solution to a conjecture on the maximal energy of bipartite bicyclic graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let c n denote the cycle of order n and p n 6 6 the graph obtained from joining two cycles c 6 by a path p n 12 with its two leaves let b n denote the class of all bipartite bicyclic graphs but not the graph r a b which is obtained from joining two cycles c a and c b a b 10 and a b 2 mod 4 by an edge in i gutman d vidovic quest for molecular graphs with maximal energy a computer experiment j chem inf sci 41 2001 1002 1005 gutman and vidovic conjectured that the bicyclic graph with maximal energy is p n 6 6 for n 14 and n 16 in x li j zhang on bicyclic graphs with maximal energy linear algebra appl 427 2007 87 98 li and zhang showed that the conjecture is true for graphs in the class b n however they could not determine which of the two graphs r a b and p n 6 6 has the maximal value of energy in b furtula s radenkovic i gutman bicyclic molecular graphs with the greatest energy j serb chem soc 73 4 2008 431 433 numerical computations up to a b 50 were reported supporting the conjecture so it is still necessary to have a mathematical proof to this conjecture this paper is to show that the energy of p n 6 6 is larger than that of r a b which proves the conjecture for bipartite bicyclic graphs for non bipartite bicyclic graphs the conjecture is still open

rigorous derivation of nonlinear scalar conservation laws from follow the leader type models via many particle limit we prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow the leader type model which is interpreted as the discrete lagrangian approximation of the nonlinear scalar conservation law more precisely we prove that the empirical measure respectively the discretised density obtained from the follow the leader system converges in the 1 wasserstein topology respectively in mathbf l 1 loc to the unique kru kov entropy solution of the conservation law the initial data are taken in mathbf l infty nonnegative and with compact support hence we are able to handle densities with a vacuum our result holds for a reasonably general class of velocity maps including all the relevant examples in the applications for example in the lighthill whitham richards model for traffic flow with a possible degenerate slope near the vacuum state the proof of the result is based on discrete mathbf bv estimates and on a discrete version of the one sided oleinik type condition in particular we prove that the regularizing effect mathbf l infty mapsto mathbf bv for nonlinear scalar conservation laws is intrinsic to the discrete model

generalization of flanders theorem to matrix triples abstract this paper deals with several ways to generalize the flanders theorem to matrix triples we consider six invertible matrices and try to write them as the possible products of three matrices initially we describe a wide set of necessary conditions so that this system be solvable showing that they are not sufficient next we study the simultaneous solvability of two equations selected appropriately among the matrix system the rest of the paper is devoted to the study of a particular case in which the six given matrices are simultaneously diagonalizable with distinct nonzero eigenvalues in this case we obtain a necessary and sufficient condition for the solvability of the full matrix system moreover an explicit solution to it is constructed certain technical results necessary for this work may be of independent interest

deterministic diffusion of particles the diffusion of the particles is described in terms of a mean motion with a speed equal to the osmotic velocity associated with the diffusion process three numerical schemes are presented the first two are based on the approximation of the gradient on an irregular mesh the third is derived from a finite element approach voronoi diagrams are used to handle the irregular grid of the particles the convergence of the schemes is studied numerically by comparing the results with the exact solution applications to the fokker planck equation and to the problem of disposing particles according to a given probability distribution are presented

difference schemes for degenerate parabolic equations construction de schemas aux differences pour les equations paraboliques degenerees engendrant des semigroupes demonstration de la convergence des algorithmes completement discrets experiences numeriques

