about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

a stochastic particle numerical method for 3d boltzmann equations without cutoff using the main ideas of tanaka the measure solution pt t of a 3 dimensional spatially homogeneous boltzmann equation of maxwellian molecules without cutoff is related to a poisson driven stochastic differential equation using this tool the convergence to pt t of solutions ptl t of approximating boltzmann equations with cutoff is proved then a result of graham meleard is used and allows us to approximate ptl t with the empirical measure tl n t of an easily simulable interacting particle system precise rates of convergence are given a numerical study lies at the end of the paper

steady states and dynamics of an autocatalytic chemical reaction model with decay abstract the dynamics and steady state solutions of an autocatalytic chemical reaction model with decay in the catalyst are considered nonexistence and existence of nontrivial steady state solutions are shown by using energy estimates upper lower solution method and bifurcation theory the effects of decay order decay rate and diffusion rates to the dynamical behavior are discussed

quenching of flames by fluid advection we consider a simple scalar reaction advection diffusion equation with ignitiontype nonlinearity and discuss the following question what kinds of velocity profiles are capable of quenching any given flame provided the velocity s amplitude is adequately large even for shear flows the answer turns out to be surprisingly subtle if the velocity profile changes in space so that it is nowhere identically constant or if it is identically constant only in a region of small measure then the flow can quench any initial data but if the velocity profile is identically constant in a sizable region then the ensuing flow is incapable of quenching large enough flames no matter how much larger the amplitude of this velocity is the constancy region must be wider across than a couple of laminar propagating front widths the proof uses a linear pde associated to the nonlinear problem and quenching follows when the pde is hypoelliptic the techniques used allow the derivation of new nearly optimal bounds on the speed of traveling wave solutions c 2001 john wiley sons inc

thermodynamics and analysis of rate independent adhesive contact at small strains we address a model for adhesive unilateral frictionless signorini type contact between bodies of heatconductive viscoelastic material in the linear kelvin voigt rheology undergoing thermal expansion the flow rule for debonding the adhesion is considered rate independent and unidirectional and a thermodynamically consistent model is derived and analysed as far as the existence of a weak solution is concerned

asymptotic of grazing collisions and particle approximation for the kac equation without cutoff the subject of this article is the kac equation without cutoff we first show that in the asymptotic of grazing collisions the kac equation can be approximated by a fokker planck equation the convergence is uniform in time and we give an explicit rate of convergence next we replace the small collisions by a small diffusion term in order to approximate the solution of the kac equation and study the resulting error we finally build a system of stochastic particles undergoing collisions and diffusion that we can easily simulate which approximates the solution of the kac equation without cutoff we give some estimates on the rate of convergence

a vanishing viscosity approach to quasistatic evolution in plasticity with softening we deal with quasistatic evolution problems in plasticity with softening in the framework of small strain associative elastoplasticity the presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate independent problems to the case of a nonconvex energy functional we argue that in this case the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view thus we analyse a different selection criterion for the solutions of the quasistatic evolution problem based on a viscous approximation this leads to a generalized formulation in terms of young measures developed in the first part of the paper in the second part we apply our approach to some concrete examples

sharp transition between extinction and propagation of reaction with x e r this equation has been extensively studied in mathematical physical and other literature starting with the pioneering works of fisher 7 and kol mogorov petrovskii piskunov 11 in these papers 1 2 was used to describe the propagation of advantageous genes in a population the main object of study in these and many subsequent works was the existence and stability of traveling fronts for 1 2 and 1 1 in recent years most of the results have been extended to in clude an advection term u vt in 1 1 and we refer to the reviews 2 16 for an extensive bibliography the above equations are used to model more than just population genetics phe nomena when f 0 0 for 9 g 0 1 then is a combustion non linearity and 1 1 1 2 model an exothermic chemical reaction in an infinite tube with a zero heat loss boundary in particular flame propagation in a premixed combustible gas without advection see zel dovich and frank kamenetskii 17 in this setting t is the normalized temperature taking values in 0 1 we note that 1 1 is usu ally obtained from a system involving both the temperature and the concentration of the reactants after the simplifying assumption of equal thermal and material diffusivities a special case of positive used particularly in chemical and biological lit erature is the kpp type with f 6 c 0 11 in combustion models the non linearity is often considered to be of arrhenius type with slow reaction rates at low temperatures modeled by f 0 e a e l 0 this case is often approximated

