on the hughes model for pedestrian flow the one dimensional case in this paper we investigate the mathematical theory of hughes model for the ow of pedestrians cf 17 consisting of a nonlinear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task we rst consider an approximation of the original model in which the eikonal equation is replaced by an elliptic approximation for such an approximated system we prove existence and uniqueness of entropy solutions in one space dimension in the sense of kru zkov 21 in which the boundary conditions are posed following the approach of bardos et al 7 we use bv estimates on the density and stability estimates on the potential in order to prove uniqueness furthermore we analyse the evolution of characteristics for the original hughes model in one space dimension and study the behaviour of simple solutions in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves the characteristic calculus is supported by numerical simulations

discrete riemann surfaces and the ising model we define a new theory of discrete riemann surfaces and present its basic results the key idea is to consider not only a cellular decomposition of a surface but the union with its dual discrete holomorphy is defined by a straightforward discretisation of the cauchy riemann equation a lot of classical results in riemann theory have a discrete counterpart hodge star harmonicity hodge theorem weyl s lemma cauchy integral formula existence of holomorphic forms with prescribed holonomies giving a geometrical meaning to the construction on a riemann surface we define a notion of criticality on which we prove a continuous limit theorem we investigate its connection with criticality in the ising model we set up a dirac equation on a discrete universal spin structure and we prove that the existence of a dirac spinor is equivalent to criticality

a well balanced scheme for the shallow water equations with topography a non negativity preserving and well balanced scheme that exactly preserves all the smooth steady states of the shallow water system including the moving ones is proposed in addition the scheme must deal with vanishing water heights and transitions between wet and dry areas a godunov type method is derived by using a relevant average of the source terms within the scheme in order to enforce the required well balance property a second order well balanced muscl extension is also designed numerical experiments are carried out to check the properties of the scheme and assess the ability to exactly preserve all the steady states

lagrangian multiforms and multidimensional consistency we show that well chosen lagrangians for a class of two dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice on the basis of this property we formulate a lagrangian description for such systems in terms of lagrangian multiforms we discuss the connection of this formalism with the notion of multidimensional consistency and the role of the lattice from the point of view of the relevant variational principle

convergence of relaxation schemes for initial boundary value problems for conservation laws abstract we consider a scalar conservation law with stiff source term in the quarter plan this equation is relaxed in a quasi linear hyperbolic system the relaxed system is approximated using an upwind scheme converging for fixed relaxation time and vanishing discretization mesh further we use chapman enskog expansion to obtain a viscous scheme for the equilibrium law and prove its convergence to the physical solution boundary information is carefully handled in an appropriate inequality linking the entropy numerical flux and the flux function an extension to general conservative schemes is also investigated

nash social and centralized solutions to consensus problems via mean field control theory the purpose of this paper is to synthesize initial mean consensus behavior of a set of agents from the fundamental optimization principles of i stochastic dynamic games and ii optimal control in the stochastic dynamic game model each agent seeks to minimize its individual quadratic discounted cost function involving the mean of the states of all other agents in this formulation we derive the limiting infinite population mean field equation system and explicitly compute its unique solution the resulting mean field mf control strategies drive each agent to track the overall population s initial state distribution mean and by applying these decentralized strategies any finite population system reaches mean consensus asymptotically as time goes to infinity furthermore these control laws possess an en nash equilibrium property where en goes to zero as the population size n goes to infinity finally the analysis is extended to the case of random mean field couplings in the social cooperative formulation the basic objective is to minimize a social cost as the sum of the individual cost functions containing mean field coupling in this formulation we show that for any individual agent the decentralized mean field social mf social control strategy is the same as the mean field nash mf nash equilibrium strategy hence mf nash controls unashn mf social controls usocn on the other hand the solution to the centralized lqr optimal control formulation yields the standard consensus sc algorithm whenever the graph representing the corresponding topology of the network is completely connected cc hence cent lqr controls ucentn sc cc controls uscn moreover a system with centralized control laws reaches consensus on the initial state distribution mean as time and population size n go to infinity hence asymptotically in time mf nash controls unashn mf social controls usocn cent lqr controls ucent sc cc controls usc finally the analysis is extended to the long time average lta cost functions case

