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

a purely lagrangian method for computing linearly perturbed flows in spherical geometry in many physical applications one wishes to control the development of multi dimensional instabilities around a one dimensional 1d complex flow for predicting the growth rates of these perturbations a general numerical approach is viable which consists in solving simultaneously the one dimensional equations and their linearized form for three dimensional perturbations in clarisse et al j m clarisse s jaouen p a raviart a godunov type method in lagrangian coordinates for computing linearly perturbed planar symmetric flows of gas dynamics j comp phys 198 2004 80 105 a class of godunov type schemes for planar symmetric flows of gas dynamics has been proposed pursuing this effort we extend these results to spherically symmetric flows a new method to derive the lagrangian perturbation equations based on the canonical form of systems of conservation laws with zero entropy flux b despres lagrangian systems of conservation laws invariance properties of lagrangian systems of conservation laws approximate riemann solvers and the entropy condition numer math 89 2001 99 134 b despres c mazeran lagrangian gas dynamics in two dimensions and lagrangian systems arch rational mech anal 178 2005 327 372 is also described it leads to many advantages first of all many physical problems we are interested in enter this formalism gas dynamics two temperature plasma equations ideal magnetohydrodynamics etc whatever is the geometry secondly a class of numerical entropic schemes is available for the basic flow 11 last linearizing and devising numerical schemes for the perturbed flow is straightforward the numerical capabilities of these methods are illustrated on three test cases of increasing difficulties and we show that due to its simplicity and its low computational cost the linear perturbations code lpc is a powerful tool to understand and predict the development of hydrodynamic instabilities in the linear regime

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

localization effects and measure source terms in numerical schemes for balance laws this paper investigates the behavior of numerical schemes for non linear conservation laws with source terms we concentrate on two significant examples relaxation approximations and genuinely nonhomogeneous scalar laws the main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using riemann based numerical schemes some illustrative and relevant computational results are provided

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

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

on the uniqueness of solutions to continuity equations abstract we obtain sufficient conditions for the uniqueness of solutions to the cauchy problem for the continuity equation in classes of measures that need not be absolutely continuous

pairwise local fisher and naive bayes improving two standard discriminants abstract the fisher discriminant is probably the best known likelihood discriminant for continuous data another benchmark discriminant is the naive bayes which is based on marginals only in this paper we extend both discriminants by modeling dependence between pairs of variables in the continuous case this is done by local gaussian versions of the fisher discriminant in the discrete case the naive bayes is extended by taking geometric averages of pairwise joint probabilities we also indicate how the two approaches can be combined for mixed continuous and discrete data the new discriminants show promising results in a number of simulation experiments and real data illustrations

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

linear transport equations with μ monotone coefficients abstract we introduce the concept of a monotone function it allows us to extend the existing theory for filippov solutions to ode linear transport equations and conservation laws for a wider range of transport velocities a 1 a d and fluxes f 1 f d

adaptive discontinuous galerkin finite element methods for nonlinear hyperbolic conservation laws we consider the a posteriori error analysis and adaptive mesh design for discontinuous galerkin finite element approximations to systems of nonlinear hyperbolic conservation laws in particular we discuss the question of error estimation for general target functionals of the solution typical examples include the outflow flux local average and pointwise value as well as the lift and drag coefficients of a body immersed in an inviscid fluid by employing a duality argument we derive so called weighted or type i a posteriori error bounds these error estimates include the product of the finite element residuals with local weighting terms involving the solution of a certain dual or adjoint problem that must be numerically approximated based on the resulting approximate type i bound we design and implement an adaptive algorithm that produces meshes specifically tailored to the efficient computation of the given target functional of practical interest the performance of the proposed adaptive strategy and the quality of the approximate type i a posteriori error bound is illustrated by a series of numerical experiments in particular we demonstrate the superiority of this approach over standard mesh refinement algorithms which employ type ii error indicators

