Gonzalo Galiano

Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model

Summary. A positivity-preserving numerical scheme for a strongly coupled cross-diffusion model for two competing species is presented, based on a semi-discretization in time. The variables are the population densities of the species. Existence of strictly positive weak solutions to the semidiscrete problem is proved. Moreover, it is shown that the semidiscrete solutions converge to a non-negative solution of the continuous system in one space dimension. The proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional.

Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion

The Boussinesq system arises in Fluid Mechanics when motion is governed by density gradients caused by temperature or concentration differences. In the former case, and when thermodynamical coefficients are regarded as temperature dependent, the system consists of the Navier-Stokes equations and the non linear heat equation coupled through the viscosity, bouyancy and convective terms. According to the balance between specific heat and thermal conductivity the diffusion term in the heat equation may lead to a singular or degenerate parabolic equation. In this paper we prove the existence of solutions of the general problem as well as the uniqueness of solutions when the spatial dimension is two.

On the Boussinesq system with nonlinear thermal diffusion

The Boussinesq system of hydrodynamics equations, arises from a zero order approximation to the coupling between the Navier-Stokes equations and the thermodynamic equation. The presence of density gradients in a fluid means that gravitational potential energy can be converted into motion through the action of bouyant forces.

A parabolic cross-diffusion system for granular materials

We analyze a cross-diffusion system of parabolic equations for the relative concentration and the dynamic repose angle of a mixture of two different granular materials in a long rotating drum. The main feature of the system is the ability to describe the axial segregation of the two granular components. The existence of global-in-time weak solutions is shown for arbitrary large cross-diffusion by using entropy-type inequalities and approximation arguments. The uniqueness of solutions is proved if cross-diffusion is not too large. Furthermore, we derive a sufficient condition on the parameters to have nonsegregation. Finally, numerical simulations illustrate the long-time coarsening of the segregation bands in the drum.

On a quasilinear degenerate system arising in semiconductors theory. Part I: existence and uniqueness of solutions

This paper is about the drift-diffusion equations for semiconductors. Existence and uniqueness of weak solutions are obtained. The existence is proved by using the regularization technique. The proof of the uniqueness is interesting.

On a chirplet transform-based method applied to separating and counting wolf howls

We propose a parametric model based in chirp decomposition to modelize wolf chorus emissions. The problem consists on estimating the phase and amplitude of chirps corresponding to each individual, as well as the number of individuals. For the first task, we use a well known technique, the Chirplet Transform, which allows us to obtain a first order approximation of the phase, improving the zero order approximation given by the Short Time Fourier Transform (STFT). This gain in accuracy allows to use criteria for a better chirp tracking, which is specially important at crossing points and in the determination of harmonics of a fundamental tone. We explore the efficiency of the method applying it to synthetic signals and to wolves choruses recordings (original motivation of this work). The results show good performance for chirps tracking even under strong noise corruption.

On a cross-diffusion population model deduced from mutation and splitting of a single species

We deduce a particular case of the population cross-diffusion model introduced by Shigesada et al. (1979) [1] by using the ideas of mutation and splitting from a single species, as described by Sanchez-Palencia for ODEs systems Sanchez-Palencia (2011) [21]. The resulting equations of the PDE system only differ in the cross-diffusion terms, the corresponding diffusion matrix being self-diffusion dominated, which implies that the well known population segregation patterns of the Shigesada et al. model do not appear in this case. We prove existence and uniqueness of solutions of the PDE system and use a finite element approximation to discuss, numerically, stability properties of solutions with respect to the parameters in comparison with related models.

Neighborhood Filters and the Decreasing Rearrangement

Nonlocal filters are simple and powerful techniques for image denoising. In this paper, we give new insights into the analysis of one kind of them, the Neighborhood filter, by using a classical although not commonly used transformation: the decreasing rearrangement of a function. Independently of the dimension of the image, we reformulate the Neighborhood filter and its iterative variants as an integral operator defined in a one-dimensional space. The simplicity of this formulation allows to perform a detailed analysis of its properties. Among others, we prove that the filtered image is a contrast change of the original image, an that the filtering procedure behaves asymptotically as a shock filter combined with a border diffusive term, responsible for the staircaising effect and the loss of contrast.Nonlocal filters are simple and powerful techniques for image denoising. In this paper, we give new insights into the analysis of one kind of them, the Neighborhood filter, by using a classical although not commonly used transformation: the decreasing rearrangement of a function. Independently of the dimension of the image, we reformulate the Neighborhood filter and its iterative variants as an integral operator defined in a one-dimensional space. The simplicity of this formulation allows to perform a detailed analysis of its properties. Among others, we prove that the filtered image is a contrast change of the original image, an that the filtering procedure behaves asymptotically as a shock filter combined with a border diffusive term, responsible for the staircaising effect and the loss of contrast.

ON A QUASILINEAR DEGENERATE SYSTEM ARISING IN SEMICONDUCTOR THEORY. PART II: LOCALIZATION OF VACUUM SOLUTIONS

The temporal and spatial localization of vacuum sets of the solutions to the drift-diffusion equations for semiconductors is studied in this paper. It is shown that if there are vacuum sets initially then there are vacuum sets for a small time, which shows the finite propagation speed of the support of the densities. It is also shown that for a certain recombination-generation rate there is no dilation of the initial support, and under some condition on the recombination-generation rate the vacuum will develop after a certain time even if there is no vacuum initially. These results are proved based on a local energy method for free boundary problems.

Implementation of a diffusive differential reassignment method for signal enhancement : An application to wolf population counting

We investigate the combination of two PDE-based image processing techniques applied to the image produced by time-frequency representations of one-dimensional signals, such as the spectrogram. Specifically, we consider the energy transport equation associated to the lagrangian coordinates corresponding to the spectrogram differential reassignment proposed by Chassandre-Mottin, Daubechies, Auger and Flandrin as a spectrogram readability improving method, together with the image restoration model proposed by Álvarez, Lions and Morel for noise reduction and edge enhancement. Our aim is to produce a transformation of the spectrogram in which the instantaneous frequency lines are easier to track, for using it as an input for a (wolves howls) counting algorithm. After presenting the model derivation, we show some analytical properties of it, such as the existence of a unique solution and a comparison principle, and perform later a discretization to numerically investigate its performance for the cases of synthetic signals and field recorded wolves choruses. We finally compare our results with those obtained from well established techniques.