viability solutions to structured hamilton jacobi equations under constraints structured hamilton jacobi partial differential equations are hamilton jacobi equations where the time variable is replaced by a vector valued variable structuring the system it could be the time age pair hamilton jacobi mckendrick equations or candidates for initial or terminal conditions hamilton jacobi cournot equations among a manifold of examples here we define the concept of viability solution which always exists and can be computed by viability algorithms this is a constructive approach allowing us to derive from the tools of viability theory dealing with sets instead of functions some known as well as new properties of its solutions regarding functions as their epigraphs we bypass the regularity issues to arrive directly at the concept of barron jensen frankowska viscosity solutions above all we take into account viability constraints and extend classical boundary conditions to other internal conditions beyond that we use the fenchel legendre transform to associate a lagrangian with the hamiltonian and uncover an underlying variational problem it is proved that the viability solution to the structured hamilton jacobi equation is the valuation function of this variational principle the optimal evolutions of which are regulated by a regulation map which is constructed from the viability solution and can be computed by viability algorithms in other words the viability solution solves all at once a class of first order partial differential equations of intertemporal optimization problems and the regulation of their optimal evolutions

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

solutions to unsolved problems on the minimal energies of two classes of trees the energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph let t n p t n d be the set of all trees of order n with p pendent vertices diameter d respectively in this paper we completely characterize the trees with second minimal and third minimal energy in t n p t n d respectively for 4 p n 9 10 d n 3 respectively which solves the problems left in ma 2014

an interface tracking algorithm for the porous medium equation we study the convergence of a finite difference scheme for the cauchy problem for the porous medium equation u m av m 1 the scheme exhibits the following two features the first is that it employs a discretization of the known interface condition for the propagation of the support of the solution we thus generate approximate interfaces as well as an approximate solution the second feature is that it contains a vanishing viscosity term this term permits an estimate of the form i 1 a viii r c t we prove that both the approximate solution and the approximate interfaces converge to the correct ones finally error bounds for both solution and free boundaries are proved in terms of the mesh parameters 1 introduction in this paper we derive and analyze a finite difference scheme for computing both the solution and the interfaces for the porous medium equation in one space dimension we demonstrate that the approximate solutions and the approximate interface curves converge to the correct ones and we obtain l00 bounds for the error in terms of the mesh parameter consider the laminar flow of a polytropic fluid of density x t u x t in a porous medium which is assumed to occupy the whole space and suppose that at time t 0 the fluid is contained in the slab 0 x fr 0 the phenomenon can be modeled by

a model for the formation and evolution of traffic jams in this paper we establish and analyze a traffic flow model which describes the formation and dynamics of traffic jams it consists of a pressureless gas dynamics system under a maximal constraint on the density and is derived through a singular limit of the aw rascle model from this analysis we deduce the particular dynamical behavior of clusters or traffic jams defined as intervals where the density limit is reached an existence result for a generic class of initial data is proved by means of an approximation of the solution by a sequence of clusters finally numerical simulations are produced

on the similarity classes among products of m nonsingular matrices in various orders abstract it is shown that for any k 1 2 m 1 there exist m invertible complex matrices such that among the m products a a 1 a 2 a m s m exactly k different similarity classes occur the cases in which the matrices a i are upper triangular or are 2 by 2 are considered in detail in the former case it is shown that any m 1 unispectral matrices with common eigenvalue may be found among the a and in the latter case it is shown explicitly how to achieve m 1 and m 1 2 similarity classes as well as any number from 1 to 6 when m 4 other particular results are given as well as a discussion of further natural questions

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

on the maximal energy tree with two maximum degree vertices abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix for 3 and t 3 denote by t a t or simply t a the tree formed from a path p t on t vertices by attaching 1 p 2 s on each end of the path p t and t b t or simply t b the tree formed from p t 2 by attaching 1 p 2 s on an end of the p t 2 and 2 p 2 s on the vertex next to the end in li et al 2009 16 proved that among trees of order n with two vertices of maximum degree the maximal energy tree is either the graph t a or the graph t b where t n 4 4 3 however they could not determine which one of t a and t b is the maximal energy tree this is because the quasi order method is invalid for comparing their energies in this paper we use a new method to determine the maximal energy tree it turns out that things are more complicated we prove that the maximal energy tree is t b for 7 and any t 3 while the maximal energy tree is t a for 3 and any t 3 moreover for 4 the maximal energy tree is t a for all t 3 but one exception that t 4 for which t b is the maximal energy tree for 5 the maximal energy tree is t b for all t 3 but 44 exceptions that t is both odd and 3 t 89 for which t a is the maximal energy tree for 6 the maximal energy tree is t b for all t 3 but three exceptions that t 3 5 7 for which t a is the maximal energy tree one can see that for most cases of t b is the maximal energy tree 5 is a turning point and 3 and 4 are exceptional cases which means that for all chemical trees whose maximum degrees are at most 4 with two vertices of maximum degree at least 3 t a has maximal energy with only one exception t a 4 4

