Institut de Mathématiques de Toulon

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An extension of the Colombo phase transition model is proposed. The congestion phase is described by a two-dimensional zone defined around a standard fundamental diagram. General criteria for building such a set-valued fundamental diagram are enumerated and instantiated on several standard fluxes with different concavity properties. The solution to the Riemann problem in the presence of phase transitions is obtained through the design of a Riemann solver, which enables the construction of the solution of the Cauchy problem using wavefront tracking. The free-flow phase is described using a Newell–Daganzo fundamental diagram, which allows for a more tractable definition of phase transition compared to the original Colombo phase transition model. The accuracy of the numerical solution obtained by a modified Godunov scheme is assessed on benchmark scenarios for the different flux functions constructed.

We consider the non-nonlinear optimal transportation problem of minimizing the cost functional $\mathcal{C}_\infty(\lambda) = \operatornamewithlimits{\lambda-ess\,sup}_{(x,y) \in \Omega^2} |y-x|$ in the set of probability measures on $\Omega^2$ having prescribed marginals. This corresponds to the question of characterizing the measures that realize the infinite Wasserstein distance. We establish the existence of “local” solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.

SUMMARY In this paper, we present the exact solution of the Riemann problem for the nonlinear one‐dimensional so‐called shallow‐water or Saint‐Venant equations with friction proposed by SAVAGE and HUTTER to describe debris avalanches. This model is based on the depth‐averaged thin layer approximation of granular flows over sloping beds and takes into account a Coulomb type friction law with a constant friction coefficient. A particular configuration of the Riemann problem corresponds to a dam of infinite length in one direction from which granular material is released from rest at a given time over an inclined rigid or erodible bed. We solve analytically and numerically the depth‐averaged long‐wave equations derived in a topography‐linked coordinate system for all the possible Riemann problems. The detailed mathematical proof of the derivation of the analytical solutions and the analysis of their structure and properties is intended, first of all, for geophysicists, mathematicians, and physicists because of the possible extension of this study to more complex problems (geometries, friction laws, …). The numerical solution of the first‐order finite‐volume method based on a Godunov‐type scheme is compared with the proposed exact Riemann problem solution. This solution is used to solve the dam‐break problem and analyze the influence of the thickness of the erodible bed on the speed of the granular front. Comparison with existing experimental results shows that, for an erodible bed, the equations lack fundamental physical significance to reproduce the observed dynamics of erosive granular flows. Copyright © 2012 John Wiley & Sons, Ltd.

We first consider the Monge problem in a convex bounded subset of Rd. The cost is given by a general norm, and we prove the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure. In the final part of the paper we show how to extend this existence result to a general open subset of Rd.

This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi-Pasta-Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.

To the aim of studying the homogenization of low-dimensional periodic structures, we identify each of them with a periodic positive measure $\mu$ on $\ren$. We introduce a new notion of two-scale convergence for a sequence of functions $v_\e \in L ^p_{\me} (\O; \re ^d)$, where $\O$ is an open bounded subset of $\ren$, and the measures $\mu _\e$ are the $\e$-scalings of $\mu$, namely, $\mu_\e (B) := \e ^n \mu (\e ^ {-1}B)$. Enforcing the concept of tangential calculus with respect to measures and related periodic Sobolev spaces, we prove a structure theorem for all the possible two-scale limits reached by the sequences $( u_\e, \nabla u _\e)$ when $\{u _\e\} \subset {\cal C} ^1_0 (\O)$ satisfy the boundedness condition $\sup _\e \int _{\O} |\ue| ^p + |\nabla \ue| ^p \, d \me < + \infty$ and when the measure $\mu$ satisfies suitable connectedness properties. This leads us to deduce the homogenized density of a sequence of energies of the form $\int _{\O} j (\xe, \nabla u) \, d \me$, where j(y,z) is a convex integrand, periodic in y, and satisfying a p-growth condition. The case of two parameter integrals is also investigated, in particular for what concerns the commutativity of the limit process.

