The fractional porous medium equation on the hyperbolic space

We consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual Lp spaces or to larger (weighted) spaces determined either in terms of a ground state of the laplacian, or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative L1- L∞ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples

Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

We study qualitative and quantitative properties of local weak solutions of the fast $p$-Laplacian equation, $\partial_t u=\Delta_{p}u$, with $1<p<2$. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of \RR^n\times [0,T]. We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when $1<p \le 2n/(n+1)$. The boundedness results may be also extended to the limit case $p=1$, while the positivity estimates cannot. We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any $1<p<2$, and point out their main properties. We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely $\partial_t u\in L^2_{\rm loc}$

Atmospheric anomalies over Mt.Etna using GPS signal delays and tomography of radio wave velocities

Due to the prominent topography of Mt. Etna, the use of satellite geodetic techniques may significantly suffer from atmospheric heterogeneities. This problem mainly affects the DInSAR technique. To overcome these drawbacks the present study attempts to make headway in measuring and interpreting atmospheric anomalies. We used the GAMIT software to obtain the ZTD (Zenith Total Delay) values for the GPS sessions performed on 1996-97, during ERS-2 passes at Mt. Etna. GAMIT software also allows to characterize the statistical behaviour of the tropospheric effects, by using residuals for each station-satellite pair, and to locate the atmospheric anomalies, present mostly at low altitudes. The attempt at using these results to produce a tomography of radio waves velocity of the troposphere suggests that the number of GPS stations used to investigate atmosphere is a critical point in such a study. The three stations are too few to invert anomalies eventually existing in the lower atmosphere. This result is a good starting point for better direct future study to verify the applicability of this tomographic technique to a geodetic network with a higher number of stations, with the aim of characterizing the lower atmosphere of Mt. Etna for a more accurate monitoring of ground deformations

Remote Sensing and Geodetic Measurements for Volcanic Slope Monitoring: Surface Variations Measured at Northern Flank of La Fossa Cone (Vulcano Island, Italy)

Abstract: Results of recent monitoring activities on potentially unstable areas of the NW volcano flank of La Fossa cone (Vulcano Island, Italy) are shown here. They are obtained by integration of data by aerial photogrammetry, terrestrial laser scanning (TLS) and GPS taken in the 1996–2011 time span. A comparison between multi-temporal models built from remote sensing data (photogrammetry and TLS) highlights areas characterized by ~7–10 cm/y positive differences (i.e., elevation increase) in the upper crown of the slope. The GPS measurements confirm these results. Areas characterized by negative differences, related to both mass collapses or small surface lowering, also exist. The higher differences, positive and negative, are always observed in zones affected by higher fumarolic activity. In the 2010–2012 time span, ground motions in the northern part of the crater rim, immediately above the upper part of observed area, are also observed. The results show different trends for both vertical and horizontal displacements of points distributed along the rim, with a magnitude of some centimeters, thus revealing a complex kinematics. A slope stability analysis shows that the safety factors estimated from these data do not OPEN ACCESS Remote Sens. 2013, 5 2239 indicate evidence of possible imminent failures. Nevertheless, new time series are needed to detect possible changes with the time of the stability conditions, and the monitoring has to go on

Ground deformations related to the effusive eruptions of Stromboli: the 2002-2003 case

Stromboli volcano erupted suddenly on 28 December 2002 after a long period of typically persistent and moderate explosive activity. Lava flows outpoured from the northern wall of the NE crater and descended into the Sciara del Fuoco (SdF). On December 30th, 2002, two landslides occurred on the northern part of the SdF, producing a tsunami that caused significant damage. This event led to the upgrading of the ground deformation monitoring system. The new requisite was the real-time detection of the deformation related both to the magma movements within the eruptive feeding system and to potential slope failures of the SdF. To this end, a remotely controlled monitoring system, based both on high-frequency (1 Hz) instantaneous GPS and terrestrial geodetic techniques (manual EDM measurements, transformed in automated terrestrial geodetic measurements) was planned and set up in a few months. During the recorded eruptive phases, the new monitoring system aided the Department of Civil Protection in making decisions related to hazards from landslides and volcanic activity and, more generally, on the evolution of volcanic phenomena throughout the eruption. The measurements carried out on the benchmarks located on the high flank allowed us to make some hypotheses on the dynamics of the craters. In particular, the behaviour of the EDM baselines, showing alternating periods of increase and periods of stop in length variation, could be linked to movements of the magmatic column within the craters. Moreover, the monitoring system gave us the opportunity to observe the effects of an effusive vent opening on February 16th. The new geodetic network provided, for the first time, useful information on ground deformations due to shallow and very shallow volcanic sources at Stromboli

On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation.Comment: To appear in DCDS-

Ground deformation modeling of flank dynamics prior to the 2002 eruption of Mt. Etna

