structural stability of solutions to the riemann problem for a scalar conservation law the aim of this paper is to study the structural stability of solutions to the riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients it is proved that the riemann solution is possibly instable when one of the riemann initial data is at the vacuum furthermore we point out that the riemann solution is also possibly instable even when the riemann initial data stay far away from vacuum in order to deal with it we perturb the riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively it is also proved that the riemann solutions are unstable in some certain situations under the local small perturbations of the riemann initial data by letting the perturbed parameter e tend to zero in addition the interaction of the delta standing wave and the contact vacuum state is considered which appear in the riemann solutions

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

superoscillations with optimum energy concentration oscillations of a bandlimited signal at a rate faster than the bandlimit are called superoscillations and have applications e g in superresolution and superdirectivity the synthesis of superoscillating signals is a numerically difficult problem minimum energy superoscillatory signals seem attractive for applications because i the minimum energy solution is unique ii it has the smallest energy cost iii it may yield a signal of the smallest possible amplitude on the negative side superoscillating functions of minimum energy depend heavily on cancellation and give rise to expressions that have very large coefficients furthermore these coefficients have to be found by solving equations that are very ill conditioned surprisingly we show that by dropping the minimum energy requirement practicality can be gained rather than lost we give a method of constructing superoscillating signals that leads to coefficients and condition numbers that are smaller by several orders of magnitude than the minimum energy solution yet yields energies close to the minimum in contrast with the minimum energy method which builds superoscillations by linearly combining functions with an ill conditioned gram matrix our method combines orthonormal functions the gram matrix of which is obviously the identity another feature of the method is that it yields the superoscillatory signal that maximises the energy concentration in a given set which may or may not include the superoscillatory segment

construction of aharonov berry s superoscillations a simple method is described for constructing functions that superoscillate at an arbitrarily chosen wavelength scale our method is based on the technique of oversampled signal reconstruction this allows us to explicitly demonstrate that the observed fragility of superoscillating wavefunctions is indeed mathematically closely connected to what in the communication theory community is known as the instability of oversampled signal reconstruction confirming a previous conjecture this is of potential interest for example concerning the understanding of the practical difficulties in experimentally producing superoscillatory wavefunctions

nonlinear resonance in systems of conservation laws the riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed the inhomogeneity is modeled by a linearly degenerate field the solution of the riemann problem determines the nature of wave interactions and thus the riemann problem serves as a canonical form for nonlinear systems of conservation laws generic conditions on the fluxes are stated and it is proved that under these conditions the solution of the riemann problem exists is unique and has a fixed structure this demonstrates that in the above sense resonant inhomogeneous systems generically have the same canonical form the wave curves for these systems are only lipschitz continuous in a neighborhood of the states where the wave speeds coincide and so in contrast to strictly hyperbolic systems the implicit function theorem cannot be applied directly to obtain existence and uniqueness here we show that existence

convergence results for the flux identification in a scalar conservation law here we study an inverse problem for a quasilinear hyperbolic equation we start by proving the existence of solutions to the problem which is posed as the minimization of a suitable cost function then we use a lagrangian formulation in order to formally compute the gradient of the cost function introducing an adjoint equation despite the fact that the lagrangian formulation is formal and that the cost function is not necessarily differentiable a viscous perturbation and a numerical approximation of the problem allow us to justify this computation when the adjoint problem for the quasi linear equation admits a smooth solution then the perturbed adjoint states can be proved to converge to that very solution the sequences of gradients for both perturbed problems are also proved to converge to the same element of the subdifferential of the cost function we evidence these results for a large class of numerical schemes and particular cost functions which can be applied to the identification of isotherms for conservation laws modeling distillation or chromatography they are illustrated by numerical examples

a branch and bound algorithm for determining locations of long term care facilities this paper focuses on the problem of determining locations for long term care facilities with the objective of balancing the numbers of patients assigned to the facilities we present a branch and bound algorithm by developing dominance properties a lower bounding scheme and a heuristic algorithm for obtaining an upper bound for the problem for evaluation of the suggested branch and bound algorithm computational experiments are performed on a number of test problems results of the experiments show that the suggested algorithm gives optimal solutions of problems of practical sizes in a reasonable amount of computation time

