generalized solutions for inextensible string equations we study the system of equations of motion for inextensible strings this system can be recast into a discontinuous system of conservation laws as well as into the total variation wave equation we prove existence of generalized young measure solutions with non negative tension after transforming the problem into a system of conservation laws and approximating it with a regularized system for which we obtain uniform estimates of the energy and the tension we also discuss sufficient conditions for non negativity of the tension for strong solutions

optimal besov differentiability for entropy solutions of the eikonal equation in this paper we study the eikonal equation in a bounded planar domain we prove the equivalence among optimal besov regularity the finiteness of every entropy production and the validity of a kinetic formulation

an existence result for conservation laws having bv spatial flux heterogeneities without concavity abstract we prove existence for an initial value problem featuring a conservation law whose flux has a discontinuous spatial dependence for the type of flux considered here where the spatial dependence occurs in a very general form a uniqueness result is known but the existence question was open until the recent work of piccoli and tournus 2018 22 piccoli and tournus proved existence using approximate solutions generated by the wave front tracking algorithm a concavity assumption plays a simplifying role in their analysis the main contribution of the present paper is an extension of this existence theorem in the absence of the concavity assumption we accomplish this via finite difference approximations

homogenization of non linear scalar conservation laws we study the limit as e 0 of the entropy solutions of the equation partial t u varepsilon rm div x left a left frac x varepsilon u varepsilon right right 0 we prove that the sequence ue two scale converges toward a function u t x y and u is the unique solution of a limit evolution problem the remarkable point is that the limit problem is not a scalar conservation law but rather a kinetic equation in which the macroscopic and microscopic variables are mixed we also prove a strong convergence result in l rm loc 1

convergence of numerical approximations to non linear continuity equations with rough force fields we prove quantitative regularity estimates for the solutions to non linear continuity equations and their discretized numerical approximations on cartesian grids when advected by a rough force field this allows us to not only recover the known optimal regularity for linear transport equations but also to obtain the convergence of a wide range of numerical schemes our proof is based on novel commutator estimates quantifying and extending to the non linear case the classical commutator approach of the theory of renormalized solutions

compactness for nonlinear continuity equations abstract we prove compactness and hence existence for solutions to a class of nonlinear transport equations the corresponding models combine the features of linear transport equations and scalar conservation laws we introduce a new method which gives quantitative compactness estimates compatible with both frameworks

periodic solution to nonlinear differential equation at resonance abstract in this paper a class of nonlinear sturm liouville problems at resonance is discussed an extended form of a non variational version of a max min principle is applied to show that the equation possesses unique 2 periodic solution under a less restrictive condition this result extends what we have already known

a function space model approach to the rational evaluation subgroups let f u x be a map from a connected nilpotent space u to a connected rational space x the evaluation subgroup g u x f which is a generalization of the gottlieb group of x is investigated the key device for the study is an explicit sullivan model for the connected component containing f of the function space of maps from u to x which is derived from the general theory of such a model due to brown and szczarba trans am math soc 349 4931 4951 1997 in particular we show that non gottlieb elements are detected by analyzing a sullivan model for the map f and by looking at non triviality of higher order whitehead products in the homotopy group of x the gottlieb triviality of a fibration in the sense of lupton and smith the evaluation subgroup of a fibre inclusion 2006 is also discussed from the function space model point of view moreover we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space in consequence the first gottlieb group of the total space of each s1 bundle over the n dimensional torus is determined explicitly in the non rational case

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

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

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

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

structure of entropy solutions for multi dimensional scalar conservation laws an entropy solution u of a multi dimensional scalar conservation law is not necessarily in bv even if the conservation law is genuinely nonlinear we show that u nevertheless has the structure of a bv function in the sense that the shock location is codimension one rectifiable this result highlights the regularizing effect of genuine nonlinearity in a qualitative way it is based on the locally finite rate of entropy dissipation the proof relies on the geometric classification of blow ups in the framework of the kinetic formulation

vanishing viscosity solutions for conservation laws with regulated flux abstract in this paper we introduce a concept of regulated function v t x of two variables which reduces to the classical definition when v is independent of t we then consider a scalar conservation law of the form u t f v t x u x 0 where f is smooth and v is a regulated function possibly discontinuous w r t both t and x by adding a small viscosity one obtains a well posed parabolic equation as the viscous term goes to zero the uniqueness of the vanishing viscosity limit is proved relying on comparison estimates for solutions to the corresponding hamilton jacobi equation as an application we obtain the existence and uniqueness of solutions for a class of 2 2 triangular systems of conservation laws with hyperbolic degeneracy

convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function we consider the scalar conservation law with flux function discontinuous in the space variable i e begin eqnarray label eq1 u t h x f u 1 h x g u x 0 quad mbox in r times r nonumber u 0 x u 0 x quad mbox in r label 0 1 end eqnarray where h is the heaviside function and f and g are smooth with the assumptions that either f is convex and g is concave or f is concave and g is convex the existence of a weak solution of ref eq1 is proved by showing that upwind finite difference schemes of godunov and enquist osher type converge to a weak solution uniqueness follows from a kruzkhov type entropy condition we also provide explicit solutions to the riemann problem for ref eq1 at the level of numerics we give easy to implement numerical schemes of godunov and enquist osher type the central feature of this paper is the modification of the singular mapping technique the main analytical tool for these types of equations which allows us to show that the numerical schemes converge equations of type ref eq1 with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux begin equation label eq2 begin array email protected l u t k x f u x 0 u 0 x u 0 x end array end equation with f convex and k of indefinite sign

