Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation

We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of initial data

Features of alterations of igneous rocks from Sierras of Guadarrama and Malagon (Spanish Central Range)

[Resumen] Se describen las características-de las alteraciones desarrolladas sobre las rocas ígneas de un sector del Sistema Central. Criterios geomorfológicos y su distinta composición mineralógica permiten suponer su origen a partir de distintas etapas de alteración. Se discuten las condiciones climáticas durante su formación, realizada siempre bajo climas poco agresivos.[Abstract] The main features of the alterations developped on igneous rocks of Central System are described. From geomorphological criteria as well as mineralogical data, different stages of weathering are been inferred. Although weathering processes have not been strong, different conditions are discussed for each stage

Morphological and structural evidences concerning the origin of sheet fractures

[Abstract] Sheet fractures are well and widely developed in massive rocks, i.e., rocks lacking other partings, and they have been discussed in the literature for more than a century. Yet there is no agreement as to their origino Two contrasted interpretations hold sway. Almost without exception, geologist adhere to the pressure release or erosional offloading hypothesis. Engineers and engineering geologist, on the other hand, interpret sheet partings as buckling, i.e., crumpling or bending out of plane, related to compressive stresses, particularly lateral stresses. After a review of nomenclature and a description of the characteristics of sheet fractures, a critique ofprevious explanations of their origin is presented. What are perceived to be critical Enes of structural and morphological evidence. bearing on the origin ofsheet fracture are next reviewed, and this is followed by a discusion of the possible origins of the structures

Numerical Simulations And Laboratory Measurements In Hydraulic Jumps

Hydraulic jump is one of the most extended and effective mechanism for hydraulic energy dissipation. Usually, hydraulic jump characteristics have been studied through physical models. Nowadays, computational fluid dynamics (CFD) are an important tool that can help to analyze and to understand complex phenomena that involve high turbulence and air entrainment cases. Free and submerged hydraulic jumps with Froude numbers from 2.9 to 5.5 are studied in a rectangular channel downstream a sluice gate. Velocity measurements with different flow rates are carried out by using Acoustic Doppler Velocimeter (ADV) and Particle Image Velocimeter (PIV) instrumentations. In this paper, laboratory measurements are used to calibrate and to validate open source and commercial CFD programs. Air-water two-phase flows are considered in the simulations. The closure problem is solved by using different turbulence models. Water depths, hydraulic jumps lengths, velocity profiles and energy dissipation rates are compared with laboratory measurements and other referenced results

Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for x\in\RR^d, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold (\RR^d,{\bf g}), with a metric ${\bf g}$ which is conformal to the standard \RR^d metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker--Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of $m$.Comment: 37 page