free fermion checkerboard and z invariant lattice models in statistical mechanics it is shown that the two dimensional free fermion model is equivalent to a checkerboard ising model which is a special case of the general z invariant ising model expressions are given for the partition function and local correlations in terms of those of the regular square lattice ising model corresponding results are given for the self dual potts model and the application of the methods to the three dimensional zamolodchikov model is discussed the paper ends with a discussion of the critical and disorder surfaces of the checkerboard potts model

compatibility conditions for time dependent partial differential equations and the rate of convergence of chebyshev and fourier spectral methods abstract compatibility conditions for partial differential equations pdes are an infinite set of relations between the initial conditions the pde and the boundary conditions which are necessary and sufficient for the solution to be c that is infinitely differentiable everywhere on the computational domain including the boundaries since the performance of chebyshev spectral and spectral element methods is dramatically reduced when the solution is not c one would expect that the compatibility conditions would be a major theme in the spectral literature instead it has been completely ignored therefore we pursue three goals here first we present a proof of the compatibility conditions in a simplified form that does not require functional analysis second we analyze the connection between the compatibility conditions and the rate of convergence of chebyshev methods lastly we describe strategies for slightly adjusting initial conditions so that the compatibility conditions are satisfied

on the controllability of the non isentropic 1 d euler equation in this paper we examine the question of the boundary controllability of the one dimensional non isentropic euler equation for compressible polytropic gas in the context of weak entropy solutions we consider the system in eulerian coordinates and the one in lagrangian coordinates we obtain for both systems a result of controllability toward constant states with the limitation gamma frac 5 3 on the adiabatic constant for the lagrangian system the solutions that we obtain remain of small total variation in space if the initial condition is itself of sufficiently small total variation

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

discrete pluriharmonic functions as solutions of linear pluri lagrangian systems pluri lagrangian systems are variational systems with the multi dimensional consistency property this notion has its roots in the theory of pluriharmonic functions in the z invariant models of statistical mechanics in the theory of variational symmetries going back to noether and in the theory of discrete integrable systems a d dimensional pluri lagrangian problem can be described as follows given a d form l on an m dimensional space m d whose coefficients depend on a function u of m independent variables called field find those fields u which deliver critical points to the action functionals s sigma int sigma l for any d dimensional manifold in the m dimensional space we investigate discrete 2 dimensional linear pluri lagrangian systems i e those with quadratic lagrangians l the action is a discrete analogue of the dirichlet energy and solutions are called discrete pluriharmonic functions we classify linear pluri lagrangian systems with lagrangians depending on diagonals they are described by generalizations of the star triangle map examples of more general quadratic lagrangians are also considered

the convergence of spectral and finite difference methods for initial boundary value problems the general theory of compatibility conditions for the differentiability of solutions to initial boundary value problems is well known this paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of high order methods explicit application of boundary conditions bcs that are independent of the initial condition ic results in the compatibility conditions not being satisfied since this is the case in most science and engineering applications it is shown that not only does the error in a spectral method as measured in the maximum norm converge algebraically but the accuracy of finite differences is also reduced for the heat equation with a parabolic ic and dirichlet bcs we prove that the fourier method converges quadratically in the neighborhood of t 0 and the boundaries and quartically for large t when the first order compatibility conditions are violated for the same problem the chebyshev method initially yields quartic convergence and exponential convergence for t 0 in contrast the wave equation subject to the same conditions results in inferior convergence rates with all spectral methods yielding quadratic convergence for all t these results naturally direct attention to finite difference methods that are also algebraically convergent in the case of the wave equation we prove that a second order finite difference method is reduced to 4 3 order convergence and numerically show that a fourth order finite difference scheme is apparently reduced to 3 2 order finally for the wave equation subject to general ics and zero bcs we give a conjecture on the error for a second order finite difference scheme showing that an o n 2log n convergence is possible