threshold solutions and sharp transitions for nonautonomous parabolic equations on mathbb r n this paper is devoted to a class of nonautonomous parabolic equations of the form ut u f t u on mathbb r n we consider a monotone one parameter family of initial data with compact support such that for small values of the parameter the corresponding solutions decay to zero whereas for large values they exhibit a different behavior either blowup in finite time or locally uniform convergence to a positive constant steady state we are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs we refer to such values as threshold values and to the corresponding solutions as threshold solutions we prove that the transition from decay to the other behavior is sharp there is just one threshold value we also describe the behavior of the threshold solution it is global bounded and asymptotically symmetric in the sense that all its limit profiles as t are radially symmetric about the same center our proofs rely on parabolic liouville theorems asymptotic symmetry results for nonlinear parabolic equations and theorems on exponential separation and principal floquet bundles for linear parabolic equations

global existence for reaction diffusion systems modelling ignition the pair of parabolic equations with a 0 and b 0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f of particular interest is the case where which appears in the frank kamenetskii approximation to arrhenius type reactions we show here that for a large choice of the nonlinearity f u v in 1 2 including the model case 3 the corresponding initial value problem for 1 2 in the whole space with bounded initial data has a solution which exists for all times finite time blow up might occur though for other choices of function f u v and we discuss here a linear example which strongly hints at such behaviour

convergence from boltzmann to landau processes with soft potential and particle approximations our aim in this paper is to show how a probabilistic interpretation of the boltzmann and landau equations gives a microscopic understanding of these equations we firstly associate stochastic jump processes with the boltzmann equations we consider then we renormalize these equations following asymptotics which make prevail the grazing collisions and prove the convergence of the associated boltzmann jump processes to a diffusion process related to the landau equation the convergence is pathwise and also implies a convergence at the level of the partial differential equations the best feature of this approach is the microscopic understanding of the transition between the boltzmann and the landau equations by an accumulation of very small jumps we deduce from this interpretation an approximation result for a solution of the landau equation via colliding stochastic particle systems this result leads to a monte carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from boltzmann to landau equations numerical results are given

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

effective relaxation for microstructure simulations algorithms and applications for a wide class of problems in continuum mechanics like those involving phase transitions or finite elastoplasticity the governing potentials tend to be not quasiconvex this leads to the occurrence of microstructures of in principle arbitrarily small scale which cannot be resolved by standard discretization schemes their effective macroscopic properties however can efficiently be recovered with relaxation theory the paper introduces the variational framework necessary for the implementation of relaxation algorithms with emphasis on problems with internal variables in a time incremental setting the methods developed are based on numerical approximations to notions of generalized convexification the focus is on the thorough analysis of numerical algorithms and their efficiency in applications to benchmark problems an outlook to time evolution of microstructures within the framework of relaxation theory concludes the paper 2004 published by elsevier b v

a variational formulation of rate independent phase transformations using an extremum principle we propose a rate independent mesoscopic model for the hysteretic evolution of phase transformations in shape memory alloys the model uses the deformation and phase indicator function as basic unknowns and the potentials for the elastic energy and for the dissipation as constitutive laws using the associated functionals admissible processes are defined to be the ones which are stable at all times and which satisfy the energy inequality

quasistatic crack growth in nonlinear elasticity in this paper we prove a new existence result for a variational model of crack growth in brittle materials proposed in 19 we consider the case of n dimensional nonlinear elasticity for an arbitrary n 1 with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads both depending on time

quasi static small strain plasticity in the limit of vanishing hardening and its numerical approximation the quasi static rate independent evolution of the prager ziegler type model of plasticity with hardening is shown to converge to the rate independent evolution of the prandtl reuss elastic perfectly plastic model based on the concept of energetic solutions we study the convergence of the solutions in the limit for hardening coefficients converging to 0 by using the abstract method of gamma convergence for rate independent systems an unconditionally convergent numerical scheme is devised and two and three dimensional numerical experiments are presented a two sided energy inequality is a posteriori verified to document experimental convergence rates

