coupling techniques for nonlinear hyperbolic equations iii the well balanced approximation of thick interfaces we continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces in the first two parts of this series we introduced a new framework for coupling problems which is based on the so called thin interface model and uses an augmented formulation and an additional unknown for the interface location this framework has the advantage of avoiding any explicit modeling of the interface structure in the present paper we pursue our investigation of the augmented formulation and we introduce a new coupling framework which is now based on the so called thick interface model for scalar nonlinear hyperbolic equations in one space variable we observe that the cauchy problem is well posed then our main achievement in the present paper is the design of a new well balanced finite volume scheme which is adapted to the thick interface model together with a proof of its convergence toward the unique entropy solution for a broad class of nonlinear hyperbolic equations

continuous dependence on initial time and spatial geometry in generalized heat conduction abstract in this paper we consider two different initial boundary value problems in generalized heat conduction we first establish continuous dependence on the initial time geometry in an exterior region for solutions of one well studied model system we then derive inequalities which imply continuous dependence on the spatial geometry in a bounded region for the same problem with an added dissipation term

one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

a geometric setting for internal motions of the quantum three body system quantum mechanics for internal motions of the three body system is set up on the basis of the complex vector bundle theory the three body system is called a triatomic molecule in the born oppenheimer approximation the internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator this bundle is equipped with a linear connection which is a natural consequence of a geometric interpretation of the so called eckart condition the coupling of the internal motion with the rotation is understood naturally in terms of this connection the internal hamiltonian operator is obtained which includes the internal motion rotation coupling and a centrifugal potential the complex vector bundle for the triatomic molecule proves to be a trivial bundle though the geometric setting for the internal motion is independent of whether the bundle is trivial or not

hydrodynamical limit for spatially heterogeneous simple exclusion processes we prove hydrodynamical limit for spatially heterogeneous asymmetric simple exclusion processes on z d the jump rate of particles depends on the macroscopic position x through some nonnegative smooth velocity profile x hydrodynamics are described by the entropy solution to a spatially heterogeneous conservation law of the form

continuous dependence on initial time geometry for a thermoelastic system with sign indefinite elasticities abstract continuous dependence of the solution to the equations of dynamical linear thermoelasticity on the initial time geometry is investigated using a lagrange identity method and assuming only the major symmetry on the elasticities but no sign definiteness we derive inequalities establishing that appropriately constrained solutions depend holder continuously on perturbations of the initial geometry

erratum to existence and strong pre compactness properties for entropy solutions of a first order quasilinear equation with discontinuous flux sequences of entropy solutions of a non degenerate first order quasilinear equation are shown to be strongly pre compact in the general case of a caratheodory flux vector existence of the weak and entropy solution to the cauchy problem for such an evolutionary equation is also established the proofs are based on the general localization principle for h measures corresponding to sequences of measure valued functions

intrinsic parameters and stability of differential delay equations through the efforts of many authors analytic solutions and explicit stability criteria have been developed for many types of linear differentialdelay equations the treatise of bellman and cooke 3 provides a good account of work done before 1963 active research in the field continues up to the present besides the concrete results obtained for specific classes of equations there are also some rough general principles that have long guided users of differential delay equations in their efforts to understand the effects of the various kinds of time delays on stability one such principle says that instability will be present when the ratio of the time delay to the relaxation time of the system is large however the stability switching phenomenon discovered as early as 1948 by ansoff and krumhansl 2 contradicts this principle and a great variety of responses to the presence of time lags are now known in their landmark paper on the subject cooke and grossman 4 included a series of key examples illustrating the things that can happen it seems unlikely that any simple general principles will be found to serve as reliable indicators of stability instead we can reasonably hope that the development of further principles of restricted applicability will gradually fill in more of the overall picture to this end we introduce a new framework for the classification and representation of differential delay equations or dde based on the simplest notions of probability theory distributed time delays are represented as probability measures whose expectations and variances can be used to construct intrinsic parameters for the classes of ddes studied intrinsic parametrization has two advantages

stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays abstract this paper is devoted to the analysis of a maturity structured system of hematopoietic stem cell hsc populations in the bone marrow the model is a system of differential equations with several time delays we discuss the stability of equilibria and perform the analysis of hopf bifurcation more precisely we first obtain a set of improved sufficient conditions ensuring the global asymptotical stability of the zero solution using the lyapunov method and the embedding technique of asymptotically autonomous semiflows then we prove that there exists at least one positive periodic solution for the n dimensional system as a time delay varies in some region this result is established by combining hopf bifurcation theory the global hopf bifurcation theorem due to wu j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 and a continuation theorem of coincidence degree theory some numerical simulations are also presented to illustrate the analytic results

stability and existence results for a time delayed nonlocal model of hematopoietic stem cells dynamics abstract in this paper we consider a time delayed nonlocal model describing the dynamics of hematopoietic stem cells hscs which represent the immature cells in the hematopoiesis process by the method of characteristics the nonlocal model is obtained from an age structured reaction diffusion system in bounded domain with dirichlet boundary conditions along this paper we focus on the mathematical analysis of it firstly we give some results on the existence uniqueness positivity and boundedness of solutions next we obtain a threshold value r s and prove that the trivial steady state is globally asymptotically stable when r s 1 when r s 1 we prove the existence and uniqueness of positive stationary solution under the respective additional conditions on the monotonicity and non monotonicity of the integral term finally we prove the uniform weak persistence of the system when r s 1 some numerical simulations are provided to verify the validity of our theoretical results

coupling of general lagrangian systems this work is devoted to the coupling of two fluid models such a s two euler systems in lagrangian coordinates at a fixed interface we define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled riemann problem in the present setting where the interface is characteristic a particula r choice of dependent variables which are transmitted ensures the overall conservativity

choice of measure source terms in interface coupling for a model problem in gas dynamics this paper is devoted to the mathematical and numerical analysis of a coupling procedure for one dimensional euler systems the two systems have different closure laws and are coupled through a thin fixed interface following the work of 5 we propose to couple these systems by a bounded vector valued dirac measure concentrated at the coupling interface which in the applications may have a physical meaning we show that the proposed framework allows to control the coupling conditions and we propose an approximate riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem numerical experiments in constrained optimization problems are then presented to assess the performances of the present method 1 introduction the study of large scale and complex problems exhibiting a wide range of physical space and time scales see for instance 62 35 14 usually requires separate solvers adapted to the resolution of specific scales this is the case of many industrial flows let us quote for example the numerical simulation of two phase flows applied to the burning liquid oxygen hydrogen gas in rocket engines 58 this kind of flow contains both separated and dispersed two phase flows due to atomization and evaporation phenomena this requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled another example concerns turbomachine flows which can be modeled by the euler equations of gas dynamics with different closure laws between the stages of the turbine where the conditions of temperature and pressure are strongly heterogeneous the coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine the method of interface coupling allows to represent the evolution of such flows where different models are separated by fixed interfaces first coupling conditions are specified at the interface to exchange information between the systems the definition of transmission conditions generally results from physical consideration e g the conservation or the continuity of given variables then the transmission conditions are represented at the discrete level the study of interface coupling for nonlinear hyperbolic systems has received attention for several years in 43 the authors study the scalar case from both mathematical and numerical points of view

uniqueness for scalar conservation laws with discontinuous flux via adapted entropies we prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes to do so we introduce a partial adaptation of kruzkov s entropies which naturally takes into account the space dependency of the flux the advantage of this approach is that the proof turns out to be a simple variant of kruzkov s original method especially we do not need traces interface condition bounded variation assumptions neither on the solution nor on the flux or convex fluxes however we use a special local uniform invertibility structure of the flux which applies to cases where different interface condiftions are known to yield different solutions

an engquist osher type scheme for conservation laws with discontinuous flux adapted to flux connections we consider scalar conservation laws with the spatially varying flux h x f u 1 h x g u where h x is the heaviside function and f and g are smooth nonlinear functions adimurthi mishra and veerappa gowda j hyperbolic differ equ 2 2005 pp 783 837 pointed out that such a conservation law admits many l 1 contraction semigroups one for each so called connection a b here we define entropy solutions of type a b involving kruzkov type entropy inequalities that can be adapted to any fixed connection a b it is proved that these entropy inequalities imply the l 1 contraction property for l infty solutions in contrast to the piecewise smooth setting of adimurthi mishra and veerappa gowda for a fixed connection these entropy inequalities include a single adapted entropy of the type used by audusse and perthame proc roy soc edinburgh sect a 135 2005 pp 253 265 we prove convergence of a new difference scheme that approximates entropy solutions of type a b for any connection a b if a few parameters are varied the scheme relies on a modification of the standard engquist osher flux is simple as no 2 times2 riemann solver is involved and is designed such that the steady state solution connecting a to b is preserved in contrast to most analyses of similar problems our convergence proof is not based on the singular mapping or compensated compactness methods but on standard spatial variation estimates away from the flux discontinuity some numerical examples are presented

