Renyi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations

We investigate the large-time asymptotics of nonlinear diffusion equations $u_t = \Delta u^p$ in dimension $n \ge 1$, in the exponent interval $p > n/(n+2)$, when the initial datum $u_0$ is of bounded second moment. Precise rates of convergence to the Barenblatt profile in terms of the relative R\'enyi entropy are demonstrated for finite-mass solutions defined in the whole space when they are re-normalized at each time $t> 0$ with respect to their own second moment. The analysis shows that the relative R\'enyi entropy exhibits a better decay, for intermediate times, with respect to the standard Ralston-Newton entropy. The result follows by a suitable use of the so-called concavity of R\'enyi entropy power

Dimensionality of Local Minimizers of the Interaction Energy

In this work we consider local minimizers (in the topology of transport distances) of the interaction energy associated to a repulsive-attractive potential. We show how the imensionality of the support of local minimizers is related to the repulsive strength of the potential at the origin.Comment: 27 page

Nonlocal interactions by repulsive-attractive potentials: radial ins/stability

In this paper, we investigate nonlocal interaction equations with repulsive-attractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some conditions on the potential, radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state. Otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state. We also investigate under which condition it is possible for a non-radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface. Finally we provide a detailed analysis of the specific case of the repulsive-attractive power law potential as well as numerical results. We point out the the conditions of radial ins/stability are sharp.Comment: 42 pages, 7 figure

Numerical Study of a Particle Method for Gradient Flows

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level, and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.Comment: 27 pages, 21 figure

Key factors and barriers to the adoption of cold ironing in europe

The first cases of successful implementation of cold ironing can be found in Alaska about twenty years ago. In that case, the energy cost was lower than in Europe where cold ironing has been developed only in the latest years at few ports. The present paper investigates the innovative process of cold ironing at European level. Firstly, its recent development in Europe is documented as well as the main concern of its corresponding legislation. Then, the adoption of this initiative by the “green ports” concept is discussed. Secondly, the technical barriers, such as lack of standardization of electricity parameters are mentioned. And given that port electrical infrastructure needed onshore represents a huge investment that not all ports are financially able to do, the financial problematic is treated explicitly taking into account the cost of energy at ports (directly provided by electric centrals or converted) against the energy cost onboard. Finally, conclusions are drawn covering the main barriers confronted by this technology and the future premises of cold ironing at European ports considering the social and environmental benefits in terms of air and noise pollution.cold ironing, energy cost, technology barrier, European ports, environmenta

Variable control tool in MATLAB for energy transformation processes

During the stages of transformation of energy in a process, exercise control over the variables that intervene in it, improve its performance, and identify undesirable conditions in these. Thus, this study is developed as a graphical interface to implement a methodology for controlling variables of energy conversion processes, such as internal combustion engines. The control tool developed in MATLAB variables is based on multivariate statistics. The methods for developing this tool of Graphic User Interface is based on the statistics of principal component analysis and failure statistics such as T! Hotelling and the Q statistic that allows the control of anomalies presented in the operation's behavior. About the methodology, first, the input data are normalized, achieving standardization of the observation matrix vs. variables, then the spectral decomposition of the normalized data is performed, reaching the generation of the matrix of auto-values, allowing the age of the projection space of the data. With this based and delimited, it is possible to establish the ranges of observation of the mentioned statisticians. The result obtained from this research corresponds to software that allows the constant observation and analysis of the behavior of each variable of the generation engine. It describes the upper limit, lower limit, arithmetic mean, principal components, graphics of the statistics, and detects the failures in real times