the boundary value problem for discrete analytic functions abstract this paper is on further development of discrete complex analysis introduced by r isaacs j ferrand r duffin and c mercat we consider a graph lying in the complex plane and having quadrilateral faces a function on the vertices is called discrete analytic if for each face the difference quotients along the two diagonals are equal we prove that the dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution in the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit this solves a problem of s smirnov from 2010 this was proved earlier by r courant k friedrichs h lewy and l lusternik for square lattices by d chelkak s smirnov and implicitly by p g ciarlet p a raviart for rhombic lattices in particular our result implies uniform convergence of the finite element method on delaunay triangulations this solves a problem of a bobenko from 2011 the methodology is based on energy estimates inspired by alternating current network theory

the boundary riemann solver coming from the real vanishing viscosity approximation we study the limit of the hyperbolic parabolic approximation left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr tilde rm ss v varepsilon t 0 equiv bar g cr v varepsilon 0 x equiv bar v 0 end array right the function tilde ss is defined in such a way as to guarantee that the initial boundary value problem is well posed even if tilde b is not invertible the data bar g and bar v 0 are constant when tilde b is invertible the previous problem takes the simpler form left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr v varepsilon t 0 equiv bar v b cr v varepsilon 0 x equiv bar v 0 end array right again the data bar v b and bar v 0 are constant the conservative case is included in the previous formulations convergence of the v varepsilon smallness of the total variation and other technical hypotheses are assumed and a complete characterization of the limit is provided the most interesting points are the following first the boundary characteristic case is considered that is one eigenvalue of tilde a can be 0 second as pointed out before we take into account the possibility that tilde b is not invertible to deal with this case we take as hypotheses conditions that were introduced by kawashima and shizuta relying on physically meaningful examples we also introduce a new condition of block linear degeneracy we prove that if this condition is not satisfied then pathological behaviors may occur

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

entropy satisfying relaxation method with large time steps for euler ibvps this paper could have been given the title how to positively and implicitly solve euler equations using only linear scalar advections the new relaxation method we propose is able to solve euler like systems as well as initial and boundary value problems with real state laws at very low cost using a hybrid explicit implicit time integration associated with the arbitrary lagrangian eulerian formalism furthermore it possesses many attractive properties such as i the preservation of positivity for densities ii the guarantee of min max principle for mass fractions iii the satisfaction of entropy inequality under an expressible bound on the cfl ratio the main feature that will be emphasized is the design of this optimal time step which takes into account data not only from the inner domain but also from the boundary conditions

etude variationnelle du probleme de riemann abstract we study the riemann problem for a system of conservation laws with a strongly convex entropy in arbitrary dimensions we show that the resolution of this problem can be reduced to solving an infinite number of entropy inequalities after this transformation we consider various applications maximum principles uniqueness of the solution of the trivial riemann problem etc these inequalities are also used for a study of the wave curves in connection with the ibvp as considered in a benabdallah et d serre problemes aux limites pour des systemes hyperboliques non lineaires de deux equations a une dimension d espace c r acad sci paris ser i math 305 1987 677 680 f dubois and p le floch boundary conditions for nonlinear hyperbolic system of conservation laws j differential equations71 1988 93 122 at last we propose a numerical scheme based on the discretization of the entropy inequalities for the solution of the monodimensional riemann problem wave curves are not involved thus multi dimensional extensions can be expected

on general balance laws with boundary this paper is devoted to general balance laws with a possibly non local source term with a non characteristic boundary basic well posedness results are obtained trying to provide sharp estimates in particular bounds tend to blow up as the boundary tends to be characteristic new uniqueness results for the solutions to conservation and or balance laws with boundary are also provided

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

singularly perturbed reaction diffusion problems with discontinuities in the initial and or the boundary data abstract numerical approximations to the solutions of three different problem classes of singularly perturbed parabolic reaction diffusion problems each with a discontinuity in the boundary initial data are generated for each problem class an analytical function associated with the discontinuity in the data is identified parameter uniform numerical approximations to the difference between the analytical function and the solution of the singularly perturbed problem are generated using piecewise uniform shishkin meshes numerical results are given to illustrate all the theoretical error bounds established in the paper