optimal results on tv bounds for scalar conservation laws with discontinuous flux abstract this paper is concerned with the total variation of the solution of scalar conservation law with discontinuous flux in one space dimension one of the main unsettled questions concerning conservation law with discontinuous flux was the boundedness of the total variation of the solution near interface in 1 it has been shown by a counter example at t 1 that the total variation of the solution blows up near interface but in that example the solution become of bounded variation after time t 1 so the natural question is what happens to the bv ness of the solution for large time here we give a complete picture of the bounded variation of the solution for all time for a uniform convex flux with only l data we obtain a natural smoothing effect in bv for all time t t 0 also we give a counter example even for a bv data to show that the assumptions which have been made are optimal

numerical solution of the high frequency asymptotic expansion for the scalar wave equation new numerical methods are derived for calculation of high frequency asymptotic expansion of the scalar wave equation the nonlinear partial differential equations defining the terms in the expansio

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

the discrete one sided lipschitz condition for convex scalar conservation laws physical solutions to convex scalar conservation laws satisfy a one sided lipschitz condition oslc that enforces both the entropy condition and their total variation boundedness consistency with this condition is therefore desirable for a numerical scheme and was proved for both the godunov and the lax friedrichs scheme also in a weakened version for the roe scheme all of them being only first order accurate a new fully second order scheme is introduced here which is consistent with the oslc the modified equation is considered and shows interesting features another second order scheme is then considered and numerical results are discussed

an hopf lax formula for a class of measurable hamilton jacobi equations abstract we consider the cauchy problem u t h x du 0 x t r n 0 u x 0 u 0 x x r n where h is measurable in x continuous convex and positive homogeneous in p we adapt the definition of viscosity solution to the measurable framework and we prove that the unique viscosity solution is given by a representation formula of hopf lax type

on the unified dispersion problem efficient formulations and exact algorithms abstract facility dispersion problems involve placing a number of facilities as far apart from each other as possible four different criteria of facility dispersal have been proposed in the literature erkut neuman 1991 despite their formal differences these four classic dispersion objectives can be expressed in a unified model called the partial sum dispersion model lei church 2013 in this paper we focus on the unweighted partial sum dispersion problem and introduce an efficient formulation for this generalized dispersion problem based on a construct by ogryczak and tamir 2003 we also present a fast branch and bound based exact algorithm

stability of a resonant system of conservation laws modeling polymer flow with gravitation abstract we prove l 1 uniqueness and stability for a resonant 2 2 system of conservation laws that arise as a model for two phase polymer flow in porous media the analysis uses the equivalence of the eulerian and lagrangian formulation of this system and the results are first established for an auxiliary scalar equation our methods are based on front tracking approximations for the auxiliary equation and the kru kov entropy condition for scalar conservation laws

the sharp form of oleĭnik s entropy condition in several space variables we investigate the conditions under which the volpert kruzkov solution of a single conservation law in several space variables with flux f will satisfy the simplified entropy condition div f u s l t and when this condition guarantees uniqueness for given lx cauchy data we show that when f is c1 our condition guarantees uniqueness iff f is isotropic and that for such f the volpert kruzkov solution always satisfies our condition

a model of continuous sedimentation of flocculated suspensions in clarifier thickener units the chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentation consolidation processes of flocculated suspensions in clarifier thickener units this model appears in two variants for cylindrical and variable cross sectional area units respectively models 1 and 2 in both cases the governing equation is a scalar strongly degenerate parabolic equation in which both the convective and diffusion fluxes depend on parameters that are discontinuous functions of the depth variable the initial value problem for this equation is analyzed for model 1 we introduce a simple finite difference scheme and prove its convergence to a weak solution that satisfies an entropy condition a limited analysis of steady states as desired stationary modes of operation is performed numerical examples illustrate that the model realistically describes the dynamics of flocculated suspensions in clarifier thickeners