hydrodynamic limit for particle systems with nonconstant speed parameter we establish the hydrodynamic limit for a class of particle systems on d with nonconstant speed parameter assuming that the speed parameter is continuously differentiable in the spatial variable if the particle system is on the one dimensional lattice and totally asymmetric we derive the hydrodynamic equation for continuous speed parameters we obtain nontrivial upper and lower bounds when either the speed parameter is discontinuous or there is a blockage at a fixed site

strong traces for solutions of multidimensional scalar conservation laws in this paper we consider multidimensional scalar conservation laws without bv estimates defined in a subset d we show that with a non degeneracy hypothesis on the flux we can define a strong notion of trace at the boundary of reached by l 1 convergence

boundary layers in weak solutions of hyperbolic conservation laws this paper is concerned with the initial boundary value problem for a nonlinear hyperbolic system of conservation laws we study the boundary layers that may arise in approximations of entropy discontinuous solutions we consider both the vanishing viscosity method and finite difference schemes lax friedrichs type schemes and the godunov scheme we demonstrate that different regularization methods generate different boundary layers hence the boundary condition can be formulated only if an approximation scheme is selected first assuming solely uniform l infty bounds on the approximate solutions and so dealing with l infty solutions we derive several entropy inequalities satisfied by the boundary layer in each case under consideration a young measure is introduced to describe the boundary trace when a uniform bound on the total variation is available the boundary young measure reduces to a dirac mass from the above analysis we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not each formulation is based on a set of admissible boundary values following the terminology of dubois lefloch 15 the local structure of these sets and the well posedness of the corresponding initial boundary value problem are investigated the results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics

growth and uniqueness in thermoelasticity a uniqueness theorem is proved for two theories of thermoelasticity capable of admitting finite speed thermal waves the theories having been proposed by green naghdi uniqueness is proved under

the numerical interface coupling of nonlinear hyperbolic systems of conservation laws i the scalar case we study the theoretical and numerical coupling of two general hyperbolic conservation laws the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model in order to analyze the convergence of the coupled numerical scheme we first revisit the approximation of the boundary value problems we then prove the convergence and characterize the limit solution of the coupled schemes in a few simple but significative coupling situations the general coupling problem is analyzed for riemann initial data and illustrated by numerical simulations resume nous nous interessons a une nouvelle forme de couplage de deux systemes hyperboliques de lois de conservation ce couplage assure de facon faible la continuite de la solution a l interface sans imposer la conservativite du modele couple pour etudier la convergence du schema d approximation numerique nous commencons par reprendre les resultats concernant l approximation du probleme aux limites nous demontrons ensuite la convergence du schema couple dans un certain nombre de cas interessants le cas general du couplage est etudie et illustre numeriquement pour une donnee initiale de riemann