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

multi time lagrangian 1 forms for families of backlund transformations relativistic toda type systems we establish the pluri lagrangian structure for families of backlund transformations of relativistic toda type systems the key idea is a novel embedding of these discrete time one dimensional systems into certain two dimensional 2d pluri lagrangian lattice systems this embedding allows us to identify the corner equations which are the main building blocks of the multi time euler lagrange equations with local superposition formulae for backlund transformations these superposition formulae in turn are key ingredients necessary to understand and to prove commutativity of the multi valued backlund transformations furthermore we discover a 2d generalization of the spectrality property known for families of backlund transformations this result produces a family of local conservations laws for 2d pluri lagrangian lattice systems with densities being derivatives of the discrete 2 form with respect to the backlund spectral parameter thus a relation of the pluri lagrangian structure with more traditional integrability notions is established

lagrangian multiform structure for the lattice kp system we present a lagrangian for the bilinear discrete kp or hirota miwa equation furthermore we show that this lagrangian can be extended to a lagrangian 3 form when embedded in a higher dimensional lattice obeying a closure relation thus we establish the multiform structure as proposed in lobb and nijhoff 2009 j phys a math theor 42 454013 in a higher dimensional case

treatment of incompatible initial and boundary data for parabolic equations in higher dimension a new method is proposed to improve the numeri cal simulation of time dependent problems when the initial and boundary data are not compatible unlike earlier methods limited to space dimension one this method can be used for any space dimension when both methods are applicable in space dimen sion one the improvements in precision are comparable but the method proposed here is not restricted by dimension

entropy satisfying flux vector splittings and kinetic bgk models we establish forward and backward relations between entropy satisfying bgk relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws classically to a kinetic bgk model that is compatible with some family of entropies we can associate an entropy flux vector splitting we prove that the converse is true any entropy flux vector splitting can be interpreted by a kinetic model and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes we deduce a new proof of discrete entropy inequalities under a sharp cfl condition that generalizes the monotonicity criterion in the scalar case in particular this gives a stability condition for numerical kinetic methods with noncompact velocity support a new interpretation of general kinetic schemes is also provided via approximate riemann solvers we deduce the construction of finite velocity relaxation systems for gas dynamics and obtain a hllc scheme for which we are able to prove positiveness of density and internal energy and discrete entropy inequalities

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data we consider the optimal control of initial boundary value problems for entropy solutions of scalar hyperbolic conservation laws in particular we consider initial boundary value problems where the initial and boundary data switch between different c 1 functions at certain switching points and both the functions and the switching points are controlled we show that the control to state mapping is differentiable in a certain generalized sense which implies frechet differentiability with respect to the control functions and the switching points for the composition with a tracking type functional even in the presence of shocks we also present an adjoint based formula for the gradient of the reduced objective functional

on the nature of initial boundary value solutions for dispersive equations if the initial and boundary data for a partial differential equation pde do not obey an infinite set of compatibility conditions singularities will arise in its solutions for dissipative equations these singularities are well localized in both time and space and an effective numerical remedy is available for accurate computation of initial transients this study analyzes the nature of similar corner discrepancies for dispersive equations such as ut uxxx 0 and iut uxxx 0

accurate numerical resolution of transients in initial boundary value problems for the heat equation if the initial and boundary data for a pde do not obey an infinite set of compatibility conditions singularities will arise in the solution at the corners of the initial time space domain for dissipative equations such as the 1 d heat equation or 1 d convection diffusion equations the impacts of these singularities are short lived however they can cause a very severe loss of numerical accuracy if we are interested in transient solutions the phenomenon has been described earlier from a theoretical standpoint here we illustrate it graphically and present a simple remedy which with only little extra cost and effort restores full numerical accuracy

traffic flow modelling with junctions abstract motivated by the modelling of a roundabout we are led to study the traffic on a road with points of entry and exit in this note we would like to describe the modellisation of a junction and solve the riemann problem for such a model more precisely between each point of discontinuity we use a multi class extension of the lwr model to describe the evolution of the density of the vehicles the multi class approach being used in order to distinguish the vehicles after their origin and destination then we treat the points of entry and exit thanks to special boundary conditions that give bounds on the flows of the different types of vehicles in the case of the one t road we obtain a result of existence and uniqueness this first step allows us to obtain a similar result for the n t road we describe these results and also some properties of the obtained solutions in order to see how long this model is valid

