Steady nearly incompressible vector fields in 2D: chain rule and renormalization

Given bounded vector field $b : \mathbb R^d \to \mathbb R^d$, scalar field $u : \mathbb R^d \to \mathbb R$ and a smooth function $\beta : \mathbb R \to \mathbb R$ we study the characterization of the distribution $\mathrm{div}(\beta(u)b)$ in terms of $\mathrm{div}\, b$ and $\mathrm{div}(u b)$. In the case of $BV$ vector fields $b$ (and under some further assumptions) such characterization was obtained by L. Ambrosio, C. De Lellis and J. Mal\'y, up to an error term which is a measure concentrated on so-called \emph{tangential set} of $b$. We answer some questions posed in their paper concerning the properties of this term. In particular we construct a nearly incompressible $BV$ vector field $b$ and a bounded function $u$ for which this term is nonzero. For steady nearly incompressible vector fields $b$ (and under some further assumptions) in case when $d=2$ we provide complete characterization of $\mathrm{div}(\beta(u) b)$ in terms of $\mathrm{div}\, b$ and $\mathrm{div}(u b)$. Our approach relies on the structure of level sets of Lipschitz functions on $\mathrm R^2$ obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique we obtain new sufficient conditions when any bounded weak solution $u$ of $\partial_t u + b \cdot \nabla u=0$ is \emph{renormalized}, i.e. also solves $\partial_t \beta(u) + b \cdot \nabla \beta(u)=0$ for any smooth function $\beta : \mathbb R \to \mathbb R$. As a consequence we obtain new uniqueness result for this equation.Comment: 50 pages, 8 figure

Rotating Globular Clusters

Internal rotation is considered to play a major role in the dynamics of some globular clusters. However, in only few cases it has been studied by quantitative application of realistic and physically justified global models. Here we present a dynamical analysis of the photometry and three-dimensional kinematics of omega Cen, 47 Tuc, and M15, by means of a recently introduced family of self-consistent axisymmetric rotating models. The three clusters, characterized by different relaxation conditions, show evidence of differential rotation and deviations from sphericity. The combination of line-of-sight velocities and proper motions allows us to determine their internal dynamics, predict their morphology, and estimate their dynamical distance. The well-relaxed cluster 47 Tuc is very well interpreted by our model; internal rotation is found to explain the observed morphology. For M15, we provide a global model in good agreement with the data, including the central behavior of the rotation profile and the shape of the ellipticity profile. For the partially relaxed cluster omega Cen, the selected model reproduces the complex three-dimensional kinematics; in particular the observed anisotropy profile, characterized by a transition from isotropy, to weakly-radial anisotropy, and then to tangential anisotropy in the outer parts. The discrepancy found for the steep central gradient in the observed line-of-sight velocity dispersion profile and for the ellipticity profile is ascribed to the condition of only partial relaxation of this cluster and the interplay between rotation and radial anisotropy.Comment: 19 pages, 14 figures, accepted for publication in the Astrophysical Journa

The boundary Riemann solver coming from the real vanishing viscosity approximation

We study a family of initial boundary value problems associated to mixed hyperbolic-parabolic systems: v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x = \epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx} The conservative case is, in particular, included in the previous formulation. We suppose that the solutions $v^{\epsilon}$ to these problems converge to a unique limit. Also, it is assumed smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $A$ can be $0$. Second, we take into account the possibility that $B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.Comment: 84 pages, 6 figures. Text changes in Sections 1 and 3.2.3. Added Section 3.1.2. Minor changes in other section

Method to measure off-axis displacements based on the analysis of the intensity distribution of a vortex beam

We study the properties of the Fraunhofer diffraction patterns produced by Gaussian beams crossing spiral phase plates. We show, both analytically and numerically, that off-axis displacements of the input beam produce asymmetric diffraction patterns. The intensity profile along the direction of maximum asymmetry shows two different peaks. We find that the intensity ratio between these two peaks decreases exponentially with the off-axis displacement of the incident beam, the decay being steeper for higher strengths of the optical singularity of the spiral phase plate. We analyze how this intensity ratio can be used to measure small misalignments of the input beam with a very high precision.Comment: 8 pages, 4 figures. Accepted for publication in PR

