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

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

Compagni !! tutti insieme cresciamo.

I nuovi compiti della Geomatica non sono disgiunti dallo impegno di fronte ai problemi attuali del mondo intero e, sempre più, richiedono di essere capaci di “sporcarsi le mani” e rischiare di persona. Questo lavoro sviluppa i temi del misurare la qualità, andando oltre una etica della convinzione e della responsabilità, e ripercorre il lungo cammino verso la modernità, ponendosi correttamente solo domande penultime. Lo approccio adottato si rifà allo scetticismo ed al relativismo moderati, ben consci del fallimento totale delle soluzioni “in grande”, tutte le filosofie, ideologie e religioni

A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space W1,p, where p is larger than the space dimension, under the assumption that the so-called "forward-backward integral curves" associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable

Artificially depleted plasmas are not necessarily commutable with native patient plasmas for International Sensitivity Index calibration and International Normalized Ratio derivation: a rebuttal

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

We consider the mixing behavior of the solutions to the continuity equation associated with a divergence-free velocity field. In this note 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

IMAGE-BASED RECONSTRUCTION AND ANALYSIS OF DYNAMIC SCENES IN A LANDSLIDE SIMULATION FACILITY

The application of image processing and photogrammetric techniques to dynamic reconstruction of landslide simulations in a scaled-down facility is described. Simulations are also used here for active-learning purpose: students are helped understand how physical processes happen and which kinds of observations may be obtained from a sensor network. In particular, the use of digital images to obtain multi-temporal information is presented. On one side, using a multi-view sensor set up based on four synchronized GoPro 4 Black® cameras, a 4D (3D spatial position and time) reconstruction of the dynamic scene is obtained through the composition of several 3D models obtained from dense image matching. The final textured 4D model allows one to revisit in dynamic and interactive mode a completed experiment at any time. On the other side, a digital image correlation (DIC) technique has been used to track surface point displacements from the image sequence obtained from the camera in front of the simulation facility. While the 4D model may provide a qualitative description and documentation of the experiment running, DIC analysis output quantitative information such as local point displacements and velocities, to be related to physical processes and to other observations. All the hardware and software equipment adopted for the photogrammetric reconstruction has been based on low-cost and open-source solutions

Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation

In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation \phi_y+ [\phi^{2}/2]_t=w, where w is a bounded function depending on \phi

A história da certificação ISO 9001 da Embrapa Meio Ambiente.

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