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

incomplete sampling series and the recovery of missing samples from oversampled band limited signals it is well known that a band limited oversampled signal is completely determined even if an arbitrary finite number of samples is lost it is shown that an alternative simple proof of this fact carries over to generalized sampling expansions more precisely it is shown that any finite number of missing samples can be recovered from the remaining ones in the case of generalized kramer sampling expansions if an appropriate oversampling constraint is satisfied the recovery can be accomplished either iteratively or noniteratively

solutions of hamilton jacobi equations and scalar conservation laws with discontinuous space time dependence abstract we establish a unique stable solution to the hamilton jacobi equation u t h k x t u x 0 x t 0 with lipschitz initial condition where k x t is allowed to be discontinuous in the x t plane along a finite number of possibly intersecting curves parameterized by t we assume that for fixed k h k p is convex in p and lim p h k p p the solution is determined by showing that if k is made smooth by convolving k in the x direction with the standard mollifier then the control theory representation of the viscosity solution to the resulting hamilton jacobi equation must converge uniformly as the mollification decreases to a lipschitz continuous solution with an explicit control theory representation this also defines the unique stable solution to the corresponding scalar conservation law u t f k x t u x 0 x t 0 with k discontinuous

central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions in this paper two existing one dimensional mathematical models one for continuous sedimentation of monodisperse suspensions and one for settling of polydisperse suspensions are combined into a model of continuous separation of polydisperse mixtures this model can be written as a first order system of conservation laws for the local concentrations of each particle species with a flux vector that depends discontinuously on the space variable this application motivates the extension of the kurganov tadmor central difference scheme to systems with discontinuous flux the new central schemes are based on discretizing an enlarged system in which the discontinuous coefficients are viewed as additional conservation laws these additional conservation laws can either be discretized and the evolution of the discontinuity parameters is calculated in each time step or solved exactly that is the discontinuity parameters are kept constant with respect to time numerical examples and an l1 error study show that the kurganov tadmor scheme with first order in time discretization produces spurious oscillations whereas its semi discrete version discretized by a second order runge kutta scheme produces good results the scheme with discontinuity parameters kept constant is slightly more accurate than when these are evolved numerical examples illustrate the application to separation of polydisperse suspensions

godunov type approximation for a general resonant balance law with large data abstract we consider the cauchy problem for the 2 2 nonstrictly hyperbolic system a t 0 u t f a u x g a u a x 0 a u t 0 a 0 u 0 for possibly large discontinuous and resonant data the generalized solution to the riemann problem is introduced interaction estimates are carried out using an original change of variables and the convergence of godunov approximations is shown uniqueness is addressed relying on a suitable extension of kru kov s techniques

existence and nonexistence of tv bounds for scalar conservation laws with discontinuous flux for the scalar conservation laws with discontinuous flux an infinite family of a b interface entropies are introduced and each one of them is shown to form an l1 contraction semigroup see 2 one of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution away from the interface boundedness of total variation of the solution has been proved in a recent paper 6 in this paper we discuss this particular issue in detail and produce a counterexample to show that the solution in general has unbounded total variation near the interface in fact this example illustrates that smallness of the bv norm of the initial data is immaterial we hereby settle the question of determining for which of the aforementioned a b pairs the solution will have bounded total variation in the case of strictly convex fluxes 2010 wiley periodicals inc

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

a difference scheme for conservation laws with a discontinuous flux the nonconvex case in a previous work by the author convergence was established for a simple difference scheme approximating a scalar conservation law where the flux was concave and had a discontinuous spatially varying coefficient j d towers siam j numer anal 38 2000 pp 681 698 the main result of this paper is an extension of that convergence theorem to the situation where the flux may have any finite number of critical points additionally spatially varying source terms are allowed the spatially varying numerical flux is also shown to satisfy maximum and minimum principles and to be total variation decreasing tvd in time

direct construction of superoscillations oscillations of a bandlimited signal at a rate faster than its maximum frequency are called superoscillations and have been found useful e g in connection with superresolution and superdirectivity we consider signals of fixed bandwidth and with a finite or infinite number of samples at the nyquist rate which are regarded as the adjustable signal parameters we show that this class of signals can be made to superoscillate by prescribing its values on an arbitrarily fine and possibly nonuniform grid the superoscillations can be made to occur at a large distance from the nonzero samples of the signal we give necessary and sufficient conditions for the problem to have a solution in terms of the nature of the two sets involved in the problem since the number of constraints can in general be different from the number of signal parameters the problem can be exactly determined underdetermined or overdetermined we describe the solutions in each of these situations the connection with oversampling and variational formulations is also discussed