existence and convergence for quasi static evolution in brittle fracture this paper investigates the mathematical well posedness of the variational model of quasi static growth for a brittle crack proposed by francfort and marigo in 15 the starting point is a time discretized version of that evolution which results in a sequence of minimization problems of mumford and shah type functionals the natural weak setting is that of special functions of bounded variation and the main difficulty in showing existence of the time continuous quasi static growth is to pass to the limit as the time discretization step tends to 0 this is performed with the help of a jump transfer theorem which permits under weak convergence assumptions for a sequence un of sbv functions to its bv limit u to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence un in particular it is shown that the notion of minimizer of a mumford and shah type functional for its own jump set is stable under weak convergence assumptions furthermore our analysis justifies numerical methods used for computing the time continuous quasi static evolution 2003 wiley periodicals inc

existence and nonexistence of travelling waves of a reaction diffusion system abstract in this paper we study the existence and nonexistence of travelling waves of the reaction diffusion system a t a x x e a x a b k a b q b t d b x x d e b x a b k a b q where a b are concentration densities of the two chemical species d 0 is the ratio of diffusion coefficients of the two chemical species involved e is the strength of the electric field applied k 0 is a parameter and q 1 the system arises from the study of two dimensional iodate arsenous acid iaa reaction with a constant electric field applied iaa reaction is very much studied in experiments and demonstrates the interesting phenomenon of instability driven by buoyancy of travelling waves by using a novel approach explicit bounds c d e and c d e are derived such that there is a unique travelling wave of every speed c c d e and there does not exist any travelling wave of speed c c d e with q being fixed in addition c d e bears a close resemblance of the minimum travelling wave speed of the classical case of d 1 and e 0 when 0 e 1 and d 1

ground states of simple bodies that may undergo brittle fractures abstract equilibrium states of elastic brittle solids that may suffer nucleation of cracks in finite deformation setting are analyzed crack patterns are described in terms of appropriate radon measures namely curvature varifolds with boundary a new form of the energy is presented it includes terms associated with the curvature of margins and tips of possible cracks existence of minima of the energy is established in classes of pairs of deformation and families of varifolds configurational balances in weak form are determined with reference to generic curvature varifolds with boundary they include non standard terms associated with the curvatures involved in the energy pointwise balances of configurational actions are also evaluated in a special case new pointwise balances at the tips and along the margins of the crack pattern emerge

a density result in two dimensional linearized elasticity and applications we show that in a two dimensional bounded open set whose complement has a finite number of connected components the vector fields u h 1 2 are dense in the space of fields whose symmetrized gradient e u is in l 2 4 this allows us to show the continuity of some linearized elasticity problems with respect to variations of the set with applications to shape optimization or the study of crack evolution

the existence of minimum speed of traveling wave solutions to a non kpp isothermal diffusion system abstract the reaction diffusion system a t a x x a b n b t d b x x a b n where n 1 and d 0 arises from many real world chemical reactions whereas n 1 is the kpp type nonlinearity which is much studied and very important results obtained in literature not only in one dimensional spatial domains but also multi dimensional spaces but n 1 proves to be much harder one of the interesting features of the system is the existence of traveling wave solutions in particular for the traveling wave solution a x t a x v t b x t b x v t where v 0 if we fix lim x a b 0 1 it was proved by many authors with different bounds v n d 0 such that a traveling wave solution exists for any v v when n 1 for the latest progress see 7 that is the traveling wave problem exhibits the mono stable phenomenon for traveling wave of scalar equation u t u x x f u with f 0 f 1 0 f u 0 in 0 1 and u 0 is unstable and u 1 is stable a natural and significant question is whether like the scalar case there exists a minimum speed that is whether there exists a minimum speed v m i n 0 such that traveling wave solution of speed v exists iff v v m i n this is an open question in spite of many works on traveling wave of the system in last thirty years this is duo to the reason unlike the kpp case the minimum speed cannot be obtained through linear analysis at equilibrium points a b 0 1 and a b 1 0 in this work we give an affirmative answer to this question

