Exponential self-similar mixing and loss of regularity for continuity equations

We consider the mixing behaviour of the solutions of the continuity equation associated with a divergence-free velocity field. In this announcement we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the continuity equation with Sobolev but non-Lipschitz velocity field exhibiting instantaneous loss of any fractional Sobolev regularity.Comment: 8 pages, 3 figures, statement of Theorem 11 slightly revise

Singular kernels, multiscale decomposition of microstructure, and dislocation models

We consider a model for dislocations in crystals introduced by Koslowski, Cuiti\~no and Ortiz, which includes elastic interactions via a singular kernel behaving as the $H^{1/2}$ norm of the slip. We obtain a sharp-interface limit of the model within the framework of $\Gamma$-convergence. From an analytical point of view, our functional is a vector-valued generalization of the one studied by Alberti, Bouchitt\'e and Seppecher to which their rearrangement argument no longer applies. Instead we show that the microstructure must be approximately one-dimensional on most length scales and exploit this property to derive a sharp lower bound

Exponential self-similar mixing by incompressible flows

We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case $s=1$ and $1 \leq p \leq \infty$ (including the case of Lipschitz continuous velocities, and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.Comment: To appear in Journal of the American Mathematical Society. Some results were announced in G. Alberti, G. Crippa, A. L. Mazzucato, "Exponential self-similar mixing and loss of regularity for continuity equations", C. R. Math. Acad. Sci. Paris, 352(11):901--906, 2014, arXiv:1407.2631v

Damage as Gamma-limit of microfractures in anti-plane linearized elasticity

A homogenization result is given for a material having brittle inclusions arranged in a periodic structure. <br/> According to the relation between the softness parameter and the size of the microstructure, three different limit models are deduced via Gamma-convergence. <br/> In particular, damage is obtained as limit of periodically distributed microfractures

A simple abstraction of arrays and maps by program translation

We present an approach for the static analysis of programs handling arrays, with a Galois connection between the semantics of the array program and semantics of purely scalar operations. The simplest way to implement it is by automatic, syntactic transformation of the array program into a scalar program followed analysis of the scalar program with any static analysis technique (abstract interpretation, acceleration, predicate abstraction,.. .). The scalars invariants thus obtained are translated back onto the original program as universally quantified array invariants. We illustrate our approach on a variety of examples, leading to the " Dutch flag " algorithm

Coherent transport of atomic wave packets in amplitude-modulated vertical optical lattices

We report on the realization of dynamical control of transport for ultra-cold Sr88 atoms loaded in an accelerated and amplitude-modulated 1D optical lattice. We tailor the energy dispersion of traveling wave packets and reversibly switch between Wannier-Stark localization and driven transport based on coherent tunneling. Within a Loschmidt-echo scheme where the atomic group velocities are reversed at once, we demonstrate a novel mirror for matter waves working independently of the momentum state and discuss possible applications to force measurements at micrometric scales

Reduction on characteristics for continuous solutions of a scalar balance law

We consider continuous solutions u to the balance equation ∂t u(t, x) + ∂x [f (u(t, x))] = g(t, x) f ∈ C 2 (R), g ∈ L∞ (R) for a bounded source term g. Continuity improves to H ̈lder continuity o when f is uniformly convex, but it is not more regular in general. We discuss the reduction to ODEs on characteristics, mainly based on the joint works [5, 1]. We provide here local regularity results holding in the region where f (u)f (u) = 0 and only in the simpler case of autonomous sources g = g(x), but for solutions u(t, x) which may depend on time. This corresponds to a local regularity result, in that region, for the system of ODEs γ(t) = f (u(t, γ(t))) ̇ d u(t, γ(t)) = g(t, γ(t)). d

SBV regularity for Hamilton-Jacobi equations in Rn\mathbb R^n

In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$\partial_t u + H(D_{x} u)=0 \qquad \textrm{in} \Omega\subset \mathbb R\times \mathbb R^{n} .$$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb R^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.Comment: 15 page

Poblaciones y comunidades de algas bentónicas en la costa catalana

Las poblaciones de algas bentónicas forman comunidades que en unos lugares están poco diferenciadas y en otros constituyen comunidades definidas de un cierto valor indicativo. En este trabajo se comentan diversos perfiles de la costa catalana, confeccionadas según esquemas tomados sobre el terreno y auxiliados por fotografías submarinas.Los perfiles, tomados en varias localidades de la costa, revelan algunos horizontes y facies característicos del Mediterráneo occidental, lo que nos permite a modo de síntesis, tabular según su exposición al oleaje y a la luz, las facies más conspicuas y mejor caracterizadas.The populations of benthic algae form communities which are little differenciated in some spots while in others they constitute definite communities with some indicative value. Several profiles of the catalan coast are treated in this paper they have been drawn in accordance with outlines taken on the ground and with the aid of submarine pictures.The profiles, taken at several spots of the coast, reveal some horizonts and facieswhich are characteristic of the west Mediterranean, this allows us to tabulate the facies more evidents and better characterized according to their exposure to the swell and to the light

Experiences of health communication within the family: Parent and adolescent perspectives

Background: Adolescence is a critical period for the development of health and well-being and is a pivotal time for transition toward independent health decision making. Family health communication patterns, can influence behaviors and attitudes of adolescents. Health literacy is a pre-requisite for acquiring and comprehending health information. However, minimal research exists examining the connection between health literacy and family communication. Purpose: To examine the risk of limited health literacy and explore the experiences of health communication among families. Methods: 9 Adolescents and 8 parents completed demographic questionnaires, the Newest Vital Sign and private semi-structured qualitative interviews. Glazer’s grounded theory approach was used to identify thematic categories and theoretic cores of the qualitative data. Results: Limited health literacy risk was 12.5% for parents and 22/2% for adolescents. Narrow definitions of health, evaluation methods of information, opportunity and interpretation were the identified themes. Conclusions: Results from our study suggest that within families who are not at risk for limited health literacy, health communication is occurring related to topics of interest to adolescents. These conversations persist despite parental perceptions of these conversations being negative. Further research should investigate parent motivation related to health communication and the role that health literacy may play in motivation toward and ability to engage in family health communication