a theory of l 1 dissipative solvers for scalar conservation laws with discontinuous flux we propose a general framework for the study of l1 contractive semigroups of solutions to conservation laws with discontinuous flux u t mathfrak f x u x 0 qquad mathfrak f x u left begin array ll f l u x 0 end array right quad quad quad rm cl where the fluxes fl fr are mainly assumed to be continuous developing the ideas of a number of preceding works baiti and jenssen in j differ equ 140 1 161 185 1997 towers in siam j numer anal 38 2 681 698 2000 towers in siam j numer anal 39 4 1197 1218 2001 towers et al in skr k nor vidensk selsk 3 1 49 2003 adimurthi et al in j math kyoto university 43 1 27 70 2003 adimurthi et al in j hyperbolic differ equ 2 4 783 837 2005 audusse and perthame in proc roy soc edinburgh a 135 2 253 265 2005 garavello et al in netw heterog media 2 159 179 2007 burger et al in siam j numer anal 47 1684 1712 2009 we claim that the whole admissibility issue is reduced to the selection of a family of elementary solutions which are piecewise constant weak solutions of the form c x c l11 left x 0 right we refer to such a family as a germ it is well known that cl admits many different l1 contractive semigroups some of which reflect different physical applications we revisit a number of the existing admissibility or entropy conditions and identify the germs that underly these conditions we devote specific attention to the vanishing viscosity germ which is a way of expressing the condition of diehl j hyperbolic differ equ 6 1 127 159 2009 for any given germ we formulate germ based admissibility conditions in the form of a trace condition on the flux discontinuity line x 0 in the spirit of vol pert math ussr sbornik 2 2 225 267 1967 and in the form of a family of global entropy inequalities following kruzhkov math ussr sbornik 10 2 217 243 1970 and carrillo arch ration mech anal 147 4 269 361 1999 we characterize those germs that lead to the l1 contraction property for the associated admissible solutions our approach offers a streamlined and unifying perspective on many of the known entropy conditions making it possible to recover earlier uniqueness results under weaker conditions than before and to provide new results for other less studied problems several strategies for proving the existence of admissible solutions are discussed and existence results are given for fluxes satisfying some additional conditions these are based on convergence results either for the vanishing viscosity method with standard viscosity or with specific viscosities adapted to the choice of a germ or for specific germ adapted finite volume schemes

initial boundary value problem for conservation laws this paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions the main assumption is that the system under study admits a convex entropy extension it is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either godunov schemes or by suitable viscous approximations furthermore in the case that the weak interior solutions are generated by godunov schemes any lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values this in particular gives a trace formula for the flux functions in terms of the numerical boundary data we also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws it is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary riemann problems and the converse is not true in general the general theory is then applied to some specific examples first several new facts are obtained for convex scalar conservation laws for example we give example which show that godunov schemes produce numerical boundary layers it is shown that any continuous functions of density have traces on the boundary instead of only entropy fluxes we also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data a generalized oleinik entropy condition is also obtained next we prove the existence of a weak solution to the initial boundary value problem for a family of quadratic system with a uniformly characteristic boundary condition

multi dimensional scalar balance laws with discontinuous flux abstract we consider scalar balance laws with a dissipative source term the flux function may be discontinuous with respect to both the space variable x and the unknown quantity u we formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem the problem is formulated in the framework of multi valued mappings the notion of entropy measure valued solutions is used to prove the so called contraction principle and comparison principle

explosive instabilities in heat transmission a global non existence result is proved for a theory of heat propagation governed by a third order in time partial differential equation an upper bound for the length of the existence interval is derived

on the existence uniqueness and stability of entropy solutions to scalar conservation laws abstract we consider one dimensional scalar conservation laws with and without viscosity where the flux function f x t u is only assumed to be absolutely continuous in x locally integrable in t and continuous in u the existence and uniqueness of entropy solutions for the associated initial value problem are obtained through the vanishing viscosity method and the doubling variables technique we also prove the stability of entropy solutions in c 0 t l loc 1 r and in c 0 t l 1 r with respect to both initial data and flux functions

a mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia this paper is devoted to the analysis of a mathematical model of blood cell production in the bone marrow hematopoiesis the model is a system of two age structured partial differential equations integrating these equations over the age we obtain a system of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration this system describes the evolution of the total cell populations by constructing a lyapunov functional it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium it is also shown that the nontrivial equilibrium the most biologically meaningful one can become unstable via a hopf bifurcation numerical simulations are carried out to illustrate the analytical results the study may be helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases

stratified reduction of classical many body systems with symmetry the centre of mass system for n bodies admits a natural so 3 action and thereby is stratified into strata according to the orbit types for the so 3 action the principal stratum consists of non singular configurations for which the isotropy subgroup is trivial so that it is made into an so 3 principal fibre bundle the strata of lower dimension consist of singular configurations at each of which the isotropy subgroup is not trivial practically singular configurations are collinear ones and simultaneous multiple collision classical lagrangian and hamiltonian systems are defined on the tangent and the cotangent bundles over each stratum respectively the euler lagrange and the hamilton equations for the n body system are derived on the variational principle according to the stratification of the centre of mass system the reduction procedure will be accordingly performed for the lagrangian and the hamiltonian systems with symmetry respectively by the rotational symmetry the euler lagrange and the hamilton equations are reduced to those defined on reduced bundles from the tangent and the cotangent bundles respectively

hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems abstract we study the following class of scalar hyperbolic conservation laws with discontinuous fluxes 0 1 t x f x 0 the main feature of such a conservation law is the discontinuity of the flux function in the space variable x kruzkov s approach for the l 1 contraction does not apply since it requires the lipschitz continuity of the flux function in x an additional jump wave may occur in the solution besides the classical waves and entropy solutions even for the riemann problem are not unique under the classical entropy conditions on the other hand it is known that in statistical mechanics some microscopic interacting particle systems with discontinuous speed parameter x in the hydrodynamic limit formally lead to scalar hyperbolic conservation laws with discontinuous fluxes of the form 0 2 t x x h 0 the natural question arises which entropy solution the hydrodynamic limit selects thereby leading to a suitable physical relevant notion of entropy solutions of this class of conservation laws this paper is a first step and provides an answer to this question for a family of discontinuous flux functions in particular we identify the entropy condition for 0 1 and proceed to show the well posedness by combining our existence result with a uniqueness result of audusse and perthame 2005 for the family of flux functions we establish a compactness framework for the hydrodynamic limit of large particle systems and the convergence of other approximate solutions to 0 1 which is based on the notion and reduction of measure valued entropy solutions and we finally establish the hydrodynamic limit for a zrp with discontinuous speed parameter governed by an l entropy solution to 0 2

convergence and structural stability in thermoelasticity we analyze the equations of thermoelasticity in a linear approximation first we prove that the solution depends continuously on changes in two thermal constants next we investigate how the solution of thermoelasticity of type iii converges to the solution of the usual thermoelasticity and to the solution of thermoelasticity without energy dissipation consequently we can also obtain an estimate of the approximation when we compare the classical thermoelasticity with thermoelasticity without energy dissipation last section investigates the structural stability in the class of thermoelasticity without energy dissipation

coupling techniques for nonlinear hyperbolic equations iv well balanced schemes for scalar multi dimensional and multi component laws this series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations the coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations in an earlier work this strategy allowed us to develop a regularization method based on a thick interface model in one space variable for coupling scalar equations in the present paper we significantly extend this framework and in addition encompass equations in several space variables this new formulation includes the coupling of several distinct scalar conservation laws and allows for a possible covering in space our main contributions are on one hand the design and analysis of a well balanced finite volume method on general triangulations and on the other hand a proof of convergence of this method toward entropy solutions extending coquel cockburn and lefloch s theory restricted to a single conservation law without coupling the core of our analysis is first the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and second a proof of convergence toward the entropy solution of the coupling problem

reduction of quantum systems on riemannian manifolds with symmetry and application to molecular mechanics this paper deals with a general method for the reduction of quantum systems with symmetry for a riemannian manifold m admitting a compact lie group g as an isometry group the quotient space q m g is not a smooth manifold in general but stratified into a collection of smooth manifolds of various dimensions if the action of the compact group g is free m is made into a principal fiber bundle with structure group g in this case reduced quantum systems are set up as quantum systems on the associated vector bundles over q m g this idea of reduction fails if the action of g on m is not free however the peter weyl theorem works well for reducing quantum systems on m when applied to the space of wave functions on m the peter weyl theorem provides the decomposition of the space of wave functions into spaces of equivariant functions on m which are interpreted as hilbert spaces for reduced quantum systems on q the concept of connection on a principal fiber bundle is generalized to be defined well on the

initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws we consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case the entropy solutions we address assume their initial data only in the sense of weak star in l as t 0 and satisfy the entropy inequality in the sense of distributions for t 0 we prove that if the flux function has weakly genuine nonlinearity then the entropy solutions are always unique and the initial layers do not appear we also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation

on nonconservative algorithms for grid interfaces in computations of fluid flows by domain decomposition methods the necessity of conservation at grid interfaces is now widely claimed in this paper we consider nonconservative algorithms for the interfaces our investigation begins with discussions about typical nonconservative interface treatments for one dimensional calculations the analysis shows that the conservation error of a numerical solution caused by a nonconservative interface matching has an upper bound when the solution itself is bounded furthermore if the numerical solution converges as the mesh size goes to zero it converges to a weak solution of the problem under certain conditions that may be detected numerically also we acquire similar results for two dimensional calculations on grids intersecting with each other in an arbitrary way in order to illustrate the theoretical results we present numerical examples and demonstrate that under those conditions conservation error reduces and accuracy for jumps as well as locations of discontinuities improves as the mesh size decreases

periodic solutions for a class of nonlinear partial differential equations in higher dimension we prove the existence of periodic solutions in a class of nonlinear partial differential equations including the nonlinear schrodinger equation the nonlinear wave equation and the nonlinear beam equation in higher dimension our result covers cases of completely resonant equations where the bifurcation equation is infinite dimensional such as the nonlinear schrodinger equation with zero mass for which solutions which at leading order are wave packets are shown to exist

stratified reduction of many body kinetic energy operators the center of mass system of many bodies admits a natural action of the rotation group so 3 according to the orbit types for the so 3 action the center of mass system is stratified into three types of strata the principal stratum consists of nonsingular configurations for which the isotropy subgroup is trivial and the other two types of strata consist of singular configurations for which the isotropy subgroup is isomorphic with either so 2 or so 3 depending on whether the isotropy subgroup is isomorphic with so 2 or so 3 the stratum in question consists of collinear configurations or of a single configuration of the multiple collision it is shown that the kinetic energy operator is expressed as the sum of rotational and vibrational energy operators on each stratum except for the stratum of multiple collision the energy operator for nonsingular configurations has singularity at singular configurations however the singularity is not essential in the sense that both of the rotational and vibrational energy integrals have a finite value this can be proved by using the boundary conditions of wave functions at singular configurations for three body systems for simplicity it is shown in addition that the energy operator for collinear configurations has also singularity at the multiple collision but the singularity is not essential either in the sense that the kinetic energy integral is not divergent at the multiple collision reduction procedure is applied to the respective energy operators for the nonsingular and the collinear configurations to obtain respective reduced operators both of which are expressed in terms of internal coordinates

energy bounds for some non standard problems in thermoelasticity a e green f r s and p m naghdi developed two theories of thermoelasticity called type ii and type iii which are likely to be more natural candidates for the identification of a thermoelastic body than the usual theory we here derive energy bounds for a class of problem in which the initial data are given as a combination of data at time t 0 and at a later time t t such problems are in vogue in the mathematical literature and may be used for example to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time

numerical integration of a mathematical model of hematopoietic stem cell dynamics a mathematical model of hematopoiesis describing the dynamics of stem cell population is investigated this model is represented by a system of two nonlinear age structured partial differential equations describing the dynamics of resting and proliferating hematopoietic stem cells it differs from previous attempts to model the hematopoietic system dynamics by taking into account cell age dependence of coefficients that prevents a usual reduction of this system to an unstructured delay differential system we prove the existence and uniqueness of a solution to our problem and we investigate the existence of stationary solutions a numerical scheme adapted to the problem is presented we show the effectiveness of this numerical technique in the simulation of the dynamics of the solution numerical simulations show that long period oscillations can be obtained in this model corresponding to a destabilization of the system these oscillations can be related to observations of some periodical hematological diseases such as chronic myelogenous leukemia

blockage hydrodynamics of one dimensional driven conservative systems we consider an arbitrary one dimensional conservative particle system with finite range interactions and finite site capacity governed on the hydrodynamic scale by a scalar conservation law with lipschitz continuous flux h a finite size perturbation restricts the local current to some maximum value we show that the perturbed hydrodynamic behavior is entirely determined by if inf h is first nondecreasing and then nonincreasing which we believe is a necessary condition

reduction of quantum systems with symmetry continuous and discrete reduction of dynamical systems is closely related with symmetry the purpose of this article is to show that fourier analysis both on compact lie groups and on finite groups serves as a reduction procedure for quantum systems with symmetry on an equal footing the reduction procedure is applied to systems of many identical particles lying in r3 which admit the action of a rotation group so 3 and of a symmetric or permutation group