rapidly convergent representations for 2d and 3d green s functions for a linear periodic array of dipole sources hybrid spectral spatial representations are introduced to rapidly calculate periodic scalar and dyadic green s functions of the helmholtz equation for 2d and 3d configurations with a 1d linear periodicity the presented schemes work seamlessly for any observation location near the array and for any practical array periodicities including electrically small and large periodicities the representations are based on the expansion of the periodic green s functions in terms of the continuous spectral integrals over the transverse to the array spectral parameters to achieve high convergence and numerical efficiency the introduced integral representations are cast in a hybrid form in terms of i a small number of contributions due to sources located around the unit cell of interest ii a small number of symmetric combinations of the floquet modes and iii an integral evaluated along the steepest descent path sdp the sdp integral is regularized by extracting the singular behavior near the saddle point of the integrand and integrating the extracted components in closed form efficient quadrature rules are established to evaluate this integral using a small number of quadrature nodes with arbitrary small error for a wide range of structure parameters strengths of the introduced approach are demonstrated via extensive numerical examples

renormalized solutions to the vlasov equation with coefficients of bounded variation we prove that weak bounded solutions to the vlasov equation with bv coefficients have the renormalization property and we show that when the renormalization property holds for a general transport equation it also holds for only lipschitz nonlinearities

on bandwidth choice for spatial data density estimation bandwidth choice is crucial in spatial kernel estimation in exploring non gaussian complex spatial data the paper investigates the choice of adaptive and non adaptive bandwidths for density estimation given data on a spatial lattice an adaptive bandwidth depends on local data and hence adaptively conforms with local features of the spatial data we propose a spatial cross validation scv choice of a global bandwidth this is done first with a pilot density involved in the expression for the adaptive bandwidth the optimality of the procedure is established and it is shown that a non adaptive bandwidth choice comes out as a special case although the cross validation idea has been popular for choosing a non adaptive bandwidth in data driven smoothing of independent and time series data its theory and application have not been much investigated for spatial data for the adaptive case there is little theory even for independent data conditions that ensure asymptotic optimality of the scv selected bandwidth are derived actually also extending time series and independent data optimality results further for the adaptive bandwidth with an estimated pilot density oracle properties of the resultant density estimator are obtained asymptotically as if the true pilot were known numerical simulations show that finite sample performance of the scv adaptive bandwidth choice works quite well it outperforms the existing r routines such as the rule of thumb and the so called second generation sheather jones bandwidths for moderate and big data sets an empirical application to a set of spatial soil data is further implemented with non gaussian features significantly identified

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

uniqueness of continuous solutions for bv vector fields we consider a vector field whose coefficients are functions of bounded variation with a bounded divergence we prove the uniqueness of continuous solutions for the cauchy problem

pseudo maximum likelihood estimation in two classes of semiparametric diffusion models abstract two classes of semiparametric diffusion models are considered where either the drift or the diffusion term is parameterized while the other term is left unspecified we propose a pseudo maximum likelihood estimator pmle of the parametric component that maximizes the likelihood with a preliminary estimator of the unspecified term plugged in it is demonstrated how models and estimators can be used in a two step specification testing strategy of semiparametric and fully parametric models and shown that approximate simulated versions of the pmle inherit the properties of the actual but infeasible estimator a simulation study investigates the finite sample performance of the pmle

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

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

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

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

a global unique solvability of entropic weak solution to the one dimensional pressureless euler system with a flocking dissipation abstract we present a unique global solvability and flocking estimate of an entropic weak solution to the one dimensional pressureless euler system with a flocking dissipation in all to all coupling setting this model appears naturally as a quasi equilibrium model for hydrodynamic description of cucker smale flocking for the unique global solvability we adopt the variation approach from ding and huang s work 19 on the inhomogeneous pressureless gas dynamic model when initial mass and velocity are locally integrable and bounded measurable functions respectively we give explicit representations for the global entropic weak solutions our results do not require any smallness of initial data except that initial mass density is almost everywhere positive

lagrangian flows and the one dimensional peano phenomenon for odes we consider the one dimensional ordinary differential equation with a vector field which is merely continuous and nonnegative and satisfies a condition on the amount of zeros although it is classically known that this problem lacks uniqueness of classical trajectories we show that there is uniqueness for the so called regular lagrangian flow by now usual notion of flow in nonsmooth situations as well as uniqueness of distributional solutions for the associated continuity equation the proof relies on a space reparametrization argument around the zeros of the vector field