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

a numerical procedure for the porous media equation introduction in this paper we contmue a study mltlated m 11 of numencal procedures for the porous media problem u u2l 0 x u x u x 2 0 t 0 x e r p it 1s well known 2 6 that p has a umque solution u which for each t is supported m a fimte interval c f t t inside the support u 1s a classical solution of p but across tf u 1s generally dlscontmuous in fact one has

on the tree with diameter 4 and maximal energy the energy of a simple graph is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix jianping found the structure of trees which have maximal energy among all trees of diameter 4 whose centers are of degree t this paper generalizes the result we determine the structure of the general tree of diameter 4 and maximal energy

chaperoning state evolutions by variable durations mckendrick partial differential equations in demography provide the state of a population depending on chronological time t and age d and involve two time evolutions the chronological one t in 0 t mapsto t geq 0 and the calendar age t in t d t mapsto t t d geq 0 both with constant velocity equal to 1 the calendar age evolution chaperons so to speak the evolution t in t d t mapsto x t of the state of a system some physical biological and economic problems motivate the introduction of variable durations with variable velocities representing the fluidity of time offering mathematical metaphors of a subjective fleeting specious time passing more or less slowly variable durations are no longer prescribed but chosen among those available and regulated the joint evolution of the variable evolution and the state is assumed to be governed by a differential inclusion or control system and provides as a byproduct the unknown temporal windows on which they evolve toge

an oleinik type estimate for a convection diffusion equation and convergence to n waves in this article we propose an oleinik type estimate for sign changing solutions to a convection diffusion equation ut u x uxx u x 0 u0 x u x r 1 0 since the oleinik entropy inequality holds for nonnegative solutions or inviscid case 0 only the theoretical progress for the case was limited in this paper we show that its solution satisfies an oleinik type estimate t2 ux c 1 0 where c c u0 0 using this estimate the convergence to an n wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled

a holder continuous ode related to traffic flow this paper is devoted to the proof of the well posedness of a class of ordinary differential equations odes the vector field depends on the solution to a scalar conservation law forward uniqueness of filippov solutions is obtained as well as their holder continuous dependence on the initial data of the ode furthermore we prove the continuous dependence in c 0 of the solution to the ode from the initial data of the conservation law in l 1 this problem is motivated by a model of traffic flow

derivation of continuum traffic flow models from microscopic follow the leader models in this paper we establish a connection between a microscopic follow the leader model based on ordinary differential equations and a semidiscretization of a macroscopic continuum model based on a c

sticky particles and scalar conservation laws one dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line which stick under collision with conservation of mass and momentum

complete solution to a problem on the maximal energy of unicyclic bipartite graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix denote by c n the cycle and p n 6 the unicyclic graph obtained by connecting a vertex of c 6 with a leaf of p n 6 caporossi et al conjectured that the unicyclic graph with maximal energy is p n 6 for n 8 12 14 and n 16 in hou et al 2002 y hou i gutman c woo unicyclic graphs with maximal energy linear algebra appl 356 2002 27 36 the authors proved that e p n 6 is maximal within the class of the unicyclic bipartite n vertex graphs differing from c n and they also claimed that the energies of c n and p n 6 is quasi order incomparable and left this as an open problem in this paper by utilizing the coulson integral formula and some knowledge of real analysis especially by employing certain combinatorial techniques we show that the energy of p n 6 is greater than that of c n for n 8 12 14 and n 16 which completely solves this open problem and partially solves the above conjecture