on the chain rule formulas for divergences and applications to conservation laws abstract in this paper we prove a nonautonomous chain rule formula for the distributional divergence of the composite function v x b x u x where b t is a divergence measure vector field and u is a function of bounded variation as an application we prove a uniqueness result for scalar conservation laws with discontinuous flux

on the rational homotopy type of function spaces the main result of this paper is the construction of a minimal model for the function space f x y of continuous functions from a finite type finite dimensional space x to a finite type nilpotent space y in terms of minimal models for x and y for the component containing the constant map 7r f x y 0 q 7r y 0 h x q in positive dimensions when x is formal there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for y and the coproduct of h x q we also give a version of the main result for the space of cross sections of a fibration

structure of solutions of multidimensional conservation laws with discontinuous flux and applications to uniqueness we investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux we show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized kato inequality for any pair of solutions applications to uniqueness of solutions are then given

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

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

existence results for hughes model for pedestrian flows in this paper we prove two global existence results for hughes model for pedestrian flows under assumptions that ensure that the traces of the solutions along the turning curve are zero for any positive times

an extension of the pairing theory between divergence measure fields and bv functions abstract in this paper we introduce a nonlinear version of the notion of anzellotti s pairing between divergence measure vector fields and functions of bounded variation motivated by possible applications to evolutionary quasilinear problems as a consequence of our analysis we prove a generalized gauss green formula

periodic solutions for 2kth order ordinary differential equations with resonance in this paper a non variational version of a max min principle is extended and some unique existence results are obtained for the periodic boundary value problem of the higher order ordinary differential equations under a resonant condition

odd periodic solutions for 2nth order ordinary differential equations this paper deals with the existence of periodic solutions for the 2nth order ordinary differential equation u 2n t f t u t u t u 2n 2 t where the nonlinear term f r rn r is a continuous odd function and f t x0 x1 xn 1 is 2 periodic in t some existence results for odd 2 periodic solutions are obtained under the condition that f satisfies some linear superlinear or sublinear growth conditions on x0 x1 xn 1

on periodic solutions for even order differential equations abstract a class of sufficient conditions are obtained for the existence of a unique 2 periodic solution of even order differential equations employing an initial value problem

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

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

periodic solutions for 2kth boundary value problem with resonance abstract in this paper a non variational version of a max min principle is extended and some unique existence results for the periodic boundary value problem of the higher order nonconservative systems with resonance u 2 k t j 1 k a j u 2 j 1 t 1 k 1 g u t e t are obtained

on the periodic solution to the newtonian equation of motion on considere l equation du mouvement newtonienne d un systeme mecanique soumis a des forces internes conservatives et a des forces externes periodiques u t grad g u t p t ou p r r 11 est continue et de periode 2 et g a des derivees partielles d ordre 2 continues

existence and uniqueness results for periodic solution of nonlinear differential equations the existence and uniqueness of solutions for a class of 2nd order differential equations with periodic conditions are discussed

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

convergence of a godunov scheme to an audusse perthame adapted entropy solution for conservation laws with bv spatial flux in this article we consider the initial value problem for a scalar conservation law in one space dimension with a spatially discontinuous flux there may be infinitely many flux discontinuities and the set of discontinuities may have accumulation points thus the existence of traces cannot be assumed in audusse and perthame proc r soc edinb sect a 135 253 265 2005 proved a uniqueness result that does not require the existence of traces using adapted entropies we generalize the godunov type scheme of adimurthi et al siam j numer anal 42 1 179 208 2004 for this problem with the following assumptions on the flux function i the flux is bv in the spatial variable and ii the critical point of the flux is bv as a function of the space variable we prove that the godunov approximations converge to an adapted entropy solution thus providing an existence result and extending the convergence result of adimurthi jaffre and gowda

existence of solutions of the hyperbolic keller segel model we are concerned with the hyperbolic keller segel model with quorum sensing a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities this is a first order quasilinear equation its flux depends on space and time via the solution to an elliptic pde in which the right hand side is the solution to the hyperbolic equation this model lacks strong compactness or contraction properties our purpose is to prove the existence of an entropy solution obtained as usual in passing to the limit in a sequence of solutions to the parabolic approximation the method consists in the derivation of a kinetic formulation for the weak limit the specific structure of the limiting kinetic equation allows for a rigidity theorem which identifies some property of the solution which might be nonunique to this kinetic equation this is enough to deduce a posteriori the strong convergence of a subsequence

a note on global inversion theorems and applications to differential equations analysing the proof of plastock s theorem given by m and s rbdulescu 4 we prove that for 1 o s cx we may obtain estimates of the balls on which diffeomorphisms take place secondly we shall use these results to find estimates for the existence interval of the solution of cauchy s problem for differential equation these results will generalize a theorem of m and s radulescu 3 throughout this paper e f g will denote banach spaces

periodic solutions of 2n 1 th order ordinary differential equations abstract the goal of this paper is to prove the existence and uniqueness of periodic solutions for a class of the odd order differential equations here the leray schauder degree theory and the schwarz inequality are used

