A Riemann solver at a junction compatible with a homogenization limit

We consider a junction regulated by a traffic lights, with n incoming roads and only one outgoing road. On each road the Phase Transition traffic model, proposed in [6], describes the evolution of car traffic. Such model is an extension of the classic Lighthill-Whitham-Richards one, obtained by assuming that different drivers may have different maximal speed. By sending to infinity the number of cycles of the traffic lights, we obtain a justification of the Riemann solver introduced in [9] and in particular of the rule for determining the maximal speed in the outgoing road.Comment: 19 page

The Godunov Method for a 2-Phase Model

We consider the Godunov numerical method to the phase-transition traffic model, proposed in [6], by Colombo, Marcellini, and Rascle. Numerical tests are shown to prove the validity of the method. Moreover we highlight the differences between such model and the one proposed in [1], by Blandin, Work, Goatin, Piccoli, and Bayen.Comment: 13 page

NonLocal Systems of Balance Laws in Several Space Dimensions with Applications to Laser Technolog

For a class of systems of nonlinear and nonlocal balance laws in several space dimensions, we prove the local in time existence of solutions and their continuous dependence on the initial datum. The choice of this class is motivated by a new model devoted to the description of a metal plate being cut by a laser beam. Using realistic parameters, solutions to this model obtained through numerical integrations meet qualitative properties of real cuts. Moreover, the class of equations considered comprises a model describing the dynamics of solid particles along a conveyor belt

Smooth and discontinuous junctions in the p-system

Consider the p-system describing the subsonic flow of a fluid in a pipe with section a = a(x). We prove that the resulting Cauchy problem generates a Lipschitz semigroup, provided the total variation of the initial datum and the oscillation of a are small. An explicit estimate on the bound of the total variation of a is provided, showing that at lower fluid speeds, higher total variations of a are acceptable. An example shows that the bound on TV(a) is mandatory, for otherwise the total variation of the solution may grow arbitrarily.Comment: 27 pages, 4 figure

Coupling conditions for the 3x3 Euler system

This paper is devoted to the extension to the full $3\times3$ Euler system of the basic analytical properties of the equations governing a fluid flowing in a duct with varying section. First, we consider the Cauchy problem for a pipeline consisting of 2 ducts joined at a junction. Then, this result is extended to more complex pipes. A key assumption in these theorems is the boundedness of the total variation of the pipe's section. We provide explicit examples to show that this bound is necessary.Comment: 21 pages, 6 figure

Modeling and analysis of pooled stepped chutes

We consider an application of pooled stepped chutes where the transport in each pooled step is described by the shallow--water equations. Such systems can be found for example at large dams in order to release overflowing water. We analyze the mathematical conditions coupling the flows between different chutes taken from the engineering literature. We present the solution to a Riemann problem in the large and also a well--posedness result for the coupled problem. We finally report on some numerical experiments.Comment: 17 pages, 31 figure

Balance laws with integrable unbounded sources

We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$\{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | \lambda_i(u)| \geq c > 0 {for all} i\in \{1,...,n\}, \|g(x,\cdot)\|_{\mathbf{C}^2}\leq \tilde M(x) \in L1, {array}.$$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that the $\mathbf{L}^1$ norm of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and \|u_o\|_{BV(\reali)} are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [1] to unbounded (in $L^\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.Comment: 26 pages, 4 figure

Dynamics and ordering of weakly Brownian particles in directional drying

Drying of particle suspensions is an ubiquitous phenomenon with many natural and practical applications. In particular, in unidirectional drying, the evaporation of the solvent induces flows which accumulate particles at the liquid/air interface. The progressive build-up of a dense region of particles can be used, in particular, in the processing of advanced materials and architectures while the development of heterogeneities and defects in such systems is critical to their function. A lot of attention has thus been paid to correlate the flow and particles dynamics to the ordering of particles. However, dynamic observation at the particle scale and its correlation with local particle ordering are still missing. Here we show by measuring the particle velocities with high frame rate laser scanning confocal microscopy that the ordering of weakly Brownian particles during directional drying in a Hele-Shaw cell opened on one side depends on the particle velocity. Under the ambient and experimental conditions presented in the following, the particle velocities accumulate in two branches. A higher degree of ordering is found for the branch of faster particle velocity which we explain by an increase in the pressure drop which drags the particles into a denser packing as the flow velocity increases. This counter-intuitive behaviour is the opposite to what is found with Brownian particles, which can reorganize by Brownian motion into denser packing during drying, as long as the flow velocity is not too high. These results show that different kinetic conditions can be used to obtain dense, defect-free regions of particles after drying. In particular, it suggests that rapid, directional drying could be used to control the crystallinity of particle deposits.Comment: 10 pages, 12 figure

Autonomous Vehicles Driving Traffic: The Cauchy Problem

This paper deals with the Cauchy Problem for a PDE-ODE model, where a system of two conservation laws, namely the Two-Phase macroscopic model, is coupled with an ordinary differential equation describing the trajectory of an autonomous vehicle (AV), which aims to control the traffic flow. Under suitable assumptions, we prove a global in time existence result.Comment: 32 page