existence and regularity of a solution of a kac equation without cutoff using the stochastic calculus of variations we consider a generalized kac equation without cutoff with which we associate a non standard nonlinear stochastic differential equation we adapt the techniques in bichteler and jacod 2 to prove that the law of a solution of the stochastic differential equation has a density which is a solution of the kac equation the initial law is very general we only assume it has second order moments and is not the dirac mass at 0 we thus generalize the analytical results of existence of a solution of this equation if we furthermore assume existence of all moments for the initial law we obtain as a corollary using the proof in desvillettes 6 that the density is smooth we prove a slightly better regularity result under more stringent assumptions using the stochastic calculus of variations adapting the methods in 1

the gilbert equation with dry friction type damping a modified gilbert equation for micromagnetics is considered obtained by augmenting the standard viscous like dissipation with a rate independent term we prove existence of a weak solution both with and without viscous dissipation by scaling time we show that if the applied field varies very slowly then gyromagnetic effects and viscous dissipation become negligible in the infinitesimally slow loading limit the system thus becomes fully rate independent

hair triggered instability of radial steady states spread and extinction in semilinear heat equations we first study the initial value problem for a general semilinear heat equation we prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low n 10 or if the steady state is flat enough at infinity the solution of the heat equation either becomes unbounded as t approaches the lifespan or eventually stays above or below another bounded radial steady state depending on if the initial value is above or below the first steady state moreover the second steady state must be a constant if n 10 using this instability result we then prove that every nonconstant radial steady state of the generalized fisher equation is a hair trigger for two kinds of dynamical behavior extinction and spreading we also prove more criteria on initial values for these types of behavior similar results for a reaction diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained

maximally dissipative local solutions to rate independent systems and application to damage and delamination problems abstract the system of two inclusions u e t u t z t 0 and r z z e t u t z t 0 with the dissipation potential r degree 1 homogeneous and with the stored energy e t separately convex is considered the relation between conventional weak solutions and local solutions is shown and a suitably integrated maximal dissipation principle is devised to select force driven local solutions and eliminate solutions with too early jumps as it may occur in energy driven ones this is illustrated on scalar examples an approximation by a simple and efficient semi implicit time discretization of the fractional step type is shown to converge to local solutions on the scalar examples the approximate solutions are shown to satisfy the integrated maximal dissipation principle asymptotically while in general it is devised only to serve as an a posteriori tool to justify or possibly adaptively adjust thus obtained approximate solutions as in fact force driven applications of such solutions are illustrated on specific examples from continuum mechanics at small strains involving inelastic processes in a bulk or on a surface namely damage and delamination

regularization properties of the 2d homogeneous boltzmann equation without cutoff we consider the 2 dimensional spatially homogeneous boltzmann equation for hard potentials we assume that the initial condition is a probability measure that has some exponential moments and is not a dirac mass we prove some regularization properties for a class of very hard potentials the solution instantaneously belongs to hr for some r in 1 2 depending on the parameters of the equation our proof relies on the use of a well suited malliavin calculus for jump processes

crack growth in polyconvex materials we discuss a model for crack propagation in an elastic body where the crack path is described a priori in particular we develop in the framework of finite strain elasticity a rate independent model for crack evolution which is based on the griffith fracture criterion due to the nonuniqueness of minimizing deformations the energy release rate is no longer continuous with respect to time and the position of the crack tip thus the model is formulated in terms of the clarke differential of the energy generalizing the classical crack evolution models for elasticity with strictly convex energies we prove the existence of solutions for our model and also the existence of special solutions where only certain extremal points of the clarke differential are allowed

travelling waves of iodate arsenous acid reaction with a constant electric field the effects of applying a constant electric field e to an iodate arsenous acid reaction of mixed order are considered where the reacting species a and b are present the resulting equations are studied in detail by obtaining conditions for the existence and nonexistence of travelling waves by using a novel approach explicit bounds c d e and c d e are derived such that there is a unique travelling wave of every speed c c d e and there does not exist any travelling wave of speed c c d e where c d e and c d e are two functions of constant electric field e and d the ratio of diffusion coefficients of the species a and b numerical simulations of the resulting equations confirm our main results

