strong traces for solutions to scalar conservation laws with general flux in this paper we consider bounded weak solutions u of scalar conservation laws not necessarily of class bv defined in a subset omega subset mathbb r times mathbb r we define a strong notion of trace at the boundary of omega reached by l1 convergence for a large class of functionals of u g u the functionals g depend on the flux function of the conservation law and on the boundary of omega the result holds for a general flux function and a general subset

nonautonomous impulsive systems with unbounded nonlinear terms in fenner and pinto 1999 23 proved that if the linear impulsive system satisfies the is condition see definition 2 1 with bounded nonlinear term f t x then the perturbed nonlinear impulsive system has a unique bounded solution see theorem a the method used to prove theorem a cannot be applied to the unbounded case in this paper we prove that if f t x e t m then the perturbed nonlinear impulsive system has a unique solution

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

godunov type methods for conservation laws with a flux function discontinuous in space scalar conservation laws with a flux function discontinuous in space are approximated using a godunov type method for which a convergence theorem is proved the case where the flux functions at the interface intersect is emphasized a very simple formula is given for the interface flux a numerical comparison between the godunov numerical flux and the upstream mobility flux is presented for two phase flow in porous media a consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration furthermore for regular solutions uniqueness has been shown

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

convergence of a difference scheme for conservation laws with a discontinuous flux convergence is established for a scalar finite difference scheme based on the godunov or engquist osher eo flux for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient other works in this direction have established convergence for methods employing the solution of 2 2 riemann problems the algorithm discussed here uses only scalar riemann solvers satisfaction of a set of kruzkov type entropy inequalities is established for the limit solution from which geometric entropy conditions follow assuming a piecewise constant coefficient it is shown that these conditions imply l1 contractiveness for piecewise c1 solutions thus extending a well known theorem

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

algebraic synchronization criterion and computing reset words we refine a uniform algebraic approach for deriving upper bounds on reset thresholds of synchronizing automata we express the condition that an automaton is synchronizing in terms of linear algebra and obtain new upper bounds for automata with a short word of small rank the results are applied to make several improvements in the area in particular we improve the upper bound for reset thresholds of finite prefix codes huffman codes we show that an n state synchronizing decoder has a reset word of length at most o nlog3n in addition to that we prove that the expected reset threshold of a uniformly random synchronizing binary n state decoder is at most o nlog n we prove the ern conjecture for n state automata with a letter of rank 6 n 6 3 in another corollary we show that the probability that the ern conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states and that the expected value of the reset threshold is at most n 3 2 o 1 moreover all of our bounds are constructible we present suitable polynomial algorithms for the task of finding a reset word of length within our bounds

finite volume schemes for locally constrained conservation laws this paper is devoted to the numerical analysis of the road traffic model proposed by colombo and goatin j differ equ 234 2 654 675 2007 the model involves a standard conservation law supplemented by a local unilateral constraint on the flux at the point x 0 modelling a road light a toll gate etc we first show that the problem can be interpreted in terms of the theory of conservation laws with discontinuous flux function as developed by adimurthi et al j hyperbolic differ equ 2 4 783 837 2005 and burger et al siam j numer anal 47 3 1684 1712 2009 we reformulate accordingly the notion of entropy solution introduced by colombo and goatin j differ equ 234 2 654 675 2007 and extend the well posedness results to the l framework then starting from a general monotone finite volume scheme for the non constrained conservation law we produce a simple scheme for the constrained problem and show its convergence the proof uses a new notion of entropy process solution numerical examples modelling a green wave are presented

scalar conservation laws with discontinuous flux function i the viscous profile condition the equation whereh is heaviside s step function appears for example in continuous sedimentation of solid particles in a liquid in two phase flow in traffic flow analysis and in ion etching the discontinuity of the flux function atx 0 causes a discontinuity of a solution which is not uniquely determined by the initial data the equation can be written as a triangular 2 2 non strictly hyperbolic system this augmentation is non unique and a natural definition is given by means of viscous profiles by a viscous profile we mean a stationary solution ofu t f x u xx wheref is a smooth approximation of the discontinuous flux i e h is smoothed in terms of the 2 2 system the discontinuity atx 0 is either a regular lax an under or overcompressive a marginal under or overcompressive or a degenerate shock wave in some cases depending onf andg there is a unique viscous profile e g undercompressive and regular lax waves and in some cases there are infinitely many e g overcompressive waves the main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution atx 0 and the viscous profile condition

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

scalar conservation laws with general boundary condition and continuous flux function abstract we introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition u t div u f on q 0 t u 0 u 0 on and u a on some part of the boundary 0 t existence and uniqueness of the entropy solution is established for any c r r n u 0 l f l q a l 0 t in the l 1 setting a corresponding result is proved for the more general notion of renormalised entropy solution

a new comparison principle for impulsive differential systems with variable impulsive perturbations and stability theory in this paper the stability properties in terms of two measures for impulsive differential systems with variable impulses are investigated a new comparison principlem which allows trajectories strike the same hypersurface finite times is established and then be applied to obtain a stability criterion finally one example is worked out to show the advantage of our results

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

existence of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic parabolic problems associated with the equation b t div a d f existence of renormalized solutions is established for general l 1 data uniqueness of renormalized solutions has already been shown in a previous work

new results on pulse phenomena for impulsive differential systems with variable moments abstract consider the impulsive differential systems with variable moments x f t x t k x x t 0 x 0 t 0 0 x i k x t k x some sufficient conditions that guarantee the absence or presence of pulse phenomena are obtained our results in this paper generalize and improve several known results

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

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

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

measure valued solutions of scalar conservation laws with boundary conditions we define a solution concept for measure valued solutions to scalar conservation laws with initial conditions and boundary conditions and prove a uniqueness theorem for such solutions this result may be used to prove convergence towards the unique solution for approximate solutions which are uniformly bounded in l weakly consistent with certain entropy inequalities and strongly consistent with the initial condition i e without using derivative estimates as an example convergence of a finite element method is demonstrated

pulse phenomena of second order impulsive differential equations with variable moments abstract this paper obtains sufficient conditions for presence or absence of pulse phenomena of second order impulsive differential equations with variable times

pulse accumulation in impulsive differential equations with variable times abstract we study pulse accumulation phenomena in first order impulsive differential equations and give necessary and sufficient conditions to ensure pulse accumulation in such equations

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

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

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

l1 theory of scalar conservation law with continuous flux function abstract uniqueness of a generalized entropy solution g e s to the cauchy problem for n dimensional scalar conservation laws u t div x u g u 0 f with continuous flux function is still an open problem for data f g vanishing at infinity we show that there exist a maximal and a minimal g e s to the cauchy problem and to the associated stationary problem u div x u f in the case of l 1 data using the nonlinear semigroup theory we prove that there is uniqueness for all data of a g e s to the cauchy problem if and only if there is uniqueness for all data of a g e s to the related stationary problem applying this result and an induction argument on the dimension n we extend uniqueness results of benilan kruzhkov 1996 nonlinear differential equations appl 3 395 419 for flux having some monotonicity properties

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

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