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

existence of global weak solutions to one component vlasov poisson and fokker planck poisson systems in one space dimension with measures as initial data we consider cauchy problems for the 1 d one component vlasov poisson and fokker planck poisson equations with the initial electron density being in the natural space of arbitrary non negative finite measures in particular the initial density can be a dirac measure concentrated on a curve which we refer to as electron sheet initial data these problems resemble both structurally and functional analytically cauchy problems for the 2 d euler and navier stokes equations in vorticity formulation with vortex sheet initial data here we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved we give a natural definition of the product establish its weak stability and existence of weak solutions follows our concept of weak solutions through the newly defined product is justified since solutions to the fokker planck poisson equation the analogue of navier stokes equation are shown to converge to weak solutions of the vlasov poisson equation as the fokker planck term vanishes the main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations this is needed because the problem studied here is beyond the range of applicability of the velocity averaging compactness methods of diperna lions 1994 john wiley sons inc

a godunov type method in lagrangian coordinates for computing linearly perturbed planar symmetric flows of gas dynamics linear stability studies of complex flows require that efficient numerical methods be devised for predicting growth rates of multi dimensional perturbations for one dimensional 1d basic flow i e of planar cylindrical or spherical symmetry a general numerical approach is viable which consists in solving simultaneously the one dimensional equations of gas dynamics and their linearized forms for three dimensional perturbations extensions of artificial viscosity methods have thus been used in the past more recently equations aux derivees partielles et applications articles dedies a j l lions 1998 godunov type schemes for single fluid flows of gas dynamics and magnetohydrodynamics have been proposed pursuing this effort we introduce within the lagrangian perturbation approach a class of godunov type schemes which is well suited for solving multi material problems of gas dynamics these schemes tire developed here for the planar symmetric case and comprise two second order extensions the numerical capabilities of these methods are illustrated by computations of richtmyer meshkov instabilities occurring at a single material interface a systematic comparison of numerically computed growth rates with results of the linear theory for the richtmyer meshkov instability is provided

the cauchy problem for 1 d linear nonconservative hyperbolic systems with possibly expansive discontinuity of the coefficient a viscous approach abstract in this paper we consider linear nonconservative cauchy systems with discontinuous coefficients across the noncharacteristic hypersurface x 0 for the sake of simplicity we restrict ourselves to piecewise constant hyperbolic operators of the form t a x x with a x a 1 x 0 a 1 x 0 where a m n r under assumptions incorporating a sharp spectral stability assumption we prove that a unique solution is successfully singled out by a vanishing viscosity approach due to our framework which includes systems with expansive discontinuities of the coefficient the selected small viscosity solution satisfies an unusual hyperbolic problem which is well posed even though it does not satisfy in general a uniform lopatinski condition in addition based on a detailed analysis of our stability assumption explicit examples of 2 2 systems checking our assumptions are given our result is new and contains both a stability result and a description of the boundary layers forming at any order two kinds of boundary layers form each polarized on specific linear subspaces in direct sum

an optimal penalty method for a hyperbolic system modeling the edge plasma transport in a tokamak the penalization method is used to take account of obstacles such as the limiter in a tokamak because of the magnetic confinement of the plasma in a tokamak the transport occurs essentially in the direction parallel to the magnetic field lines we study a 1d nonlinear hyperbolic system as a simplified model of the plasma transport in the area close to the wall a penalization which cuts the flux term of the momentum is studied we show numerically that this penalization creates a dirac measure at the plasma limiter interface which prevents us from defining the transport term in the usual distribution sense hence a new penalty method is proposed for this hyperbolic system for this penalty method an asymptotic expansion and numerical tests give an optimal rate of convergence without spurious boundary layer another two fields penalization has also been implemented and the numerical convergence analysis when the penalization parameter tends to 0 reveals the presence of a boundary layer

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

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

numerical high field limits in two stream kinetic models and 1d aggregation equations numerical resolution of two stream kinetic models in a strong aggregative setting is considered to illustrate our approach we consider a one dimensional kinetic model for chemotaxis in hydrodynamic scaling and the high field limit of the vlasov poisson fokker planck system a difficulty is that in this aggregative setting weak solutions of the limiting model blow up in finite time and therefore the scheme should be able to handle dirac measures it is overcome thanks to a careful discretization of the macroscopic velocity resulting of vol pert calculus accordingly a new well balanced and asymptotic preserving numerical scheme is provided numerical simulations confirm a good behavior of solutions