existence and nonexistence of traveling waves and reaction diffusion front propagation in periodic media we review the existence results of traveling wave solutions to the reaction diffusion equations with periodic diffusion convection coefficients and combustion bistable nonlinearities we prove that whenever traveling waves exist the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions in the case of bistable nonlinearity and one space dimension we give an example of nonexistence of traveling waves which causes quenching localization of wavefront propagation quenching localization only occurs when the variations of the media from their constant mean values are large enough our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example finally we comment on the role of the effective wave speeds in determining the effective wavefront equation hamilton jacobi equation of the reactiondiffusion equations under the small diffusion fast reaction limit with a formal geometric optics expansion

a variational view of partial brittle damage evolution under time dependent loading an elastic material undergoes the simplest form of damage in which it passes from its original state to a weaker elastic state elaborating on prior work 14 we establish the existence of a relaxed variational evolution where at each time the two states of the material combine to form a fine mixture optimal from the standpoint of the applied load at that time yet preserving the irreversibility of the damaging process

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

threshold based quasi static brittle damage evolution we introduce models for static and quasi static damage in elastic materials based on a strain threshold and then investigate the relationship between these threshold models and the energy based models introduced in francfort and marigo eur j mech a solids 12 149 189 1993 and francfort and garroni ration mech anal 182 1 125 152 2006 a somewhat surprising result is that while classical solutions for the energy models are also threshold solutions this is shown not to be the case for nonclassical solutions that is solutions with microstructure a new and arguably more physical definition of solutions with microstructure for the energy based model is then given in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors rather than by the tensors themselves we prove existence for this energy based problem and show that these solutions are also threshold solutions a by product of this analysis is that all local minimizers in both the classical setting and for the new microstructure definition are also global minimizers

stability results for doubly nonlinear differential inclusions by variational convergence we present a stability result for a wide class of doubly nonlinear equations featuring general maximal monotone operators and possibly nonconvex and nonsmooth energy functionals the limit analysis consists in the reformulation of the differential evolution as a scalar energy conservation equation with the aid of the so called fitzpatrick theory for the representation of monotone operators in particular our result applies to the vanishing viscosity approximation of rate independent systems

exact multiplicity of positive solutions for a class of semilinear problems we establish an exact multiplicity result of positive solutions of u f u 0inbn u 0on bn wherebnis the unit ball fsatisfiesf changes sign only once and asymptotically sublinear or linear the nonlinearities which we are concerned with here includef u u u b c u for 0 2b candf u up uqfor 1 p q precise global bifurcation diagrams are obtained

approximation of rate independent optimal control problems this work introduces an abstract approximation scheme for optimal control problems in the energetic theory of rate independent systems this scheme builds upon the usual temporal semidiscretization and a gamma convergence approximation of the functionals in a full discretization this approximation consists of a sequence of discrete functionals defined on finer and finer discrete spaces the main new result of this work is a rigorous convergence analysis for this scheme interestingly it turns out that the usual time discretization is incomplete as a basis for a fully discrete approximation of the optimal control problem and a new nonlocal condition has to be introduced since this new condition is necessary and sufficient for a one to one correspondence of discrete time and continuous time solutions to rate independent systems we also gain a new theoretical insight which might be of independent interest at the heart of the derivation are so called reverse approximation results for which we provide extensions and refinements to give a concrete example and to demonstrate how the abstract theory can be applied in a concrete problem we present a full discretization with finite elements of the optimal control problem for a rate independent partial differential inclusion

deriving new evolution equations for microstructures via relaxation of variational incremental problems we study material models for rate independent inelasticity in situations where no internal length scale is given and formation of microstructure for the deformation gradients and the internal variables may occur we develop a rational procedure for deriving consistent macroscopic models which allow for the computation of nontrivial effective quantities without resolving the fine scales the method involves the relaxation of variational incremental problems which are derived from an elastic and a plastic potential we use young measures to describe the microscopic distribution of the internal variables as well as the quasiconvexification of the elastic stored energy density with respect to the deformation gradient the resulting model provides a new rate independent model in terms of the deformation and the young measure the approach is based on a derivative free energetic formulation using one functional for elastic energy storage and one for the dissipation distance the latter is derived from a dissipation potential defining the dissipation distance between internal states and hence the wasserstein distance between young measures this approach is strongly linked to an associated time incremental problem which is a minimization problem of the type used for several years now in the engineering literature the update algorithm for the incremental problem is discussed in detail and two simple examples are given