on the attainable set for temple class systems with boundary controls consider the initial boundary value problem for a strictly hyperbolic genuinely nonlinear temple class system of conservation laws begin equation label eq 1 u t f u x 0 qquad u 0 x ov u x qquad left begin array ll u t a widetilde u a t 2pt u t b widetilde u b t end array right end equation on the domain omega t x in r 2 t geq 0 a le x leq b we study the mixed problem ref eq 1 from the point of view of control theory taking the initial data overline u fixed and regarding the boundary data widetilde u a widetilde u b as control functions that vary in prescribed sets u a u b of li boundary controls in particular we consider the family of configurations a t doteq u t cdot u mbox is a sol to ref eq 1 widetilde u a in u a widetilde u b in u b that can be attained by the system at a given time t 0 and we give a description of the attainable set a t in terms of suitable oleinik type conditions we also establish closure and compactness of the set a t in the lu topology

lagrangian multiform structure for the lattice gel fand dikii hierarchy the lattice gel fand dikii hierarchy was introduced by nijhoff et al 1992 inverse problems 8 597 621 as the family of partial difference equations generalizing to higher rank the lattice korteweg de vries systems and includes in particular the lattice boussinesq system we present a lagrangian for the generic member of the lattice gel fand dikii hierarchy and show that it can be considered as a lagrangian 2 form when embedded in a higher dimensional lattice obeying a closure relation thus the multiform structure proposed by lobb and nijhoff 2009 j phys a math theor 42 454013 is extended to a multi component system

hydrodynamics and hydrostatics for a class of asymmetric particle systems with open boundaries we consider attractive particle systems in mathbb z d with product invariant measures we prove that when particles are restricted to a subset of mathbb z d with birth and death dynamics at the boundaries the hydrodynamic limit is given by the unique entropy solution of a conservation law with boundary conditions in the sense of bardos et al comm part diff equ 4 1017 1034 1979 for the hydrostatic limit between parallel hyperplanes we prove a multidimensional version of the phase diagram conjectured in popkov and schutz europhys lett 48 257 263 1999 and show that it is robust with respect to perturbations of the boundaries

inverse problem for non viscous mean field control example from traffic this paper presents an inverse problem for mean field games where we find the mean field problem statement for which the given dynamics is the solution we use distributed traffic as an example and derive the classic lighthill whitham richards lwr model as a solution of the non viscous mean field game we also derive the same model by choosing a different problem where we use travel time which is a distributed parameter as the cost for the optimal control we then study the stationary versions of these two problems and provide numerical solutions for the same

a mean field game of evacuation in multilevel building this paper puts forward a simple mean field game that captures some of the key dynamic features of crowd and pedestrian flows in multilevel building evacuations it considers both microscopic and macroscopic route choice by strategic agents to achieve this we use mean field differential game with local congestion measure based on the location of the agent in the building including the local mean field term and its evolution along the path causes a sort of dispersion of the flow the agents will try to avoid high density areas in order to reduce their overall walking costs and queuing costs at the stairs and exits each agent state is represented by a center of a box that follows a simple first order dynamical system in an euclidean space each agent will move to one of the closest exits that is safer and with less congested path first we formulate the problem and derive optimality equations using maximum principle and dynamic programming with boundary conditions second well posedness and existence results are provided numerics and simulations are carried out to illustrate mean field equilibria of a safer evacuation process finally the methodology is shown to be flexible enough to include movement noises and stochastic structural component of the building

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

global structure stability of riemann solutions for general hyperbolic systems of conservation laws in the presence of a boundary abstract this work is a continuation of our previous work z q shao global structure instability of riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary j math anal appl 330 1 2007 511 540 in the present paper we study the global structure stability of the riemann solution u u x t containing only shocks and contact discontinuities of general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary we prove the existence and uniqueness of global piecewise c 1 solution containing only shocks and contact discontinuities to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws in the half space t x t 0 x 0 our result indicates that this kind of riemann solution u u x t mentioned above for general n n quasilinear hyperbolic systems of conservation laws in a half space possesses a global structural stability some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines are also given