spatial behavior for some non standard problems in linear thermoelasticity without energy dissipation in the present paper we consider a prismatic cylinder occupied by an anisotropic and homogeneous compressible linear thermoelastic material within the framework of the linear theory of thermoelasticity without energy dissipation the cylinder is subject to zero body force and heat supply and zero lateral specific boundary conditions and the motion is induced by a time dependent displacement and thermal displacement specified pointwise over the base further the motion is constrained such that the displacement thermal displacement velocity and temperature variation at points in the cylinder and at a prescribed time are in given proportions to but not identical with their respective initial values it is shown that certain integrals of the solution spatially evolve with respect to the axial variable conditions are derived that show the integrals exhibit alternative behavior and in particular for the semi infinite cylinder that there is either at least exponential growth or at most exponential decay provided the elasticity tensor is positive definite or strongly elliptic

stratified dynamical systems and their boundary behaviour for three bodies in space with insight into small vibrations the centre of mass system for many particles is stratified into strata by the rotation group action the principal stratum consists of nonlinear configurations the collinear configurations form a lower dimensional stratum classical mechanics for many particles with nonlinear configurations and for those with collinear configurations are set up on the tangent or cotangent bundles over respective strata and can be reduced by the use of rotational symmetry a question arises as to how a many body system behaves in a neighbourhood of a collinear configuration the system may make a vibration to bend its collinear configuration which is a motion taking place across the boundary of the principal stratum this paper deals with the behaviour of those boundaries for three bodies in space the equations of motion for small vibrations as boundary behaviour at a collinear configuration will be given as a limit of those equations of motion for nonlinear configurations by means of two key facts that the isotropy subgroup may act non trivially on the tangent space at the collinear configuration and that vibrations take place in a constant plane in space further the perturbation of small vibrations is studied by applying moser s averaging method to show that small vibrations may give rise to finite rotations after a sufficiently large number of periods

discontinuous galerkin methods for weakly coupled hyperbolic multidomain problems in this paper we develop and analyze the runge kutta discontinuous galerkin rkdg method to solve weakly coupled hyperbolic multidomain problems such problems involve transfer type boundary conditions with discontinuous fluxes between different domains calling for special techniques to prove stability of the rkdg methods we prove both stability and error estimates for our rkdg methods on simple models and then apply them to a biological cell proliferation model n echenim d monniaux m sorine and f clement math biosci 198 2005 pp 57 79 numerical results are provided to illustrate the good behavior of our rkdg methods

processus des misanthropes we consider a system of identical interacting particles moving on the lattice d the rate at which a particle at the site x jumps to the site y is p y x b x y where p is an irreducible probability on d and b x y is an increasing resp decreasing function of the number x resp y of particles at site x resp y we study the convergence of the system to equilibrium and describe the invariant measures

a numerical method for transport equations with discontinuous flux functions application to mathematical modeling of cell dynamics in this work we propose a numerical method to handle discontinuous fluxes arising in transport like equations more precisely we study hyperbolic pdes with flux transmission conditions at interfaces between subdomains where coefficients are discontinuous a dedicated finite volume scheme with a limited high order enhancement is adapted to treat the discontinuities arising at interfaces the validation of the method is done on one and two dimensional toy problems for which exact solutions are available allowing us to do a thorough convergence study we then apply the method to a biological model focusing on complex cell dynamics that initially motivated this study and illustrates the full potentialities of the scheme

the stability chart for the linearized cushing equation with a discrete delay and with gamma distributed delays in a proper setting the problem of the title amounts to the determination of all pairs d t for which the function z z 1 exp zt d and the functions fn z z 1 1 ztn n d n n d t 0 have all their zeros in re z 0 of the complex z plane correct versions of results and proofs found in papers of k l cooke and j m grossman j math anal appl 86 1982 592 627 and s p blythe r m nisbeth w s c gurney and n macdonald j math anal appl 109 1985 388 396 are given the argumentation in the proofs of the present paper differs from that of the studies mentioned in contrast to blythe et al the stability behaviours of both models are seen as similar stability charts and stability conditions for some other classes of systems are also given in addition to the gamma distribution for the lags other distributions which are similar in shape but differ greatly in their stability behaviour are also considered

