Fractional semilinear Neumann problems arising from a fractional Keller--Segel model

We consider the following fractional semilinear Neumann problem on a smooth bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases} (-\varepsilon\Delta)^{1/2}u+u=u^{p},&\hbox{in}~\Omega,\\ \partial_\nu u=0,&\hbox{on}~\partial\Omega,\\ u>0,&\hbox{in}~\Omega, \end{cases}$$ where $\varepsilon>0$ and $1<p<(n+1)/(n-1)$. This is the fractional version of the semilinear Neumann problem studied by Lin--Ni--Takagi in the late 80's. The problem arises by considering steady states of the Keller--Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variational techniques, we prove existence of nonconstant smooth solutions for small $\varepsilon$, which are obtained by minimizing a suitable energy functional. In the case of large $\varepsilon$ we obtain nonexistence of nonconstant solutions. It is also shown that as $\varepsilon\to0$ the solutions $u_\varepsilon$ tend to zero in measure on $\Omega$, while they form spikes in $\overline{\Omega}$. The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest

Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics

We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as $t\to\infty$

Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics

We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as t → ∞

The BLEMAB European project: Muon radiography as an imaging tool in the industrial field

The European project called BLEMAB (BLast furnace stack density Estimation through on-line Muons ABsorption measurements), provides for the application of the muon radiography technique in the industrial environment. The project represents a non-invasive way of monitoring a blast furnace and in particular aims to study the geometric and density development of the so-called “cohesive zone”, which is important for the performance of the blast furnace itself. The installation of the detectors is expected in 2022 at the ArcelorMittal site in Bremen (Germany). This paper describes the status of the project, the experimental setup and the first results obtained with preliminary simulations. © 2022 Societa Italiana di Fisica. All rights reserved

Imaging of the Inner Zone of Blast Furnaces Using MuonRadiography: The BLEMAB Project

The aim of the BLEMAB project (BLast furnace stack density Estimation through online Muons ABsorption measurements) is the application of muon radiography techniques, to image a blast furnace’s inner zone. In particular, the goal of the study is to characterize the geometry and size of the so-called “cohesive zone”, i.e., the spatial region where the slowly downward-moving material begins to soften and melt, which plays such an important role in the performance of the blast furnace itself. Thanks to the high penetration power of natural cosmic-ray muon radiation, muon transmission radiography could be an appropriate non invasive methodology for the imaging of large high-density structures such as a blast furnace, whose linear dimensions can be up to a few tens of meters. A state-of-the-art muon tracking system is currently in development and will be installed at a blast furnace on the ArcelorMittal site in Bremen (Germany), where it will collect data for a period of various months. In this paper, the status of the project and the expectations based on preliminary simulations are presented and briefly discussed

Retention of chromium by modified Al-Bentonite Retenção de cromo por Al-bentonita modificada

Retention of chromium (III) from a tanning wastewater by modified Al-bentonites was studied. One bentonite from San Juan province, Argentina, was used. Al-bentonite was prepared by contact of bentonite with hydrolyzed OH-Al solutions (0.10 M in Al) for 24 hours. The modified Al-bentonites were obtained by: a) treatment with 0.5 M sodium chloride; b) with 0.5 M sodium chloride adjusted at pH 8; and c) treatment with an hexametaphosphate solution after sodium addition. Then, the samples were dried at 100 °C and heated at 500 °C. The chromium (III) retention by samples was carried out in batch system putting in contact the material with a 2000 ppm Cr tannery waste at different times. The retained chromium was characterized by analyzing the supernatant using UV-visible spectroscopy. The different chromium retention was correlated with structural characteristics of the solids.<br>Foi estudada a retenção de cromo (III) de águas residuais por meio de Al-bentonitas modificadas. Foi usada uma bentonita da província de San Juan, Argentina. As bentonitas-Al forma preparadas colocando-as em contato com soluções (0,10 M Al) hydrolizadas de OH-Al durante 24 horas. As bentonitas-Al modidicadas foram obtidas por meio de: a) tratamento com cloreto de sódio 0,5 M; b) com cloreto de sódio 0,5 M sodium com pH ajustado para 8; e c) tratamento com uma solução de hexametafosfato após a adição de sódio. As amostras foram então secas a 100 °C e aquecidas a 500 °C. A retenção do cromo (III) pelas amostras foi feita em lotes colocando o material em contato em diferentes tempos com um resíduo contendo 2000 ppm de cromo. O cromo retido foi caracterizado por meio de análise do sobrenadante usando espectroscopia UV-visível. As diferentes retenções de cromo foram correlacionadas com características estruturais dos sólidos

SYMMETRIZATION FOR FRACTIONAL NONLINEAR ELLIPTIC PROBLEMS

In this note we prove a new symmetrization result, in the form of mass concentration comparison, for solutions of nonlocal nonlinear Dirichlet problems involving fractional p Laplacians. Some regularity estimates of solutions will be established as a direct application of the main result

Comparison results for solutions of nonlinear parabolic equations

We prove a comparison result for the solutions of Cauchy–Dirichlet problems for a nonlinear parabolic equation. Using Schwarz spherical symmetrization, we compare the concentration of solutions to such problems with the concentration of solutions to conveniently symmetrized problems. The result takes into account, in a sharp form, the influence of the zero-order term

Comparison and regularity results for the fractional Laplacian via symmetrization methods

In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well-known ones for the classical Laplacian. (C) 2012 Elsevier Inc. All rights reserved