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

convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws part 1 linearized approximations and linearized output functionals this paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one dimensional hyperbolic equation with a convex flux function a simple modified lax friedrichs discretization is used on a uniform grid and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined it is proved that this gives pointwise convergence almost everywhere for the solution to the linearized discrete equations with smooth initial data and also convergence in the discrete approximation of linearized output functionals in part 2 m giles and s ulbrich siam j numer anal 48 2010 pp 905 921 we extend the results to dirac initial data for the linearized equation and will prove the pointwise convergence almost everywhere for the solution of the adjoint discrete equations in addition we present numerical results illustrating the asymptotic behavior which is analyzed

continuity equations and ode flows with non smooth velocity in this paper we review many aspects of the well posedness theory for the cauchy problem for the continuity and transport equations and for the ordinary differential equation ode in this framework we deal with velocity fields that are not smooth but enjoy suitable weak differentiability assumptions we first explore the connection between the partial differential equation pde and the ode in a very general non smooth setting then we address the renormalization property for the pde and prove that such a property holds for sobolev velocity fields and for bounded variation velocity fields finally we present an approach to the ode theory based on quantitative estimates

sensitivity analysis of 1 d steady forced scalar conservation laws abstract we analyze 1 d forced steady state scalar conservation laws we first show the existence and uniqueness of entropy solutions as limits as t of the corresponding solutions of the scalar evolutionary hyperbolic conservation law we then linearize the steady state equation with respect to perturbations of the forcing term this leads to a linear first order differential equation with possibly discontinuous coefficients we show the existence and uniqueness of solutions in the context of duality solutions we also show that this system corresponds to the steady state version of the linearized evolutionary hyperbolic conservation law this analysis leads us to the study of the sensitivity of the shock location with respect to variations of the forcing term an issue that is relevant in applications to optimal control and parameter identification problems

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

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

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

adjoint based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws we propose a rigorous procedure to obtain the adjoint based gradient representation of cost functionals for the optimal control of discontinuous solutions of conservation laws hereby it is not necessary to introduce adjoint variables for the shock positions our approach is based on stability properties of the adjoint equation we give a complete analysis for the case of convex scalar conservation laws the adjoint equation is a transport equation with discontinuous coefficients and special reversible solutions must be considered to obtain the correct adjoint based gradient formula reversible solutions of the adjoint transport equation and the required stability properties are analyzed in detail

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

nonlinear additive arx models abstract we consider in this article a class of nonlinear additive autoregressive models with exogenous variables for nonlinear time series analysis and propose two modeling procedures for building such models the procedures proposed use two backfitting techniques the ace and bruto algorithms to identify the nonlinear functions involved and use the methods of best subset regression and variable selection in regression analysis to determine the final model simulated and real examples are used to illustrate the analysis

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

stability of the riemann solutions for a nonstrictly hyperbolic system of conservation laws abstract we prove that the riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the riemann initial data the proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves during the interaction process of the delta shock wave with the rarefaction wave a new kind of nonclassical wave namely a delta contact discontinuity is discovered here which is a dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations m nedeljkov and m oberguggenberger interactions of delta shock waves in a strictly hyperbolic system of conservation laws j math anal appl 344 2008 1143 1157 moreover the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case

on a regularization of a scalar conservation law with discontinuous coefficients this paper is devoted to a scalar conservation law with a linear flux function involving discontinuous coefficients it is clear that the delta standing wave should be introduced into the riemann solution in some nonclassical situation in order to study the formation of delta standing wave we consider a regularization of the discontinuous coefficient with the helmholtz mollifier and then obtain a regularized system which depends on a regularization parameter 0 the regularization mechanism is a nonlinear bending of characteristic curves that prevents their finite time intersection it is proved rigorously that the solutions of regularized system converge to the delta standing wave solution in the 0 limit compared with the classical method of vanishing viscosity here it is clear to see how the delta standing wave forms naturally along the characteristics

hammerstein system identification with the nearest neighbor algorithm the nonlinear characteristic in a hammerstein system i e a system in which a nonlinear memoryless subsystem and a linear dynamic are connected in a cascade is recovered with the nonparametric nearest neighbor regression estimate the a priori information is nonparametric both the nonlinear characteristic and the impulse response are completely unknown and can be of any form local and global properties of the estimate are examined whatever the probability density of the input signal the estimate converges at every continuity point of the characteristic as well as in the global sense we derive the asymptotic bias and variance of the proposed estimate as a result the optimal rate of convergence is established that additionally is independent of the shape of the input density results of numerical simulations are also presented