global higher integrability of minimizers of variational problems with mixed boundary conditions abstract we consider integral functionals with densities of p growth with respect to gradients on a lipschitz domain with mixed boundary conditions the aim of this paper is to prove that under uniform estimates within certain classes of p growth and coercivity assumptions on the density the minimizers are of higher integrability order meaning that they belong to the space of first order sobolev functions with an integrability of order p for a uniform 0 the results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys

non convex potentials and microstructures in finite strain plasticity a mathematical model for a finite strain elastoplastic evolution problem is proposed in which one time step of an implicit time discretization leads to generally non convex minimization problems the elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations the results for a single slip system and von mises plasticity illustrate that finite strain elastoplasticity generates reduced problems with non quasiconvex energy densities and so allows for non attainment of energy minimizers and microstructures

a pure jump markov process associated with smoluchowski s coagulation equation the aim of the present paper is to construct a stochastic process whose law is the solution of the smoluchowski s coagulation equation we introduce first a modified equation dealing with the evolution of the distribu tion q t dx of the mass in the system the advantage we take on this is that we can perform an unified study for both continuous and discrete models the integro partial differential equation satisfied by q t t 0 can be interpreted as the evolution equation of the time marginals of a markov pure jump process at this end we introduce a nonlinear poisson driven stochastic differential equation related to the smoluchowski equation in the following way if x t satisfies this stochastic equation then the law of x t satisfies the modified smoluchowski equation the nonlinear process is richer than the smoluchowski equation since it provides historical information on the particles existence uniqueness and pathwise behavior for the solution of this sde are studied finally we prove that the nonlinear process x can be obtained as the limit of a marcus lushnikov procedure 1 introduction the coagulation model governs various phenomena as for example polymerization aggregation of colloidal particles formation of stars and planets behavior of fuel mixtures in engines etc smoluchowski s coagulation equation models the dynamic of such phenomena and describes the evolution of a system of clusters which coalesce in order to form bigger clusters each cluster is identified by its size the only mechanism taken into account is the coalescence of two clusters other effects as multiple coagulation are neglected we assume also that the rate of these reactions depends on the sizes of clusters involved in the coagulation denoting by n k t the nonnegative concentration of clusters of size k at time t the discrete

damage gradient of damage and principle of virtual power a theory of continuum damage mechanics is developed within the framework of the principle of virtual power because the damage in a solid results from microscopic movements we decided to include the power of these microscopic movements in the power of the internal forces the microscopic velocities are related to the damage rate the power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions models issued from this theory are presented they are coherent from the mechanical and mathematical points of view the numerical computations show no mesh sensitivity they describe with good agreement the main experimental properties concrete is chosen as an example to illustrate the theory a model using two damage quantities is also presented it yields the correct description of the unilateral phenomenon observed in concrete finally an extension of the existing models is proposed to describe the fatigue damage behaviour

global existence results for viscoplasticity at finite strain we study a model for rate dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor and investigate the existence of global in time solutions to the related pde system we reveal its underlying structure as a generalized gradient system where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite strain elasticity as well as the multiplicative decomposition of finite strain plasticity moreover the dissipation potential depends on the left invariant plastic rate and thus depends on the plastic state variable the existence theory is developed for a class of abstract nonsmooth and nonconvex gradient systems for which we introduce suitable notions of solutions namely energy dissipation balance and energy dissipation inequality solutions hence we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory

quenching of combustion by shear flows we consider a model describing premixed combustion in the presence of fluid flow reaction diffusion equation with passive advection and ignition type nonlinearity what kinds of velocity profiles are capable of quenching suppressing any given flame provided the velocity s amplitude is adequately large even for shear flows the solution turns out to be surprisingly subtle in this paper we provide a sharp characterization of quenching for shear flows the flow can quench any initial data if and only if the velocity profile does not have an interval larger than a certain critical size where it is identically constant the efficiency of quenching depends strongly on the geometry and scaling of the flow we discuss the cases of slowly and quickly varying flows proving rigorously scaling laws that have been observed earlier in numerical experiments the results require new estimates on the behavior of the solutions to advectionenhanced diffusion equation also known as passive scalar in physical literature a classical model describing a wealth of phenomena in nature the technique involves probabilistic and pde estimates in particular applications of malliavin calculus and central limit theorem for martingales

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

a markov process associated with a boltzmann equation without cutoff and for non maxwell molecules tanaka 18 showed a way to relate the measure solution pt t of a spatially homogeneous boltzmann equation of maxwellian molecules without angular cutoff to a poisson driven stochastic differential equation pt is the flow of time marginals of the solution of this stochastic equation in the present paper we extend this probabilistic interpretation to much more general spatially homogeneous boltzmann equations then we derive from this interpretation a numerical method for the concerned boltzmann equations by using easily simulable interacting particle systems

a model for the quasi static growth of brittle fractures existence and approximation results we give a precise mathematical formulation of a variational model for the irreversible quasi static evolution of brittle fractures proposed by g a francfort and j j marigo and based on griffith s theory of crack growth in the two dimensional case we prove an existence result for the quasi static evolution and show that the total energy is an absolutely continuous function of time although we cannot exclude the possibility that the bulk energy and the surface energy may present some jump discontinuities this existence result is proved by a time discretization process where at each step a global energy minimization is performed with the constraint that the new crack contains all cracks formed at the previous time steps this procedure provides an effective way to approximate the continuous time evolution

traveling waves solutions of isothermal chemical systems with decay this article studies propagating traveling waves in a class of reaction diffusion systems which include a model of microbial growth and competition in a flow reactor proposed by smith and zhao 17 and isothermal autocatalytic systems in chemical reaction of order m with a decay order n where m and n are positive integers and m n a typical system in autocatalysis is a 2b 3b with rate k1ab2 and b c with rate k2b where m 2 and n 1 involving two chemical species a reactant a and an auto catalyst b whose diffusion coefficients da and db are unequal due to different molecular weights and or sizes here a is the concentration density of a b that of b and c an inert chemical species the two constants k1 and k2 are material constants measuring the relative strength of respective reactions it is shown that there exist traveling waves when m 1 and n 1 with suitable relation between the ratio db da traveling speed c and rate constants k1 k2 on the other hand it is proved that there exists no traveling wave when one of the chemical species is immobile db 0 or n m for all choices of other parameters

numerical simulation of ductile fracture with the rousselier constitutive law this contribution aims at developing robust and reliable tools dedicated to predictive numerical simulation of ductile fracture description of damage mechanisms relies here on the rousselier constitutive law to achieve robustness and objectivity several points need then to be addressed reduction of the set of internal variables so as to recast the model into the framework of generalised standard materials proper expression of the normal plastic flow rule at the vertex of the yield surface finite strain formulation based on a multiplicative split of the deformation gradient fully implicit integration of the constitutive equations leading to a single scalar equation that admits a unique root use of mixed finite elements to avoid volumetric locking strain gradient based nonlocal model to control strain localisation and remedy spurious mesh dependency the capabilities of the resulting numerical formulation are demonstrated through simulating the cup cone fracture of a notched tensile specimen

gradient damage models toward full scale computations abstract the description of localisation and structural failure through local damage constitutive laws leads to ill posed problems to retrieve a well posed problem the material state is described through the gradients of damage in addition to the strain and the damage under some assumptions it is shown that these gradient laws can be cast into a variational formulation its numerical treatment is reduced to the design of a mixed finite element so that algorithms such as newton s method and path following techniques are still available possible convergence difficulties related to the reduction of the damage band width and the appearance of micro instabilities are removed thanks to the design of a new class of brittle laws finally numerical applications show the convergence with mesh refinement the compatibility with coarser approaches such as griffith s theory and cohesive crack models the robustness in case of unstable multi cracking the applicability to full scale 3d situations and the physical relevance compared to experimental data