initial boundary value problems for conservation laws with source terms and the degasperis procesi equation we consider conservation laws with source terms in a bounded domain with dirichlet boundary conditions we first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition equipped with this formulation we go on to establish the well posedness of entropy solutions to the initial boundary value problem the proof utilizes the kinetic formulation and the averaging lemma finally we make use of these results to demonstrate the well posedness in a class of discontinuous solutions to the initial boundary value problem for the degasperis procesi shallow water equation which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term

on the lagrangian formulation of multidimensionally consistent systems multidimensional consistency has emerged as a key integrability property for partial difference equations p es defined on the space time lattice it has led among other major insights to a classification of scalar affine linear quadrilateral p es possessing this property leading to the so called adler bobenko suris abs list recently a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete lagrangian multi forms this description is based on a fundamental and highly non trivial property of lagrangians for those integrable lattice equations namely the fact that on the solutions of the corresponding p e the lagrange forms are closed i e they obey a closure relation here we extend those results to the continuous case it is known that associated with the integrable p es there exist systems of partial differential equations pdes in fact differential equations with regard to the parameters of the lattice as independent variables which equally possess the property of multidimensional consistency in this paper we establish a universal lagrange structure for affine linear quad lattices alongside a universal lagrange multi form structure for the corresponding continuous pdes and we show that the lagrange forms possess the closure property

upwind schemes and boundary conditions with applications to euler equations in general geometries abstract a unified treatment of several upwind shock capturing algorithms is presented each algorithm has a riemann initial value problem as its basis the treatment of boundaries involves solving the associated riemann initial boundary value problem the first author s algorithm applied to multidimensional euler equations in general geometries is then presented its worth is verified by various calculations which include mach 8 supersonic flow past a circular cylinder

multi dimensional initial boundary value problems with strong nonlinearities initial boundary value problems for the general scalar singularly perturbed parabolic equation in a cylindrical domain are studied the one dimensional reaction diffusion equation is considered in the absence of convection then the general one dimensional and n dimensional reaction convection diffusion equations are analyzed

a discrete bgk approximation for strongly degenerate parabolic problems with boundary conditions abstract we consider a class of bgk systems with a finite number of velocities depending on a positive relaxation parameter that approximate strongly degenerate hyperbolic parabolic equations with initial boundary conditions we prove a priori estimates for the solutions of the systems showing that these functions converge towards the entropy solutions of strongly degenerate problems when the relaxation parameter goes to zero

a well posed conservation law with a variable unilateral constraint this paper considers the cauchy problem for a conservation law with a variable unilateral constraint its motivation being for instance the modeling of a toll gate along a highway this problem is solved by means of nonclassical shocks and its well posedness is proved then the solutions so obtained are shown to coincide with the limits of the classical solutions to suitable conservation laws with discontinuous flux function that approximate the constrained problem

on backlund transformations for many body systems using the n particle periodic toda lattice and the relativistic generalization due to ruijsenaars of the elliptic calogero moser system as examples we revise the basic properties of the backlund transformations bts from the hamiltonian point of view the analogy between the bt and baxter s quantum q operator pointed out by pasquier and gaudin is exploited to produce a conjugated variable for the parameter of the bt such that belongs to the spectrum of the lax operator as a consequence the generating function of the composition of n bts gives rise to another canonical transformation separating variables for the model for the toda lattice the dual bt parametrized by is introduced

on the lagrangian structure of 3d consistent systems of asymmetric quad equations recently the first named author gave a classification of 3d consistent 6 tuples of quad equations with the tetrahedron property several novel asymmetric 6 tuples have been found due to 3d consistency these 6 tuples can be extended to discrete integrable systems on we establish lagrangian structures and closeness of discrete lagrangian 2 forms for the class of discrete integrable systems involving equations for which some but not all of the biquadratics are non degenerate this class covers among others some of the above mentioned novel systems

discrete time toda systems in this paper we discuss several concepts of the modern theory of discrete integrable systems including time discretization based on the notion of backlund transformation symplectic realizations of multi hamiltonian structures interrelations between discrete 1d systems and lattice 2d systems multi dimensional consistency as integrability of discrete systems interrelations between integrable systems of quad equations and integrable systems of laplace type pluri lagrangian structure as integrability of discrete variational systems all these concepts are illustrated by the discrete time toda lattices and their relativistic analogs