extremal laplacian energy of threshold graphs let g be a connected threshold graph of order n with m edges and trace t in this paper we give a lower bound on laplacian energy in terms of n m and t of g from this we determine the threshold graphs with the first four minimal laplacian energies moreover we obtain the threshold graphs with the largest and the second largest laplacian energies

spectral properties of geometric arithmetic index the concept of geometric arithmetic index was introduced in the chemical graph theory recently but it has shown to be useful one of the main aims of algebraic graph theory is to determine how or whether properties of graphs are reflected in the algebraic properties of some matrices the aim of this paper is to study the geometric arithmetic index ga1 from an algebraic viewpoint since this index is related to the degree of the vertices of the graph our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices moreover using this matrix we define a ga laplacian matrix which determines the geometric arithmetic index of a graph and satisfies properties similar to the ones of the classical laplacian matrix

deterministic particle approximation for nonlocal transport equations with nonlinear mobility abstract we construct a deterministic lagrangian many particle approximation to a class of nonlocal transport pdes with nonlinear mobility arising in many contexts in biology and social sciences the approximating particle system is a nonlocal version of the follow the leader scheme we rigorously prove that a suitable discrete piece wise density reconstructed from the particle scheme converges strongly in l l o c 1 towards the unique entropy solution to the target pde as the number of particles tends to infinity the proof is based on uniform bv estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles as part of the proof we also prove uniqueness of entropy solutions we further provide a specific example of non uniqueness of weak solutions and discuss the interplay of the entropy condition with the steady states finally we produce numerical simulations supporting the need of a concept of entropy solution in order to get a well posed semigroup in the continuum limit and showing the behaviour of solutions for large times

a note on companion matrices abstract we show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices n 1 of them being the identity matrix in which a 2 2 identity submatrix in two consecutive rows and columns is replaced by an appropriate 2 2 matrix the remaining being the identity matrix with the last entry replaced by possibly different entry by a certain similarity transformation we obtain a simple new companion matrix in a pentadiagonal form some generalizations are also possible

order invariant spectral properties for several matrices abstract the collections of m n by n matrices over a field such that the products in any of the m orders share a common similarity class resp spectrum trace are studied the spectral and trace order invariant properties are characterized and the similarity invariant one is related to them in several cases a complete explicit description is given in case m 3 and n 2

solution of porous medium type systems by linear approximation schemes the convergence of semi discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type the enthalpy formulation and variational technique are used the semi discretization used reduces the original parabolic p d e to linear elliptic p d e the algebraic correction arising from nonlinearities is treated by newton like iterations in finite steps some numerical experiments are discussed and compared with the analytical solutions

dirichlet problems for some hamilton jacobi equations with inequality constraints we use viability techniques for solving dirichlet problems with inequality constraints obstacles for a class of hamilton jacobi equations the hypograph of the solution is defined as the capture basin under an auxiliary control system of a target associated with the initial and boundary conditions viable in an environment associated with the inequality constraint from the tangential condition characterizing capture basins we prove that this solution is the unique upper semicontinuous solution to the hamilton jacobi bellman partial differential equation in the barron jensen frankowska sense we show how this framework allows us to translate properties of capture basins into corresponding properties of the solutions to this problem for instance this approach provides a representation formula of the solution which boils down to the lax hopf formula in the absence of constraints

macroscopic traffic models shifting from densities to celerities abstract the advent of techniques to measure velocities of gps equipped vehicles using satellite technology replacing the density of the road traffic sensors motivates and justifies the revision of conceptual mathematical algorithms and software based models this paper summarizes studies on the traffic evolutions achieving the minimum of a congestion function controlled macroscopic traffic velocities called celerities instead of founding traffic regulation on the measures of traffic densities the flux valued function is the fenchel transform of the fundamental diagram and is a convex decreasing function we use the properties of capture basins investigated in viability theory specifically a lax hopf formula characterizing them and the max plus morphism of capture basins for deriving the statements proved in this paper traffic conditions involve as well boundary conditions or conditions on trajectories inside the domain

on tricyclic graphs of a given diameter with minimal energy abstract the energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph let g n d be the class of tricyclic graphs g on n vertices with diameter d and containing no vertex disjoint odd cycles c p c q of lengths p and q with p q 2 mod 4 in this paper we characterize the graphs with minimal energy in g n d

