 0 case while it may satisfy the modified scaling form in other cases moreover it is found that the scaling exponents are nonuniversal and dependent on the value of index in most cases on the other hand we also investigate the scaling behaviour of the mutually catalysis driven aggregate growth for the system with the rate kernel and its kinetics depends crucially on the values of the indices 1 and 2 either species scales according to the conventional or modified form in most cases while the system may undergo a gelation transition in some special cases

monte carlo studies on three species two particle diffusion limited reactions abstract we studied two models of three species two particle diffusion limited reactions with and without a drift model a involves annihilations a b 0 b c 0 and c a 0 with hard core interactions between like species and assumes equal initial densities of the species the density decays according to a power law c t t the critical exponent was estimated as 0 353 6 without the drift and 0 390 4 with the maximum drift model b is a coagulation model as a b c b c a and c a b in this model we observed the critical exponent 1 2 regardless of the drift we argue that model b belongs to the same universality class as single species annihilation or coagulation diffusion limited reactions we also studied the average distance of the nearest neighbor species the average domain size and the average number of the species pairs

bourgain brezis inequalities on symmetric spaces of non compact type abstract let m be a global riemannian symmetric space of non compact type we prove a duality estimate for pairings of divergence free l 1 vector fields with vector fields in a critical sobolev space on m m f d v c f l 1 d v l m d v this estimate provides a remedy for the failure of a critical sobolev embedding on such symmetric spaces

fast diffusion mean field theory for k body reactions in one dimension the authors derive an improved mean field approximation for k body annihilation reactions ka to inert for hard core diffusing particles on a line annihilating in groups of k neighbours with probability 0 q or 1 the hopping and annihilation processes are correlated to mimic chemical reactions their new mean field theory accounts for hard core particle properties and has a larger region of applicability than the standard chemical rate equation especially for large k values criteria for validity of the mean field theory and its use in phenomenological data fits are derived numerical tests are reported for k 3 4 5 6

an optimal sobolev embedding for l1 abstract in this paper we establish an optimal lorentz space estimate for the riesz potential acting on curl free vectors there is a constant c c d 0 such that i f l d d 1 r d r d c f l 1 r d r d for all fields f l 1 r d r d such that curl f 0 in the sense of distributions this is the best possible estimate on this scale of spaces and completes the picture in the regime p 1 of the well established results for p 1

form boundedness of the general second order differential operator we give explicit necessary and sufficient conditions for the boundedness of the general second order differential operator i j ln aij i j j 1nbj j c with real or complex valued distributional coefficients aij bj and c acting from the sobolev space w1 2 n to its dual w 1 2 n this enables us to obtain analytic criteria for the fundamental notions of relative form boundedness compactness and infinitesimal form boundedness of c with respect to the laplacian on l 2 n in particular we establish a complete characterization of the form boundedness of the schrodinger operator i a 2 q with magnetic vector potential a lloc2 n n and q d n 2006 wiley periodicals inc

gauss green theorem for weakly differentiable vector fields sets of finite perimeter and balance laws we analyze a class of weakly differentiable vector fields f w r n r n with the property that f 2 l 1 and div f is a signed radon measure these fields are called bounded divergence measure fields the primary focus of our inves tigation is to introduce a suitable notion of the normal trace of any divergence measure field f over the boundary of an arbitrary set of finite perimeter that en sures the validity of the gauss green theorem to achieve this we first establish a fundamental approximation theorem which states that given a signed radon measurethat is absolutely continuous with respect to h n 1 on r n a ny set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure theoretic interior of the set with respect to the mea sure kk the total variation measure we employ this approximation theorem to derive the normal trace of f on the boundary of any set of finite perimeter e as the limit of the normal traces of f on the boundaries of the approximate sets with smooth boundary so that the gauss green theorem for f holds one with these results we analyze the cauchy flux that is bounded by a nonnegative radon measure over any oriented surface i e an n 1 dimensional surface that is a part of the boundary of a set of finite perimeter and thereby develop a general mathematical formulation of the physical principle of the balance law through the cauchy flux finally we apply this framework to the derivation of systems of balance laws with measure valued source terms from the formulation of the balance law this framework also allows the recovery of cauchy entropy flux through the lax entropy inequality for entropy solutions of hyperbolic con servation laws c 2008 wiley periodicals inc

scalar parabolic pdes and braids the comparison principle for scalar second order parabolic pdes on functions u t x admits a topological interpretation pairs of solutions u 1 t 1 and u 2 t evolve so as to not increase the intersection number of their graphs we generalize to the case of multiple solutions u t n 1 by lifting the graphs to legendrian braids we give a global version of the comparison principle the curves u t evolve so as to weakly decrease the algebraic length of the braid we define a morse type theory on legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic pdes this is done via discretization to a finite dimensional system and a suitable conley index for discrete braids the result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic pdes we give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time periodic solutions

a poincare bendixson theorem for scalar reaction diffusion equations for scalar equations u t u xx f x u u x with x e s 1 and f e c 2 we show that the classical theorem of poincare and bendixson holds the limit set of any bounded solution satisfies exactly one of the following alternatives this is surprising because the system is genuinely infinite dimensional