boundary conditions for scalar conservation laws from a kinetic point of view boundary conditions for multidimensional scalar conservation laws are obtained in the context of hydrodynamic limits from a kinetic point of view the initial boundary value kinetic problem is well posed since inward and outward characteristics of the domain can be distinguished the convergence of the first momentum of the distribution function to an entropy solution of the conservation law is established boundary conditions are obtained the equivalence with the bardos leroux and nedelec conditions is studied

formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary we study nonlinear hyperbolic conservation laws posed on a differential n 1 manifold with boundary referred to as a spacetime and defined from a prescribed flux field of n forms depending on a parameter the unknown variable a class of equations proposed by lefloch and okutmustur far east j math sci 31 49 83 2008 our main result is a proof of the convergence of the finite volume method for weak solutions satisfying suitable entropy inequalities a main difference with previous work is that we allow for slices with a boundary and in addition introduce a new formulation of the finite volume method involving the notion of total flux functions under a natural global hyperbolicity condition on the flux field and the spacetime and by assuming that the spacetime admits a foliation by compact slices with boundary we establish an existence and uniqueness theory for the initial and boundary value problem and we prove a contraction property in a geometrically natural l 1 type distance

the pontryagin maximum principle from dynamic programming and viscosity solutions to first order partial differential equations on demontre le principe de maximum de pontryagin pour le probleme de lagrange de commande optimale en utilisant le fait que la fonction valeur du probleme est la solution viscosite de l equation de hamilton jacobi bellman associee

global structure instability of riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary abstract this work is a continuation of our previous work z q shao d x kong y c li shock reflection for general quasilinear hyperbolic systems of conservation laws nonlinear anal tma 66 1 2007 93 124 in this paper we study the global structure instability of the riemann solution u u x t containing shocks at least one rarefaction wave for general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary we prove the nonexistence of global piecewise c 1 solution to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws on the quarter plane our result indicates that this kind of riemann solution u u x t mentioned above for general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given

the laplacian and dirac operators on critical planar graphs on a periodic planar graph whose edge weights satisfy a certain simple geometric condition the discrete laplacian and dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph we obtain explicit expressions for the logarithms of the normalized determinants as well as the inverses of these operators we relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron in the associated dimer and spanning tree models for which the determinants of the dirac operator and the laplacian respectively play the role of the partition function this allows us to compute the entropy and correlations in terms of the local geometry in addition we define a continuous family of special discrete holomorphic functions which via convolutions gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs

well posedness of balance laws with boundary consider the initial boundary value problem for a temple system of balance laws aim of this paper is to prove the well posedness of this problem for large times and without requiring the total variation of the initial data be small

discrete complex analysis on isoradial graphs abstract we study discrete complex analysis and potential theory on a large family of planar graphs the so called isoradial ones along with discrete analogues of several classical results we prove uniform convergence of discrete harmonic measures green s functions and poisson kernels to their continuous counterparts among other applications the results can be used to establish universality of the critical ising and other lattice models

a conservation law with point source and discontinuous flux function modelling continuous sedimentation continuous sedimentation of solid particles in a liquid takes place in a clarifier thickener unit which has one feed inlet and two outlets the aim of the paper is to formulate and analyse a dynamic model for this process under idealized physical conditions the conservation of mass yields the scalar conservation law frac partial u x t partial t frac partial partial x f u x t x s t delta x where delta is the dirac measure s is a source and f is a flux function which is discontinuous at three points in the one dimensional space coordinate x under certain regularity assumptions a procedure for constructing a solution locally in time is presented the nonlinear phenomena are complicated and so is the general construction of a solution the problem of nonuniqueness due to the discontinuities of f u cdot is handled by a generalized entropy condition an advantage of this approach is that the a priori boundary conditions at the discontinuities

data assimilation and inverse problem for fluid traffic flow models and algorithms this article deals with traffic data assimilation and algorithms that are able to predict the traffic flow on a road section the traffic flow is modellized by the aw rascle hyperbolic system we have to minimize a functional whose optimization variables are initial condition we use the roe method to compute the solution to the traffic flow modelling system then we compute the gradient of the functional by an adjoint method this gradient will be used to optimize the functional

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

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved