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

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

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

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

Chromatic Pupillometry in Children

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

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

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

Recent results on the singular local limit for nonlocal conservation laws

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.

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

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

Clearance of the mutant androgen receptor in motoneuronal models of spinal and bulbar muscular atrophy.

Spinal and bulbar muscular atrophy (SBMA) is an X-linked motoneuron disease caused by an abnormal expansion of a tandem CAG repeat in exon 1 of the androgen receptor (AR) gene that results in an abnormally long polyglutamine tract (polyQ) in the AR protein. As a result, the mutant AR (ARpolyQ) misfolds, forming cytoplasmic and nuclear aggregates in the affected neurons. Neurotoxicity only appears to be associated with the formation of nuclear aggregates. Thus, improved ARpolyQ cytoplasmic clearance, which indirectly decreases ARpolyQ nuclear accumulation, has beneficial effects on affected motoneurons. In addition, increased ARpolyQ clearance contributes to maintenance of motoneuron proteostasis and viability, preventing the blockage of the proteasome and autophagy pathways that might play a role in the neuropathy in SBMA. The expression of heat shock protein B8 (HspB8), a member of the small heat shock protein family, is highly induced in surviving motoneurons of patients affected by motoneuron diseases, where it seems to participate in the stress response aimed at cell protection. We report here that HspB8 facilitates the autophagic removal of misfolded aggregating species of ARpolyQ. In addition, though HspB8 does not influence p62 and LC3 (two key autophagic molecules) expression, it does prevent p62 bodies formation, and restores the normal autophagic flux in these cells. Interestingly, trehalose, a well-known autophagy stimulator, induces HspB8 expression, suggesting that HspB8 might act as one of the molecular mediators of the proautophagic activity of trehalose. Collectively, these data support the hypothesis that treatments aimed at restoring a normal autophagic flux that result in the more efficient clearance of mutant ARpolyQ might produce beneficial effects in SBMA patients