uniqueness via the adjoint problems for systems of conservation laws we prove a result of uniqueness of the entropy weak solution to the cauchy problem for a class of nonlinear hyperbolic systems of conservation laws that includes in particular the p system of isentropic gas dynamics our result concerns weak solutions satisfying the as we call it wave entropy condition or wec for short introduced in this paper the main feature of this condition is that it concerns both shock waves and rarefaction waves present in a solution for the proof of uniqueness we derive an existence result respectively a uniqueness result for the backward respectively forward adjoint problem associated with the nonlinear system our method also applies to obtain results of existence or uniqueness for some linear hyperbolic systems with discontinuous coefficients 1993 john wiley sons inc

using k branch entropy solutions for multivalued geometric optics computations this paper is devoted to a numerical simulation of the classical wkb system arising in geometric optics expansions it contains the nonlinear eikonal equation and a linear conservation law whose coefficient can be discontinuous we address the problem of treating it in such a way that superimposed signals can be reproduced by means of the kinetic formulation of multibranch solutions originally due to brenier and corrias some existence and uniqueness results are given together with computational test cases of increasing difficulty displaying up to five multivaluations

how to preserve the mass fractions positivity when computing compressible multi component flows abstract we are interested in the numerical investigation of the compressible flow of a gaseous mixture considering a hyperbolic system including the euler equations for the mixture and a mass conservation equation for each species we propose a new approximation scheme for the convective term of the species equations this approximation relies on some properties of the exact solution of the riemann problem for the multi component system and applies when an upwind godunov type scheme is used for the euler equations its main interest lies in the fact that it preserves the positivity and monotonicity of the mass fractions of all species

multi phase computations in geometrical optics in this work we propose a new set of partial differential equations pdes which can be seen as a generalization of the classical eikonal and transport equations to allow for solutions with multiple phases the traditional geometrical optics pair of equations suffer from the fact that the class of physically relevant solutions is limited in particular it does not include solutions with multiple phases corresponding to crossing waves our objective has been to generalize these equations to accommodate solutions containing more than one phase the new equations are based on the same high frequency approximation of the scalar wave equation as the eikonal and the transport equations however they also incorporate a finite superposition principle the maximum allowed number of intersecting waves in the solution can be chosen arbitrarily but a higher number means that a larger system of pdes must be solved the pdes form a hyperbolic system of conservation laws with source terms although the equations are only weakly hyperbolic and thus not well posed in the strong sense several examples show the viability of solving the equations numerically the technique we use to capture multivalued solutions is based on a closure assumption for a system of equations representing the moments

the gravity p median model abstract in this paper we propose a new model for the p median problem in the standard p median problem it is assumed that each demand point is served by the closest facility in many situations for example when demand points are communities of customers and each customer makes his own selection of the facility demand is divided among the facilities each customer selects a facility which is not necessarily the closest one in the gravity p median problem it is assumed that customers divide their patronage among the facilities with the probability that a customer patronizes a facility being proportional to the attractiveness of that facility and to a decreasing utility function of the distance to the facility the model is analyzed and heuristic solution procedures are proposed computational experiments using a set of test problems provide excellent results

convergence results for an inhomogeneous system arising in various high frequency approximations this paper is devoted to both theoretical and numerical study of a system involving an eikonal equation of hamilton jacobi type and a linear conservation law as it comes out of the geometrical optics expansion of the wave equation or the semiclassical limit for the schrodinger equation we first state an existence and uniqueness result in the framework of viscosity and duality solutions then we study the behavior of some classical numerical schemes on this problem and we give sufficient conditions to ensure convergence as an illustration some practical computations are provided

stability of conservation laws with discontinuous coefficients we prove l1 contractivity of weak solutions to a conservation law with a flux function that may depend discontinuously on the space variable furthermore we show that the l1 difference between solutions to conservation laws with different flux functions is bounded by the total variation with respect to the space variable of the difference between the flux functions