numerical viscosity and convergence of finite volume methods for conservation laws with boundary conditions the authors study the convergence of finite volume schemes for general multidimensional conservation laws with boundary conditions a unique result for a measure valued solution which generalizes those of diperna for unbounded domains and szepessy for bounded domains is proved it gives us sufficient conditions to get convergence by studying carefully the entropy production for one dimensional e schemes we are able to prove convergence of finite volume e schemes under general assumption

well posedness for a class of 2 2 conservation laws withl data abstract the cauchy problem for a special class of 2 2 systems of conservation laws with data in l 1 l is considered in the strictly hyperbolic case we prove the existence of a weak solution which depends continuously on the initial data with respect to the l 1 norm this solution can be characterized in terms of a kru kov type entropy condition which is introduced here

conservation laws with discontinuous coefficients abstract in order to model phenomena arising in matter flows in electromagnetic fields engineers join systems of conservation laws with discontinuous coefficients the simplest quasi linear equation is u t a f u x 0 where a is a given discontinuous coefficient function and f is a smooth function we begin our study by solving the riemann problem and then we construct a numerical scheme of godunov type

coupling techniques for nonlinear hyperbolic equations i self similar diffusion for thin interfaces we investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and in particular we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows the main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost as a consequence non uniqueness of weak solutions is observed for the initial value problem and these solutions need to be supplemented with further admissibility conditions this paper is devoted to investigating these issues in the setting of self similar vanishing viscosity approximations to the riemann problem for general hyperbolic systems following earlier works by joseph lefloch and tzavaras we establish an existence theorem for the riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization our main contribution consists of nonlinear wave interaction estimates for solutions that apply to resonant wave patterns

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

boundary conditions on wavefunctions for three bodies at singular configurations singular configurations have long been out of consideration in the study of many particle systems in this paper we show that a group theoretical method can provide boundary conditions on wavefunctions at singular configurations of many particles the consequences of this method are qualitatively stated as follows i if the projection of angular momentum on an axis does not vanish the particles will not be aligned on the axis i e the wavefunction vanishes at collinear configurations along the axis ii if the total angular momentum does not vanish the simultaneous collision at a point will not take place i e the wavefunction vanishes at the configuration of simultaneous collision furthermore on the assumption of analyticity of the wavefunction at singular configurations the behaviour of the wavefunction around the singular configurations shows that a the larger the projection of angular momentum on an axis becomes the less the particles are likely to be aligned on the line and that b the more the total angular momentum grows the less likely the particles are to collide at a point the proof is carried out for three body systems by transforming the power series expansion of a wavefunction into a fourier series expansion in terms of the angular momentum eigenfunctions in both cases of collinear configurations and of the triple collision without reference to the hamiltonian operator these results are described quantitatively in terms of the angular momentum quantum numbers in propositions 1 and 3 in the text

relaxation oscillations in a class of delay differential equations we study a class of delay differential equations which have been used to model hematological stem cell regulation and dynamics under certain circumstances the model exhibits self sustained oscillations with periods which can be significantly longer than the basic cell cycle time we show that the long periods in the oscillations occur when the cell generation rate is small and we provide an asymptotic analysis of the model in this case this analysis bears a close resemblance to the analysis of relaxation oscillators such as the van der pol oscillator except that in our case the slow manifold is infinite dimensional despite this a fairly complete analysis of the problem is possible

rotational vibrational energy spectra of triatomic molecules near relative equilibria complete hamiltonian operators have been obtained in terms of internal coordinates on the basis of the fiber bundle theory in geometry j math phys 44 4411 2003 in this article the full hamiltonian is specialized for a rigid and for a semirigid molecule for the rigid molecule all internal coordinates are fixed at constants so that the hamiltonian operator comes to take an ordinary matrix form and accordingly the schrodinger equation becomes an algebraic eigenvalue equation the eigenvalues then provide rotational energy spectra of the rigid molecule for the semirigid molecule the full hamiltonian is expanded in the vicinity of an equilibrium position into a power series in an infinitesimal parameter to which the perturbation method is applied to obtain energy spectra in the form of a power series in the infinitesimal parameter indeed the energy spectra are calculated to the second order term in the infinitesimal parameter in both the cases where the unperturbed energy spectra are nondege