934 research outputs found

    An Overview on Some Results Concerning the Transport Equation and its Applications to Conservation Laws

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    We provide an informal overview on the theory of transport equations with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.Comment: 12 page

    On the singular local limit for conservation laws with nonlocal fluxes

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    We give an answer to a question posed in [P. Amorim, R. Colombo, and A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015], which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.Comment: 26 page

    Initial-boundary value problems for continuity equations with BV coefficients

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    We establish well-posedness of initial-boundary value problems for continuity equations with BV (bounded total variation) coefficients. We do not prescribe any condition on the orientation of the coefficients at the boundary of the domain. We also discuss some examples showing that, regardless the orientation of the coefficients at the boundary, uniqueness may be violated as soon as the BV regularity deteriorates at the boundary.Comment: 25 pages, 5 figure

    On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws

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    We deal with the numerical investigation of the local limit of nonlocal conservation laws. Previous numerical experiments suggest convergence in the local limit. However, recent analytic results state that (i) in general convergence does not hold because one can exhibit counterexamples; (ii) convergence can be recovered provided viscosity is added to both the local and the nonlocal equations. Motivated by these analytic results, we investigate the role of numerical viscosity in the numerical study of the local limit of nonlocal conservation laws. In particular, we show that the numerical viscosity of Lax-Friedrichs type schemes jeopardizes the reliability of the numerical scheme and erroneously detects convergence in cases where convergence is ruled out by analytic results. We also test Godunov type schemes, less affected by numerical viscosity, and show that in some cases they provide more reliable results

    Chromatic Pupillometry in Children

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    Chromatic pupillometry is a technique that is increasingly used to assess retinal disorders. As age may be one of the various factors which can influence the pupillary light reaction, this study aimed to evaluate the pupil responses to colored light stimuli in the pediatric population. Fifty-three children with normal vision and without any history of ocular disorders were tested with a portable pupillometer. Four test sequences were used: five dim blue (470 nm) stimuli presented in half log steps ranging from −3.15 to −1.15 log cd/m2 after 3 min of dark adaptation, five red (622 nm) stimuli of −1.15, −0.7, −0.15, 0.3, and 0.85 log cd/m2 after 1 min light adaptation, one bright blue stimulus of 2.2 log cd/m2 and one bright red of 2 log cd/m2. The results were grouped by age: a younger group included 27 children aged from 3 to 10 years old and an older group included 26 from 10 and 1 month to 18 years old. The younger group had a smaller pupil diameter after dark adaptation compared with the older group. A linear regression defining the photopic threshold showed that younger subjects had a higher threshold, e.g., needed a brighter red stimulus to evoke a threshold pupil response comparable that of subjects. Age thus seems to influence outer retinal sensitivity at least as evaluated by the pupillary photopic threshold intensity. The post-illumination pupillary reaction was used as a marker of intrinsic melanopsin activity and did not show any difference between the two age groups

    Recent results on the singular local limit for nonlocal conservation laws

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    We provide an informal overview of recent developments concerning the singular local limit of nonlocal conservation laws. In particular, we discuss some counterexamples to convergence and we highlight the role of numerical viscosity in the numerical investigation of the nonlocal-to-local limit. We also state some open questions and describe recent related progress.Comment: Proceeding of the "XVII International Conference on Hyperbolic Problems: Theory, Numerics, Applications.

    An overview on the local limit of non-local conservation laws, and a new proof of a compactness estimate

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    Consider a non-local (i.e., involving a convolution term) conservation law: when the convolution term converges to a Dirac delta, in the limit we formally recover a classical (or "local") conservation law. In this note we overview recent progress on this so-called non-local to local limit and in particular we discuss the case of anistropic kernels, which is extremely relevant in view of applications to traffic models. We also provide a new proof of a related compactness estimate

    Some new well-posedness results for continuity and transport equations, and applications to the chromatography system

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    We obtain various new well-posedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of nearly incompressible vector fields, possibly having a blow-up of the BV norm at the initial time. We apply these results (valid in any space dimension) to the k x k chromatography system of conservation laws and to the k x k Keyfitz and Kranzer system, both in one space dimension.Comment: 33 pages, minor change
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