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

a limiting case for the divergence equation we consider the equation text div mathbb y f with f a zero average function on the torus mathbb t d in their seminal paper bourgain and brezis j am math soc 16 2 393 426 2003 electronic proved the existence of a solution mathbb y in w 1 d cap l infty for a datum f in l d we extend their result to the critical sobolev spaces w s p with s 1 p d and p ge 2 more generally we prove a similar result in the scale of triebel lizorkin spaces we also consider the equation text div mathbb y f in a bounded domain varomega subject to zero dirichlet boundary condition

exact results for diffusion limited reactions with synchronous dynamics a method is introduced allowing one to solve exactly the reactions a a ensuremath rightarrow inert and a a ensuremath rightarrow a on the one dimensional lattice with synchronous diffusional dynamics simultaneous hopping of all particles exact connections are found relating densities and certain correlation properties of these two reactions at all times asymptotic behavior at large times as well as scaling form describing the regime of low initial density are derived explicitly

a solvable two species catalysis driven aggregation model we study the kinetics of a two species catalysis driven aggregation system in which an irreversible aggregation between any two clusters of one species occurs only with the catalytic action of another species by means of a generalized mean field rate equation we obtain the asymptotic solutions of the cluster mass distributions in a simple process with a constant rate kernel for the case without any consumption of the catalyst the cluster mass distribution of either species always approaches a conventional scaling law however the evolution behaviour of the system in the case with catalyst consumption is complicated and depends crucially on the relative data of the initial concentrations of the two species

braided connecting orbits in parabolic equations via computational homology abstract we develop and present a computational method for producing forcing theorems for stationary and periodic solutions and connecting orbits in scalar parabolic equations with periodic boundary conditions this method is based on prior work by van den berg ghrist and vandervorst on a conley index theory for solutions braided through a collection of known stationary solutions essentially the topological structure of the stationary solutions forces the existence of additional solutions with a specified topological type in particular this paper studies connecting orbits and develops and implements the algorithms required to compute the index providing sample results as illustrations

model of cluster growth and phase separation exact results in one dimension we present exact results for a lattice model of cluster growth in one dimension the growth mechanism involves interface hopping and pairwise annihilation supplemented by spontaneous creation of the stable phase 1 regions by overturning the unstable phase 1 spins with probabilityp for cluster coarsening at phase coexistence p 0 the conventional structure factor scaling applies in this limit our model falls in the class of diffusion limited reactions a a inert the 1 cluster size grows diffusively t and the two point correlation function obeys scaling however forp 0 i e for the dynamics of formation of stable phase from unstable phase we find that structure factor scaling breaks down the length scale associated with the size of the growing 1 clusters reflects only the short distance properties of the two point correlations

smoluchowski approach for three body reactions in one dimension we present an extension of the smoluchowski approach for the analysis of diffusion limited three body reactions in one dimensional systems in terms of this approach we define the evolution of the particles mean concentration over the entire time domain c 1995 the american physical society

anomalous coalescence phenomena under stationary regime condition through thompson s method abstract in this work anomalous coalescence phenomena of type a a a 0 are considered in the case of stationary regime namely the case of an external homogeneous source h of single particles a in the limit of zero external field rate the main purpose here is to pick up new critical exponents in the stationary regime for the phenomenon of anomalous coalescence of bubbles found in one dimension by josserand and rica jr phys rev lett 78 1997 1215 the critical exponents of statical concentration relaxation time and the exponent for concentration decay are evaluated in the limit h 0 zero field rate we show that such critical exponents depend on the parameter physica a 334 2004 335 which characterizes the anomalous diffusion condition in order to explain the anomalous coalescence of jr s work as already made before by the present author physica a 334 2004 335 the parameter must display an explicit dependence on the dimensionality d 4 d for d 2 we use such a result d in order to pick up new critical exponents for the special case of anomalous coalescence found by jr we find 5 4 5 and 1 4 in 1 d we also verify that the known scaling relations among such critical indexes phys rev a 32 2 1985 1129 are obeyed so that in the special case of 2 brownian diffusion condition given to mean field regime we naturally recover those results previously obtained by racz phys rev a 32 2 1985 1129 we go further to give a better explanation for such new results taking into account another exponent physica a 334 2004 335 to explain the behavior of diffusion constant which depends on concentration d d 0 n that is to say we obtain analytically new critical indexes which controll the diffusion behavior d for long time regime in the presence of an external homogeneous source