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension

We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields $$u_t + f(u)_x = 0,$$ then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families $$u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N,$$ and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure \mu_{i,\jump} is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $\mu_{i,\mathrm{jump}}$ is positive

Kinematic fingerprint of core-collapsed globular clusters

Dynamical evolution drives globular clusters toward core collapse, which strongly shapes their internal properties. Diagnostics of core collapse have so far been based on photometry only, namely on the study of the concentration of the density profiles. Here we present a new method to robustly identify core-collapsed clusters based on the study of their stellar kinematics. We introduce the \textit{kinematic concentration} parameter, $c_k$, the ratio between the global and local degree of energy equipartition reached by a cluster, and show through extensive direct $N$-body simulations that clusters approaching core collapse and in the post-core collapse phase are strictly characterized by $c_k>1$. The kinematic concentration provides a suitable diagnostic to identify core-collapsed clusters, independent from any other previous methods based on photometry. We also explore the effects of incomplete radial and stellar mass coverage on the calculation of $c_k$ and find that our method can be applied to state-of-art kinematic datasets.Comment: Accepted for publication in MNRAS Lette

Ocular hypertension in myopia: analysis of contrast sensitivity

Purpose: we evaluated the evolution of contrast sensitivity reduction in patients affected by ocular hypertension and glaucoma, with low to moderate myopia. We also evaluated the relationship between contrast sensitivity and mean deviation of visual field. Material and methods: 158 patients (316 eyes), aged between 38 and 57 years old, were enrolled and divided into 4 groups: emmetropes, myopes, myopes with ocular hypertension (IOP≥21 ±2 mmHg), myopes with glaucoma. All patients underwent anamnestic and complete eye evaluation, tonometric curves with Goldmann’s applanation tonometer, cup/disc ratio evaluation, gonioscopy by Goldmann’s three-mirrors lens, automated perimetry (Humphrey 30-2 full-threshold test) and contrast sensitivity evaluation by Pelli-Robson charts. A contrast sensitivity under 1,8 Logarithm of the Minimum Angle of Resolution (LogMAR) was considered abnormal. Results: contrast sensitivity was reduced in the group of myopes with ocular hypertension (1,788 LogMAR) and in the group of myopes with glaucoma (1,743 LogMAR), while it was preserved in the group of myopes (2,069 LogMAR) and in the group of emmetropes (1,990 LogMAR). We also found a strong correlation between contrast sensitivity reduction and mean deviation of visual fields in myopes with glaucoma (coefficient relation = 0.86) and in myopes with ocular hypertension (coefficient relation = 0.78). Conclusions: the contrast sensitivity assessment performed by the Pelli-Robson test should be performed in all patients with middle-grade myopia, ocular hypertension and optic disc suspected for glaucoma, as it may be useful in the early diagnosis of the disease. Introduction Contrast can be defined as the ability of the eye to discriminate differences in luminance between the stimulus and the background. The sensitivity to contrast is represented by the inverse of the minimal contrast necessary to make an object visible; the lower the contrast the greater the sensitivity, and the other way around. Contrast sensitivity is a fundamental aspect of vision together with visual acuity: the latter defines the smallest spatial detail that the subject manages to discriminate under optimal conditions, but it only provides information about the size of the stimulus that the eye is capable to perceive; instead, the evaluation of contrast sensitivity provides information not obtainable with only the measurement of visual acuity, as it establishes the minimum difference in luminance that must be present between the stimulus and its background so that the retina is adequately stimulated to perceive the stimulus itself. The clinical methods of examining contrast sensitivity (lattices, luminance gradients, variable-contrast optotypic tables and lowcontrast optotypic tables) relate the two parameters on which the ability to distinctly perceive an object depends, namely the different luminance degree of the two adjacent areas and the spatial frequency, which is linked to the size of the object. The measurement of contrast sensitivity becomes valuable in the diagnosis and follow up of some important eye conditions such as glaucoma. Studies show that contrast sensitivity can be related to data obtained with the visual perimetry, especially with the perimetric damage of the central area and of the optic nerve head