superoscillations with arbitrary polynomial shape we present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval these functions are obtained as the product of the polynomial with a sufficiently flat bandlimited envelope function whose fourier transform has at least continuous derivatives and an nth derivative of bounded variation n being the order of the polynomial polynomials of arbitrarily high order can be approximated if the fourier transform of the envelope is smooth i e a bump function

asymptotic preserving and well balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes abstract the original well balanced wb framework of greenberg and leroux 1996 24 and gosse 2002 18 relying on non conservative nc products see lefoch and tzavaras 1999 41 is set up in order to efficiently treat the so called cattaneo model of chemotaxis in 1d see hillen and stevens 2000 29 it proceeds by concentrating the source terms onto dirac masses this allows to handle them by nc jump relations based on steady state equations which can be integrated explicitly a riemann solver is deduced and the corresponding wb godunov scheme completed with the standard hoff smoller theory see hillen and stevens 2000 29 for the diffusion reaction equation ruling the evolution of the chemoattractant concentration is studied in detail later following former results of gosse and toscani 2002 2003 21 22 a simple rewriting of the nc jump relations allows to generate another version of the same godunov scheme which is well adapted to the parabolic scaling involving a small parameter e the standard bv framework is used to study the uniform stability of this asymptotic preserving ap scheme with respect to e allows to pass to the limit and derive a simple centered discretization of the keller segel model finally results by filbet 2006 14 permit to pass to the complementary limit when the space step h is sent to zero numerical results are included to illustrate the feasibility and the efficiency of the method

hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds in this paper we construct two classes of hamiltonian preserving numerical schemes for a liouville equation with discontinuous local wave speed this equation arises in the phase space description of geometrical optics and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics we extend our previous work in s jin x wen hamiltonian preserving schemes for the liouville equation with discontinuous potentials commun math sci 3 2005 285 315 for the semiclassical limit of the schrodinger equation into this system the designing principle of the hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition when a plane wave hits a flat interface the hamiltonian preservation is shown to be equivalent to snell s law of refraction in the case when the ratio of wave length over the width of the interface goes to zero when both length scales go to zero positivity and stabilities in both l 1 and l norms are established for both schemes the approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular discontinuous and measure valued coefficients benchmark numerical examples are given with analytic solution constructed to study the numerical accuracy of these schemes

generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics we study systems of conservation laws arising in two models of adhesion particle dynamics the first is the system of free particles which stick under collision the second is a system of gravitationally interacting particles which also stick under collision in both cases mass and momentum are conserved at the collisions so the dynamics is described by 2 2 systems of conservations laws we show that for these systems global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the lax oleinik variational principle for scalar conservation laws however this weak solution is not unique among weak solutions satisfying the standard entropy condition we also study a modified gravitational model in which instead of momentum some other weighted velocity is conserved at collisions for this model we prove both existence and uniqueness of global weak solutions we then study the qualitative behavior of the solutions with random initial data we show that for continuous but nowhere differentiable random initial velocities all masses immediately concentrate on points even though they were continuously distributed initially and the set of shock locations is dense

convergence of finite volume schemes for triangular systems of conservation laws we consider non strictly hyperbolic systems of conservation laws in triangular form which arise in applications like three phase flows in porous media we device simple and efficient finite volume schemes of godunov type for these systems that exploit the triangular structure we prove that the finite volume schemes converge to weak solutions as the discretization parameters tend to zero some numerical examples are presented one of which is related to flows in porous media

explicit hopf lax type formulas for hamilton jacobi equations and conservation laws with discontinuous coefficients we deal with a hamilton jacobi equation with a hamiltonian that is discontinuous in the space variable this is closely related to a conservation law with discontinuous flux recently an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection in this paper we characterize these infinite classes of solutions in terms of explicit hopf lax type formulas which are obtained from the viscosity solutions of the corresponding hamilton jacobi equation with discontinuous hamiltonian this also allows us to extend the framework of infinitely many classes of solutions to the hamilton jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law we have considered the case where both the hamiltonians are convex concave furthermore we also deal with the less explored case of sign changing coefficients in which one of the hamiltonians is convex and the other concave in fact in convex concave case we cannot expect always an existence of a solution satisfying rankine hugoniot condition across the interface therefore the concept of generalised rankine hugoniot condition is introduced and prove existence and uniqueness