On 22 September 2002, 1 month before the beginning of the flank eruption on the NE Rift, an M-3.7 earthquake struck the northeastern part of Mt. Etna, on the westernmost part of the Pernicana fault. In order to investigate the ground deformation pattern associated with this event, a multi-disciplinary approach is presented here. Just after the earthquake, specific GPS surveys were carried out on two small sub-networks, aimed at monitoring the eastern part of the Pernicana fault, and some baselines belonging to the northeastern EDM monitoring network of Mt. Etna were measured. The leveling route on the northeastern flank of the volcano was also surveyed. Furthermore, an investigation using SAR interferometry was performed and also the continuous tilt data recorded at a high precision sensor close to the epicenter were analyzed to constrain the coseismic deformation. The results of the geodetic surveys show a ground deformation pattern that affects the entire northeastern flank of the volcano, clearly shaped by the Pernicana fault, but too strong and wide to be related only to an M-3.7 earthquake. Leveling and DInSAR data highlight a local strong subsidence, up to 7 cm, close to the Pernicana fault. Significant displacements, up to 2 cm, were also detected on the upper part of the NE Rift and in the summit craters area, while the displacements decrease at lower altitude, suggesting that the dislocation did not continue further eastward. Three-dimensional GPS data inversions have been attempted in order to model the ground deformation source and its relationship with the volcano plumbing system. The model has also been constrained by vertical displacements measured by the leveling survey and by the deformation map obtained by SAR interferometry

Seismic potential in the Italian Peninsula from integration and comparison of seismic and geodetic strain rates.

Seismological and geodetic data provide key information about the kinematics and active tectonics of plate margins. Focal solutions enable determining the directions in which the current tectonic stress acts when fault rupturing occurs; GPS measurements provide information on the crustal velocity field and on current interseismic strain rates. The comparison of the strain rates resulting from the two datasets provides further insight into how large an area is affected by aseismic deformation, which is a valuable indicator for seismic hazard mitigation and estimating the seismic potential. In this work, we investigate both seismic and geodetic strain rates and the combined field resulting from the joint inversion of the geodetic and seismic datasets, providing a picture of the overall deformation field and its variation during the last decades. In this way, we seek to give an overview of the seismic potential distribution across the Apennines and southern Italy, as a qualitative analysis of space-time variations in the released seismic strain rate, compared to the space-time distribution of the cumulated geodetic strain rate. The results show a variable distribution of the seismic efficiency over the peninsula. The Southern Apennines shows the greatest seismic potential, highlighting a significantly lower seismicity in the last two decades over an area affected by the highest total strain rates. The Messina Straits and eastern Sicily have a significant seismic potential, together with the Calabrian arc (from the Tindari-Letojanni and central Aeolian islands to the Mt. Pollino area), as a result of seismic gaps with respect to the combined strain rates in the investigated period. This long gap highlights the longer recurrence periods for the strongest earthquakes on this area. The central-northern Apennines and off-shore northern Sicily, show a lower seismic potential than central-southern Apennines, probably due to the more recent seismicity affecting these areas

Multidisciplinary geophysical study of the NE sector of the unstable flank of Etna volcano

On volcanic areas, usually characterized by complex structural environments, a lot of independent geophysical studies are usually performed. The non-uniqueness of the geophysical inverse models, the different level of reso- lution and sensitivity of the results spurred us to integrate independent geophysical datasets and results collected on Mt. Etna volcano, in order to obtain more accurate and reliable model interpretation. Mt. Etna volcano is located along the eastern coast of Sicily and it is characterized by a complex structural set- ting. In this region, the general N-S compressive regime related to the Africa – Europe collision interacts with the WNW-ESE extensional regime associated to the Malta Escarpment dynamics, observable along the eastern coast of Sicily. At Mt Etna, a great number of studies concerns the existence of instability phenomena; a general eastward mo- tion of the eastern flank of the volcano has been measured with always increasing detail and its relationship with the eruptive and magmatic activity is being investigated. The unstable flank appears bounded to the north by the E–W-trending Provenzana - Pernicana Fault System and to the SW by the NS Ragalna Fault system. Eastwards, this area is divided by several NW–SE trending faults. Recent studies consider this area as divided into several blocks characterized by different shape and kinematics. Ground deformation studies (GPS and InSAR) define the NE portion of the unstable flank as the most mobile one. In the frame of the MEDiterranean Supersites Volcanoes (MED-SUV) project, ground deformation data (GPS and INSAR), 3D seismicity, seismic tomography and two resistivity model profiles, have been analyzed together, in order to put some constraints on the deep structure of the NE sector of the unstable flank. Seismic data come from the permanent network run by the Istituto Nazionale di Geofisica e Vulcanologia (INGV) - Sezione di Catania, Osservatorio Etneo. Ground deformation data comes from InSAR Permanent Scatterers analyses of different spaceborn sensors. The resistivity models come from a MT survey carried out on the eastern flank of the volcano and consisting of thirty broad-band soundings along N-S and NW-SE oriented profiles. We found that the NE sector of the sliding volume, modeled by ground deformation data inversions and character- ized by the highest displacement velocity, is characterized low resistivity values and it is bounded by two seismic clusters. The northern one is clearly related to the Pernicana fault and it’s not deeper than 3 km b.s.l. while the second one is located southwards, beneath the northern wall of the Valle del Bove, not related to any evident struc- ture at the surface. An evident layer with very reduced seismicity lies at 3 km of depth and well corresponds to the simplified analytic models of a sliding planar surface resulting from GPS data inversions

Kinematics of the Central Mediterranean Plate Boundary, Internal Deformation of Sicily and Interseismic Strain Accumulation Across the Messina Straits

In this work we present a new velocity field, obtained by analyzing continuous GPS (CGPS) stations operating in the Mediterranean area (updated to September 2007)and epoch GPS (EGPS) stations in the 1991-2006 time span, particularly denser in the Iblean plateau and across the Messina Straits