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

rational homotopy of the space of sections of a nilpotent bundle we show that an algebraic construction proposed by sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle in his paper uhomologie des espaces fonctionnels r thom studied the homotopy type of the space f of continuous maps of x into f homotopic to a given map starting from a postnikov decomposition of f he built the functional space f step by step he also indicated how one could construct a differential graded algebra describing the rational homotopy type of f later on sullivan gave an algebraic model which mirrors this construction in terms of a dg algebra representing x and the minimal model of f the aim of this paper is to show following the method of thom that the model of sullivan is indeed a model for the functional space under suitable restrictions as in 3 we consider the slightly more general problem of the determination of the rational homotopy type of the space of sections ts of a nilpotent fiber space p y x homotopic to a given section s in 1 we explain thorn s geometric construction in 2 we describe an algebraic model for an abelian galois covering of a nilpotent space in 3 we show how the model of sullivan fits with the geometry i thank the referee for many improvements of this paper 1 a postnikov factorization of the space of sections 1 1 let g be a finitely generated abehan group and let x be a path connected space whose cohomology groups hk x g are finitely generated for each k to avoid difficulties with the topologies cf 4 we can work in the category of simplicial sets proposition thom 4 the space k g m x of continuous maps of x in the eilenberg mac lane complex k g m is homotopically equivalent to the product nr ra fthe eilenberg mac lane spaces ki k hm x g i more precisely let x g hm k g m g be the fundamental class of k g m if e k g m x xx k g m

on the existence of periodic solutions of periodically perturbed conservative systems consider the system of ordinary differential equations u t gh u t p t where p r r is a continuous function with period 2 and g r n r has a continuous second partial derivative it is important to determine sufficient conditions for the existence of a unique 2 periodic solution in this paper the above system is related to an initial value problem in terms of which a new set of sufficient conditions can be given

on the cauchy problems for polymer flooding with gravitation abstract we study two systems of conservation laws for polymer flooding in secondary oil recovery one with gravitation force and one without for each model we prove global existence of weak solutions for the cauchy problems under rather general assumptions on the flux functions approximate solutions are constructed through a front tracking algorithm and compactness is achieved through the bound on suitably defined wave strengths as the main technical novelty we introduce some new nonlinear functionals that yield a uniform bound on the total variation of the flux function

an application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations abstract in this paper a class of systems of nonlinear differential equations at resonance is considered with the use of a global inversion theorem which is an extended form of a non variational version of a max min principle we prove that this class of equations possesses a unique 2 periodic solution under a rather weaker condition for existence and uniqueness than those given in papers j chen w li periodic solution for 2 k th boundary value problem with resonance j math anal appl 314 2006 661 671 f cong periodic solutions for 2 k th order ordinary differential equations with nonresonance nonlinear anal 32 1998 787 793 f cong periodic solutions for second order differential equations appl math lett 18 2005 957 961 w li periodic solutions for 2 k th order ordinary differential equations with resonance j math anal appl 259 2001 157 167 w li h li a min max theorem and its applications to nonconservative systems int j math math sci 17 2003 1101 1110 w li z shen a constructive proof of existence and uniqueness of 2 periodic solution to duffing equation nonlinear anal 42 2000 1209 1220 this result extends the results known so far

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

two point boundary value problem of 2mth order differential equations an existence and uniqueness result for the boundary value problem x 2 m email protected k 1 ma kx 2 k 1 1 m 1f t x e t x j 0 x j p 0 j 0 1 m 1 is given

rational l s category and its applications let s be a 1 connected cw complex of finite type and put cat0 s lusternik schnirelmann category of the localization sq this invariant is char acterized in terms of the minimal model of s it is shown that if is the fibre inclusion of a fibration it is also shown that if dim h s q oo then either dim it s q oo or the groups nk s q grow exponentially with k introduction in this paper topological space means a pointed path connected normal space s1 such that the inclusion of the basepoint is a cofibration and the rational singular homology is finite dimensional in each degree all vector spaces and algebras have q as ground field homology and cohomology of spaces is singular with rational coefficients we shall be dealing with category of spaces cat s in the sense of lusternik and schnirelmann 18 a recent survey is given in 15 but our definition differs by one from that of 15 so that for us spheres have category one not two our purpose is to provide a computationally useful characterization of the rational category cat0 5 of space 5 and to apply this in a variety of situations the definition of cat0 5 is sketched below and given precisely in 4 it satisfies cat0 s cauls when 5 is 1 connected cat0 5 coincides with the category of the localization sq which has been studied by toomer in 26 and 27 and lemaire and sigrist in 17 the characterization problem was posed by berstein 26 most of the applications depend on the following fundamental mapping theorem which we derive as a consequence of our characterization