second order lagrangian twist systems simple closed characteristics we consider a special class of lagrangians that play a fundamental role in the theory of second order lagrangian systems twist systems this subclass of lagrangian systems is defined via a convenient monotonicity property that such systems share this monotonicity property twist property allows a finite dimensional reduction of the variational principle for finding closed characteristics in fixed energy levels this reduction has some similarities with the method of broken geodesics for the geodesic variational problem on riemannian manifolds on the other hand the monotonicity property can be related to the existence of local twist maps in the associated hamiltonian flow the finite dimensional reduction gives rise to a second order monotone recurrence relation we study these recurrence relations to find simple closed characteristics for the lagrangian system more complicated closed characteristics will be dealt with in future work furthermore we give conditions on the lagrangian that guarantee the twist property

three species diffusion limited reaction with continuous density decay exponents we introduce a model of three species two particle diffusion limited reactions or b or c and or a with three persistence parameters survival probabilities in reaction of the hopping particle we consider isotropic and anisotropic diffusion hopping with a drift in one dimension we find that the particle density decays as a power law for certain choices of the persistence parameter values in the anisotropic case on one symmetric line in the parameter space the decay exponent is monotonically varying between the values close to 1 3 and 1 2 on another less symmetric line the exponent is constant for most parameter values the density does not follow a power law we also calculated various characteristic exponents for the distance of nearest particles and domain structure our results support the recently proposed possibility that one dimensional diffusion limited reactions with a drift do not fall within a limited number of distinct universality classes

renormalisation of fluctuation effects in the a a to a reaction the field theoretical description of the a a to a reaction is exactly renormalised to all orders in perturbation theory it is confirmed that it belong to the same universality class as the a a to oe reaction and that the concentration decay like t d 2 for space dimensionality d smaller than two and like ln t t for d 2 are exact results

asymptotic behavior of densities for two particle annihilating random walks consider the system of particles on d where particles are of two types a andb and execute simple random walks in continuous time particles do not interact with their own type but when ana particle meets ab particle both disappear i e are annihilated this system serves as a model for the chemical reactiona b inert we analyze the limiting behavior of the densities a t and b t when the initial state is given by homogeneous poisson random fields we prove that for equal initial densities a 0 b 0 there is a change in behavior fromd 4 where a t b t c t d 4 tod 4 where a t b t c tast for unequal initial densities a 0 b 0 a t e c l ind 1 a t e ct logt ind 2 and a t e ct ind 3 the termc depends on the initial densities and changes withd techniques are from interacting particle systems the behavior for this two particle annihilation process has similarities to those for coalescing random walks a a a and annihilating random walks a a inert the analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

a constructive proof of the fefferman stein decomposition of bmo rn let s vec r left left f r 1 f cdots r n f right f in h 1 left text r n text r right right subset h 1 left text r n text r n 1 right 22 1 begin gathered s vec r left vec g left g j right j 0 cdots n right in text bmo left text r n text r n 1 right sum limits j 0 n vec r j g j 0 text in bmo left text r n text r n 1 right left text r right end gathered 22 2 j 0

function spaces between bmo and critical sobolev spaces the function spaces d k r n are introduced and studied the definition of these spaces is based on a regularity property for the critical sobolev spaces w s w p r n where sp n obtained by j bourgain h brezis new estimates for the laplacian the div curl and related hodge systems c r math acad sci paris 338 7 2004 539 543 see also j van schaftingen estimates for l 1 vector fields c r math acad sci paris 339 3 2004 181 186 the spaces d k r n contain all the critical sobolev spaces they are embedded in bmo r n but not in vmo r n moreover they have some extension and trace properties that bmo r n does not have c 2006 elsevier inc all rights reserved

continuously varying exponents in reaction diffusion systems we propose a simple reaction diffusion model describing a class of competitive multi species reactions the model is exactly solvable at its upper critical dimension du 2 the local moments of the component concentrations follow a power law decay with exponents which vary continuously with system parameters the exponents also have an underlying multifractal spectrum the main results are supported by results from a numerical integration of the model

sharp a priori estimates for div curl systems in heisenberg groups abstract in this paper we prove a family of inequalities for differential forms in heisenberg groups h 1 and h 2 that are the natural counterpart of a class of div curl inequalities in de rham s complex proved by lanzani stein and bourgain brezis

analytical study of an asymmetric 1d statistical model for chemical reactions abstract an analytical study is presented for the asymmetric fgz reaction model involving two different species a and b only b can spontaneously desorb in contact with a bath containing a and b with the same concentration the expected values of the density and the pair correlation functions are calculated and their time behaviour is analysed in detail this also allows to define an average time dependent fragmentation index describing quantitatively the evolution of the topology connectivity of the clusters in addition when the system is doomed to end its evolution in the a poisoned state the distribution law p t of the times t k at which this occurs is investigated numerically it turns out that in the case where a single isolated b is present in the initial state this law is well enough represented by a stretched exponential in the log variable p t c ste exp 1n t s