gelfand shilov and gevrey smoothing effect for the spatially inhomogeneous non cutoff kac equation abstract we consider the spatially inhomogeneous non cutoff kac s model of the boltzmann equation we prove that the cauchy problem for the fluctuation around the maxwellian distribution enjoys gelfand shilov regularizing properties with respect to the velocity variable and gevrey regularizing properties with respect to the position variable

existence of traveling waves of auto catalytic systems with decay abstract this article establishes the existence of traveling waves of a class of reaction diffusion systems which model the pre mixed isothermal autocatalytic chemical reaction of order m m 1 between two chemical species a reactant and an auto catalyst and a linear decay moreover our result shows that the set of speed is contained in a bounded interval for any fixed initial value at x this is in strong contrast to either the reaction diffusion systems of autocatalytic chemical reaction of the order m without decay or to the systems which have the same order of decay which were shown by various authors e g 8 13 17 26 that the set of traveling wave speeds contains c for some c 0 the same systems also appear in a mathematical model of microbial growth and competition in a flow reactor see 2 24

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

allee effect and bistability in a spatially heterogeneous predator prey model a spatially heterogeneous reaction diffusion system modelling predator prey interaction is studied where the interaction is governed by a holling type ii functional response existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior it is found that while the predator population is not far from a constant level the prey population could be extinguished persist or blow up depending on the initial population distributions the various parameters in the system and the heterogeneous environment in particular our results show that when the prey growth is strong the spatial heterogeneity of the environment can play a dominant role for the presence of the allee effect our mathematical analysis relies on bifurcation theory topological methods various comparison principles and elliptic estimates we combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations though the system itself does not keep an order structure as the competition system does among other things this allows us to obtain partial descriptions of the dynamical behavior of the system

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

exact multiplicity of solutions and s shaped bifurcation curve for a class of semilinear elliptic equations abstract the set of steady state solutions to a reaction diffusion equation modeling an autocatalytic chemical reaction is completely determined when the reactor has spherical geometry and the spatial dimension is n 1 or 2 for any reaction order or n 3 for subcritical reaction order bifurcation approach and analysis of linearized problems are used to establish exact multiplicity and precise global bifurcation diagram of positive steady states

existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations we prove the existence and uniqueness of fast decay solutions and clarify the structure of positive radial solutions of the quasilinear elliptic equation mu f u 0 in rn where m denotes the m laplace operator and f has a supercritical growth for small u 0 and a subcritical growth for large u our proofs use only elementary arguments based on several variational identities and a maximum principle of peletier and serrin

an energetic formulation of a gradient damage model for concrete and its numerical implementation abstract the energetic formulation of a rate independent system assumes that the evolution of the system is driven by two scalar valued functions the storage energy functional and the dissipation pseudo potential the evolution of the material system is then characterized by two energetic principles the global stability condition and the energy balance between stored and dissipated energies with the work of external loading rate independent generalized standard materials are endowed with such structure whereas the existence of such energetic structure is not so apparent in materials with non associated flow rule in this paper we consider a rate independent gradient damage model for concrete where the evolution of the damage does not follow the normality rule we will show that such model can be nevertheless derived by a dissipation potential at the expense of having a state variable dependent potential and therefore the energetic formulation can be obtained also in this case after introducing the incremental minimization problem consistent with such formulation we obtain a discrete version of the stability condition and establish lower and upper a priori energy bounds met by the energetic solution these are fundamental results for the analysis of the formulation the actual numerical solution of the incremental minimization problem is realized by considering first a variable splitting in order to treat the gradient of the damage field as an independent variable and then by applying the augmented lagrangian method to tackle with the resulting constrained optimization problem we solve the first order stationarity conditions of the augmented lagrangian functional by a path following newton s method based on the energy dissipation rate control we show that we are able to describe highly non linear responses of the material such as softening branches and snap back responses several numerical tests are performed to verify the objectivity of the formulation and of the proposed numerical method details of the numerical implementation are also given