An overview on the approximation of boundary Riemann problems through physical viscosity

This note aims at providing an overview of some recent results concerning the viscous approximation of so-called boundary Riemann problems for nonlinear systems of conservation laws in small total variation regimes. \ua9 2016, Sociedade Brasileira de Matem\ue1tica

Orientational Effects and Random Mixing in 1-Alkanol + Alkanone Mixtures

1-Alkanol + alkanone systems have been investigated through the data analysis of molar excess functions, enthalpies, isobaric heat capacities, volumes and entropies, and using the Flory model and the formalism of the concentrationconcentration structure factor (SCC(0)). The enthalpy of the hydroxyl-carbonyl interactions has been evaluated. These interactions are stronger in mixtures with shorter alcohols (methanol-1-butanol) and 2-propanone or 2-butanone. However, effects related to the self-association of alcohols and to solvation between unlike molecules are of minor importance when compared with those which arise from dipolar interactions. Physical interactions are more relevant in mixtures with longer 1-alkanols. The studied systems are characterized by large structural effects. The variation of the molar excess enthalpy with the alcohol size along systems with a given ketone or with the alkanone size in solutions with a given alcohol are discussed in terms of the different contributions to this excess function. Mixtures with methanol show rather large orientational effects. The random mixing hypothesis is attained to a large extent for mixtures with 1-alkanols ≠ methanol and 2-alkanones. Steric effects and cyclization lead to stronger orientational effects in mixtures with 3-pentanone, 4-heptanone, or cyclohexanone. The increase of temperature weakens orientational effects. Results from SCC(0) calculations show that homocoordination is predominant and support conclusions obtained from the Flory model.Ministerio de Ciencia e Innovación, under Project FIS2010-1695

New interaction estimates for the Baiti-Jenssen system

We establish new interaction estimates for a system introduced by Baiti and Jenssen. These estimates are pivotal to the analysis of the wave front-tracking approximation. In a companion paper we use them to construct a counter-example which shows that Schaeffer\u2019s Regularity Theorem for scalar conservation laws does not extend to systems. The counter-example we construct shows, furthermore, that a wave-pattern containing infinitely many shocks can be robust with respect to perturbations of the initial data. The proof of the interaction estimates is based on the explicit computation of the wave fan curves and on a perturbation argument

Initial–boundary value problems for merely bounded nearly incompressible vector fields in one space dimension

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case where the velocity field is either nonnegative or nonpositive, one can rely on similar techniques as in the case of the Cauchy problem. Conversely, in the general case we introduce a new and more technically demanding construction, which heuristically speaking relies on a “lagrangian formulation” of the problem, albeit in a highly irregular setting. We also establish stability of the solution in weak and strong topologies, and propagation of the BV regularity. In the case of either nonnegative or nonpositive velocity fields we also establish a BV-in-time regularity result, and we exhibit a counterexample showing that the result is false in the case of sign-changing vector fields. To conclude, we establish a trace renormalization property

NEW REGULARITY RESULTS FOR SCALAR CONSERVATION LAWS, AND APPLICATIONS TO A SOURCE-DESTINATION MODEL FOR TRAFFIC FLOWS ON NETWORKS

We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function f is strictly convex and show that, for every x ∊ R, the total variation of the composite function f ∘ u(⋅, x) is controlled by the total variation of the initial datum. Next, we assume that f is monotone and, under no convexity assumption, we show that, for every x, the total variation of the left and the right trace u(⋅, x±) is controlled by the total variation of the initial datum. We also exhibit a counterexample showing that in the first result the total variation bound does not extend to the function u, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain well-posedness in view of a counterexample due to Bressan and Yu. We also establish stability and propagation of BV regularity, and this is again interesting in view of recent counterexamples

EDXRF quantitative analysis of chromophore chemical elements in corundum samples

Corundum is a crystalline form of aluminum oxide (Al2O3) and is one of the rock-forming minerals. When aluminum oxide is pure, the mineral is colorless, but the presence of trace amounts of other elements such as iron, titanium, and chromium in the crystal lattice gives the typical colors (including blue, red, violet, pink, green, yellow, orange, gray, white, colorless, and black) of gemstone varieties. The starting point for our work is the quantitative evaluation of the concentration of chromophore chemical elements with a precision as good as possible to match the data obtained by different techniques as such as optical absorption photoluminescence. The aim is to give an interpretation of the absorption bands present in the NIR and visible ranges which do not involve intervalence charge transfer transitions (Fe2+ \u2192 Fe3+ and Fe2+ \u2192 Ti4+), commonly considered responsible of the important features of the blue sapphire absorption spectra. So, we developed a method to evaluate as accurately as possible the autoabsorption effects and the secondary excitation effects which frequently are sources of relevant errors in the quantitative EDXRF analysis

A uniqueness criterion for viscous limits of boundary Riemann problems

We deal with initial-boundary value problems for systems of conservation laws in one space dimension and we focus on the boundary Riemann problem. It is known that, in general, different viscous approximations provide different limits. In this paper, we establish sufficient conditions to conclude that two different approximations lead to the same limit. As an application of this result, we show that, under reasonable assumptions, the self-similar second-order approximation and the classical viscous approximation provide the same limit. Our analysis applies to both the characteristic and the non characteristic case. We require neither genuine nonlinearity nor linear degeneracy of the characteristic fields