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

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

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

the aw rascle traffic model enskog type kinetic derivation and generalisations we study the derivation of second order macroscopic traffic models from kinetic descriptions in particular we recover the celebrated aw rascle model as the hydrodynamic limit of an enskog type kinetic equation out of a precise characterisation of the microscopic binary interactions among the vehicles unlike other derivations available in the literature our approach unveils the multiscale physics behind the aw rascle model this further allows us to generalise it to a new class of second order macroscopic models complying with the aw rascle consistency condition namely the fact that no wave should travel faster than the mean traffic flow

the large time behavior of the scalar genuinely nonlinear lax friedrichs scheme the lax friedrichs scheme approximating the scalar genuinely nonlinear conservation law u sub t f sub x u 0 where f u is say strictly convex double dot f dot a sub asterisk 0 is studied the divided differences of the numerical solution at time t do not exceed 2 t dot a sub asterisk to the 1 this one sided lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case in particular it is independent of the initial amplitude in sharp contrast to liner problems it guarantees the entropy compactness of the scheme in this case as well as providing a quantitive insight into the large time behavior of the numerical computation

note on unicyclic graphs with given number of pendent vertices and minimal energy abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let g n p denote the set of unicyclic graphs with n vertices and p pendent vertices in h hua m wang unicyclic graphs with given number of pendent vertices and minimal energy linear algebra appl 426 2007 478 489 hua and wang discussed the graphs that have minimal energy in g n p and determined the minimal energy graphs among almost all different cases of n and p in their work certain case of the values was left as an open problem in which the minimal energy species have to be chosen in two candidate graphs but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials this paper aims at solving the problem completely by using the well known coulson integral formula

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

on ordering of complements of graphs with respect to matching numbers we consider an earlier much studied quasi order defined in terms of matching numbers of graphs and apply it to graph complements we establish four transformations on the complements of graphs that increase or decrease the matching numbers accordingly several applications of these results are put forward

flandersʼ theorem for many matrices under commutativity assumptions we analyze the relationship between the jordan canonical form of products in different orders of k square matrices a 1 a k our results extend some classical results by h flanders motivated by a generalization of fiedler matrices we study permuted products of a 1 a k under the assumption that the graph of non commutativity relations of a 1 a k is a forest under this condition we show that the jordan structure of all nonzero eigenvalues is the same for all permuted products for the eigenvalue zero we obtain an upper bound on the dierence between the sizes of jordan blocks for any two permuted products and we show that this bound is attainable for k 3 we show that moreover the bound is exhaustive

lagrangian non oscillatory and fem schemes for the porous media equation abstract interface tracking schemes derived from a lagrangian formulation of the equation are considered existence and equivalence results valid even for the non smooth interface case are given and non oscillatory schemes needed in this case are discussed mixed euler lagrange fem discretization for mixed problems and numerical experiments are presented

contractivity of wasserstein metrics and asymptotic profiles for scalar conservation laws abstract the aim of this paper is to analyze contractivity properties of wasserstein type metrics for one dimensional scalar conservation laws with nonnegative l and compactly supported initial data and its implications on the long time asymptotics the flux is assumed to be convex and without any growth condition at the zero state we propose a time parameterized family of functions as intermediate asymptotics and prove the solutions after a time depending scaling converge toward this family in the d wasserstein metric this asymptotic behavior relies on the aforementioned contraction property for conservation laws in the space of probability densities metrized with the d wasserstein distance finally we also give asymptotic profiles for initial data whose distributional derivative is a probability measure