This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem.Our method is based on wave front tracking approach, see [6], and works also for boundary data and time dependent coefficients of traffic distribution at junctions, so including traffic lights.

We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equationThis equation can be regarded as a model for shallow water dynamics and its asymptotic accuracy is the same as for the Camassa-Holm equation (one order more accurate than the KdV equation). We prove existence and L 1 stability (uniqueness) results for entropy weak solutions belonging to the class L 1 ∩ BV , while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class L 2 ∩ L 4 . Finally, we extend our results to a class of generalized Degasperis-Procesi equations.

Abstract. We consider a generalized hyperelastic-rod wave equation (or generalized CamassaHolm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous semigroup of global weak solutions for any initial data from H 1 (R). We also present a "weak equals strong"uniqueness result.

Abstract.We prove the existence of a sequence of radial solutions with negative energy of the Schrödinger-Maxwell equations under the action of a negative potential. Introduction.In this paper we study the interaction between the electromagnetic field and the wave function related to a quantum nonrelativistic charged particle, which is described by the Schrödinger equation.In [2, 3, 11] the case in which the electromagnetic field is given has been studied. Here we shall assume that the unknowns of the problem are both the wave function ψ = ψ(x, t) and the gauge potentials ϕ = ϕ(x, t) and A = A(x, t) related to the electromagnetic fields E, H by the equations

We consider the vanishing viscosity approximation of the traffic model, proposed by Lighthill, Whitham, and Richards, on a network composed by a single junction with n incoming and m outgoing roads. We prove that a solution of the parabolic approximation exists and, as the viscosity vanishes, the solution of the parabolic problem converges to a solution of the original problem

Abstract. The paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system near a constant state. On the other hand, we give an example showing that exact controllability in finite time cannot be achieved, in general. 0

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Abstract. Recent work [4] has shown that the Degasperis-Procesi equation is well-posed in the class of (discontinuous) entropy solutions. In the present paper we construct numerical schemes and prove that they converge to entropy solutions. Additionally, we provide several numerical examples accentuating that discontinuous (shock) solutions form independently of the smoothness of the initial data. Our focus on discontinuous solutions contrasts notably with the existing literature on the Degasperis-Procesi equation, which seems to emphasize similarities with the Camassa-Holm equation (bi-Hamiltonian structure, integrabillity, peakon solutions, H 1 as the relevant functional space).