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

effect of dependence on stochastic measures of accuracy of density estimations in kernel density estimation those data values that make a nondegenerate contribution to the estimator computed at a given point tend to be spaced well apart this property has the effect of suppressing many of the conventional consequences of long range dependence for example slower rates of convergence which might otherwise be revealed by a traditional lossor risk based assessment of performance from that viewpoint dependence has to be very long range indeed before a density estimator experiences any first order effects however an analysis in terms of the convergence rate for a particular realization rather than the rate averaged over all realizations reveals a very different picture we show that from that viewpoint and in the context of functions of gaussian processes effects on rates of convergence can become apparent as soon as the boundary between short and long range dependence is crossed for example the distance between ise and mise optimal bandwidths is generally of larger order for long range dependent data we shed new light on cross validation too in particular we show that the variance of the cross validation bandwidth is generally larger for long range dependent data and that the first order properties of this bandwidth do not depend on how many data are left out when constructing the cross validation criterion moreover for long range dependent data the cross validation bandwidth is usually perfectly negatively correlated in the limit with the optimal stochastic bandwidth

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

on stochastic conservation laws and malliavin calculus abstract for stochastic conservation laws driven by a semilinear noise term we propose a generalization of the kru kov entropy condition by allowing the kru kov constants to be malliavin differentiable random variables existence and uniqueness results are provided our approach sheds some new light on the stochastic entropy conditions put forth by feng and nualart 17 and bauzet vallet and wittbold 2 and in our view simplifies some of the proofs

the filippov characteristic flow for the aggregation equation with mildly singular potentials existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed such a system can be written as a continuity equation with a velocity eld computed through a self consistent interaction potential in carrillo et al duke math j 2011 16 a well posedness theory based on the geometric approach of gradient ows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being convex we propose here an alternative method using classical tools from pdes we show the existence of a characteristic ow based on filippov s theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this ow uniqueness is obtained thanks to a contraction argument in transport distances using the convexity of the potential moreover we show the equivalence of this solution with the gradient ow solution finally we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their rst blow up time in l p norms

a robust moment method for evaluation of the disappearance rate of evaporating sprays in this paper we tackle a critical issue in the numerical modeling by eulerian moment methods of polydisperse multiphase systems constituted of dispersed particles or droplets a general class of systems which include aerosols their modeling starts at a mesoscopic scale with an equation on the number density function ndf of particles droplets which satisfies a population balance equation pbe also called williams equation in the spray community in order to limit the computational cost moment methods provide a system of conservation equation with an eventual closure problem which can be solved using quadrature methods in order to retrieve the unclosed terms from the considered set of moments however a drift velocity that is the rate of change due to continuous phenomena of the internal coordinate such as the size of the particles has sometimes to be taken into account it can be either positive like molecular growth or negative such as for evaporation of droplets in aerosols or oxidation of soots when negative it leads to the disappearance of droplets particles thus creating a negative flux at zero size its closure requires an evaluation of the reconstructed ndf at zero size from the knowledge of some moments the nature of this information pointwise in internal coordinate and its influence on moment dynamics results in a difficulty from both a modeling and a numerical point of view we obtain in the present contribution a comprehensive solution to this important issue since we introduce some new tools in order to resolve the flux evaluation we also introduce a new eulerian type of description which will combine both the flexibility of eulerian models for which the size phase space is discretized into sections i e size intervals and the efficiency of direct quadrature method of moments dqmom it yields a precise and stable description of moment dynamics with a minimal number of variables which should lead to a low computational cost in multi dimensional configurations

a hamiltonian preserving scheme for the liouville equation of geometrical optics with partial transmissions and reflections we construct a class of hamiltonian preserving numerical schemes for the liouville equation of geometrical optics with partial transmissions and reflections this equation arises in the high frequency limit of the linear wave equation with a discontinuous index of refraction in our previous work hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds j comput phys 214 2006 pp 672 697 we introduced the hamiltonian preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces these schemes are extended in this paper to the general case of partial transmissions and reflections the key idea is to build into the numerical flux the behavior of waves at the interface namely partial transmissions and reflections that satisfy snell s law of refraction with the correct transmission and reflection coefficients this scheme allows a hyperbolic stability condition under which positivity and stabilities in both l 1 and l infty norms are established numerical experiments are carried out to study the numerical accuracy