In some industries, a certain part can be needed in a very large number of different configurations. This is the case, e.g., for the electrical wirings in European car factories. A given configuration can be replaced by a more complete, therefore more expensive, one. The diversity management problem consists of choosing an optimal set of some given number k of configurations that will be produced, any nonproduced configuration being replaced by the cheapest produced one that is compatible with it. We model the problem as an integer linear program. Our aim is to solve those problems to optimality. The large-scale instances we are interested in lead to difficult LP relaxations, which seem to be intractable by the best direct methods currently available. Most of this paper deals with the use of Lagrangean relaxation to reduce the size of the problem in order to be able, subsequently, to solve it to optimality via classical integer optimization.

Abstract. We consider N independent stochastic processes ( X i ( t ), t ∈ [0, T i ]), i=1,…, N , defined by a stochastic differential equation with drift term depending on a random variable φ i . The distribution of the random effect φ i depends on unknown parameters which are to be estimated from the continuous observation of the processes X i . We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ i and φ i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T i = T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.

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We determine in the framework of static linear elasticity the homogenized behavior of three-dimensional periodic structures made of welded elastic bars. It has been shown that such structures can be modeled as discrete systems of nodes linked by extensional, flexural/torsional interactions corresponding to frame lattices and that the corresponding homogenized models can be strain-gradient models, i.e., models whose effective elastic energy involves components of the first and the second gradients of the displacement field. However, in the existing models, there is no coupling between the classical strain and the strain-gradient terms in the expression of the effective energy. In the present article, under some assumptions on the positions of the nodes of the unit cell, we show that classical strain and strain-gradient strain terms can be coupled. In order to illustrate this coupling we compute the homogenized energy of a particular structure that we call asymmetrical pantographic structure.

We study a steady compressible Navier–Stokes–Fourier system in a bounded three-dimensional domain. We consider a general pressure law of the form $p=(\gamma-1)\varrho e$ which includes in particular the case $p=a_1\varrho\vartheta+a_2\varrho^\gamma$. We show the existence of a variational entropy solution (i.e., a solution satisfying balance of mass, momentum, entropy inequality, and global balance of total energy) for $\gamma>\frac{3+\sqrt{41}}{8}$ which is a weak solution (i.e., also the weak formulation of total energy balance is satisfied), provided $\gamma>\frac{4}{3}$. These results cover at least two physically reasonable cases, namely, $\gamma=\frac{5}{3}$ (monoatomic gas) and $\gamma=\frac{4}{3}$ (relativistic gas).

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The classical theory of Laplace is not suitable for describing the behavior of microscopic bubbles. The theory of second gradient fluids (which are able to exert shear stresses in equilibrium conditions) allows us to obtain a new expression for surface tension and radius of these bubbles in terms of functionals of the chemical potential. This relationship allows us to generalize the results of Cahn-Hilliard and Tolman.

We study a class of optimal transport planning problems where the reference cost involves a non-linear function G ( x, p ) representing the transport cost between the Dirac measure δ x and a target probability p . This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ( $G(x,p)=\int c(x,y)dp$ ) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.

In [17] the present authors investigated the stabilization of the weak solutions to space periodic problem for barotropic compressible Navier-Stokes equations. The main goal of this paper is to show the power of the method introduced in [17] by treating other boundary conditions. In fact, the only limitation of the method is potential external force and the validity of the Poincaré inequality for the velocity.

This paper is devoted to a hyperbolic 2-phase model for traffic flow on a network. The model is rigorously described and the existence of solutions is proved, without any restriction on the network geometry.

We investigate the Navier--Stokes--Fourier system describing the motion of a compressible, viscous, and heat conducting fluid on large class of unbounded domains with no slip and slip boundary conditions. We propose a definition of weak solutions that is particularly convenient for the treatment of the Navier--Stokes--Fourier system on unbounded domains. We introduce suitable weak solutions as weak solutions that satisfy the relative entropy inequality. We prove existence of weak solutions and of suitable weak solutions for arbitrary large initial data for potential forces with an arbitrary growth at large distances. Finally we prove the weak-strong uniqueness principle, meaning that the suitable weak solutions coincide with strong solutions emanating from the same initial data (as long as the latter exist), at least when the potential force vanishes at large distances.

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