numerical approximations of one dimensional linear conservation equations with discontinuous coefficients conservative linear equations arise in many areas of application including continuum mechanics or high frequency geometrical optics approximations this kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions in this paper we study the convergence of a class of finite differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem some basic examples including computational results are also supplied

a simple model of aharonov berry s superoscillations a simple method of constructing superoscillatory functions is given it is based on a zero shifting principle of functions in paley wiener space

delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws abstract for simple models of hyperbolic systems of conservation laws we study a new type of nonlinear hyperbolic wave a delta shock wave which is a dirac delta function supported on a shock we prove that delta shock waves are w limits in l1 of solutions to some reasonable viscous perturbations as the viscosity vanishes further we solve the multiplication problem of a delta function with a discontinuous function to show that delta shock waves satisfy the equations in the sense of distributions under suitable generalized rankine hugoniot and entropy conditions we establish the existence and uniqueness of solutions involving delta shock waves for the riemann problems the existence of solutions to the cauchy problem is also investigated

analysis of superoscillatory wave functions surprisingly differentiable functions are able to oscillate arbitrarily faster than their highest fourier component would suggest the phenomenon is called superoscillation recently a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter intuitive physical effects following up on this work we here present more general methods which allow the calculation of superoscillatory wave functions with custom designed physical properties we give concrete examples and we prove results about the limits to superoscillatory behavior we also give a simple and intuitive new explanation for the exponential computational cost of superoscillations

viscosity solutions and convergence of monotone schemes for synthetic aperture radar shape from shading equations with discontinuous intensities the shape from shading sfs equation relating u y r the unknown angular height of a surface to i y r the known synthetic aperture radar sar intensity data from the surface is i frac u r 2 sqrt 1 u r 2 u y 2 where y and r are axial and radial cylindrical coordinates unlike the more common eikonal sfs equation which relates surface height in cartesian coordinates to optical photographic intensity data the above radar equation can be transformed into hamilton jacobi cauchy form ur g i uy 0 we explore the case where i is a discontinuous function which occurs commonly in radar data by considering sequences of continuous intensity functions that converge to i we obtain corresponding sequences of viscosity solutions we prove that these sequences must converge we also establish conditions that guarantee that these sequences converge to a common limit which we define as the solution to the radar equation finally we establish and demonstrate that when this common limit exists monot

the l 1 error estimates for a hamiltonian preserving scheme for the liouville equation with piecewise constant potentials we study the l 1 error of a hamiltonian preserving scheme developed in s jin and x wen commun math sci 3 2005 pp 285 315 for the liouville equation with a piecewise constant potential in one space dimension this problem has important applications in computations of the semiclassical limit of the linear schrodinger equation through barriers and of the high frequency waves through interfaces we use the l 1 error estimates established in x wen and s jin j comput math 26 2008 pp 1 22 x wen convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients ii some related binomial coefficient inequalities preprint 2008 for the immersed interface upwind scheme to the linear advection equations with piecewise constant coefficients we prove that the scheme with the dirichlet incoming boundary conditions is l 1 convergent for a class of bounded initial data and derive the half order l 1 error bounds with explicit coefficients the initial conditions can be satisfied by applying the decomposition technique proposed in s jin h l liu s osher and r tsai j comput phys 205 2005 pp 222 241 for solving the liouville equation with measure valued initial data which arises in the semiclassical limit of the linear schrodinger equation

a sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms we present a sensitivity and adjoint calculus for the control of entropy solutions of scalar conservation laws with controlled initial data and source term the sensitivity analysis is based on shift variations which are the sum of a standard variation and suitable corrections by weighted indicator functions approximating the movement of the shock locations based on a first order approximation by shift variations in l1 we introduce the concept of shift differentiability which is applicable to operators having functions with moving discontinuities as images and implies differentiability for a large class of tracking type functionals in the main part of the paper we show that entropy solutions are generically shift differentiable at almost all times t 0 with respect to the control hereby we admit shift variations for the initial data which allows us to use the shift differentiability result repeatedly over time slabs this is useful for the design of optimization methods with time domain decomposition our analysis especially of the shock sensitivity combines structural results by using generalized characteristics and an adjoint argument our adjoint based shock sensitivity analysis does not require us to restrict the richness of the shock structure a priori and admits shock generation points the analysis is based on stability results for the adjoint transport equation with discontinuous coefficients satisfying a one sided lipschitz condition as a further main result we derive and justify an adjoint representation for the derivative of a large class of tracking type functionals

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

superoscillations faster than the nyquist rate it is commonly assumed that a signal bandlimited to mu 2 hz cannot oscillate at frequencies higher than mu hz in fact however for any fixed bandwidth there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals these localized fast transients called superoscillations can only occur in signals that possess amplitudes of widely different scales this paper investigates the required dynamical range and energy squared l2 norm as a function of the superoscillation s frequency number and maximum derivative it briefly discusses some of the implications of superoscillating signals in reference to information theory and time frequency analysis for example it also shows among other things that the required energy grows exponentially with the number of superoscillations and polynomially with the reciprocal of the bandwidth or the reciprocal of the superoscillations period

high frequency wave propagation by the segment projection method geometrical optics is a standard technique used for the approximation of high frequency wave propagation computational methods based on partial differential equations instead of the traditional ray tracing have recently been applied to geometrical optics these new methods have a number of advantages but typically exhibit difficulties with linear superposition of waves in this paper we introduce a new partial differential technique based on the segment projection method in phase space the superposition problem is perfectly resolved and so is the problem of computing amplitudes in the neighborhood of caustics the computational complexity is of the same order as that of ray tracing the new algorithm is described and a number of computational examples are given including a simulation of waveguides

black holes bandwidths and beethoven it is usually believed that a function t whose fourier spectrum is bounded can vary at most as fast as its highest frequency component max this is in fact not the case as aharonov berry and others drastically demonstrated with explicit counterexamples so called superoscillations it has been claimed that even the recording of an entire beethoven symphony can occur as part of a signal with a 1 hz bandwidth bandlimited functions also occur as ultraviolet regularized fields their superoscillations have been suggested for example to resolve the trans planckian frequencies problem of black hole radiation here we give an exact proof for generic superoscillations namely we show that for every fixed bandwidth there exist functions that pass through any finite number of arbitrarily prespecified points further we show that in spite of the presence of superoscillations the behavior of bandlimited functions can be characterized reliably namely through an uncertainty relation the standard devi

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

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

the attractive traveling salesman problem in the attractive traveling salesman problem the vertex set is partitioned into facility vertices and customer vertices a maximum profit tour must be constructed on a subset of the facility vertices profit is computed through an attraction function every visited facility vertex attracts a portion of the profit from the customer vertices based on the distance between the facility and customer vertices and the attractiveness of the facility vertex a gravity model is used for computing the profit attraction the problem is formulated as an integer non linear program a linearization is proposed and strengthened through the introduction of valid inequalities and a branch and cut algorithm is developed a tabu search algorithm is also implemented computational results are reported

hydrodynamic singular regimes in 1 1 kinetic models and spectral numerical methods classical results from spectral theory of stationary linear kinetic equations are applied to efficiently approximate two physically relevant weakly nonlinear kinetic models a model of chemotaxis involving a biased velocity redistribution integral term and a vlasov fokker planck vfp system both are coupled to an attractive elliptic equation producing corresponding mean field potentials spectral de compositions of stationary kinetic distributions are recalled based on a variation of case s elementary solutions for the first model and on a sturm liouville eigenvalue problem for the second one well balanced godunov schemes with strong stability properties are deduced moreover in the stiff hydrodynamical scaling an hybridized algorithm is set up for which asymptotic preserving properties can be established under mild restrictions on the computational grid several numerical validations are displayed including the consistency of the vfp model with burgers hopf dynamics on the velocity field after blowup of the macroscopic density into dirac masses