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Discrete & Continuous Dynamical Systems - S

2014 , Volume 7 , Issue 2

Issue on workshop in fluid mechanics and population dynamics

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Preface: Workshop in fluid mechanics and population dynamics
Alain Miranville , Mazen Saad and  Raafat Talhouk
2014, 7(2): i-i doi: 10.3934/dcdss.2014.7.2i +[Abstract](139) +[PDF](77.0KB)
The Workshop in Fluid Mechanics and Population Dynamics: Models, Existence Problems, Stability and Numerical Methods took place in Hadath (Lebanon), at the Laboratory of mathematics in the Doctoral School of Science and Technology of the Lebanese University, during the week September 10 - September 14, 2012
    The workshop was held with the support of the Ecole Centrale de Nantes, the Laboratoire de mathématiques Jean Leray of Nantes university and the universities of Paris 13 and Poitiers, and was organized by Samer Israwi, Basam Kojok, Ayman Mourad, Mazen Saad, Zaynab Salloum and Raafat Talhouk. It was attended by about 60 PhD students and Mathematicians coming from universities from different countries, such as Algeria, France, Lebanon and Tunisia.
    The workshop included courses tutorials and sessions around specific themes in which participants presented their works. It was open to any scientist interested in academic or industrial applications. It was aimed at doctoral students in mathematics, as well as senior researchers. Its objective was firstly to present a synthesis of various partial differential equations involved in fluid mechanics and populations dynamics, and on the other hand to present more recent advances in research topics such as asymptotic models in fluid mechanics, homogenization and the finite volume method.
    All the contributions to this special volume were submitted to regular peer review, according to the rules of the journal Discrete and Continuous Dynamical Systems - Series S. We deeply thank all the authors for their high quality papers and the Editors of the journal for their interest in the topics of the workshop.
    We express our heartfelt thanks to the following sponsors: Ecole Central de Nantes and Laboratoire de Mathématques Jean Leray, Nantes University, represented by Mazen Saad, Université de Poitiers, represented by Alain Miranville, Université Paris 13, represented by Mikhael Balabane, Lebanese National Council for Scientific Research (CNRS) and Lebanese University.
    Finally, we express our sincere thanks to the Dean of the Doctoral School of Science and Technology at the Lebanese University, Professor Zeinab Saad, for her support throughout the workshop and to the French Embassy in Beirut, represented by its Scientific Counselor, Mr. Gilles Théodet.
Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository
Florian Caro , Bilal Saad and  Mazen Saad
2014, 7(2): 191-205 doi: 10.3934/dcdss.2014.7.191 +[Abstract](32) +[PDF](424.4KB)
Our goal is the mathematical analysis of a two phase (liquid and gas) two components (water and hydrogen) system modeling the hydrogen displacement in a storage site for radioactive waste. We suppose that the water is only in the liquid phase and is incompressible. The hydrogen in the gas phase is supposed compressible and could be dissolved into the water with the Henry law. The flow is described by the conservation of the mass of each components. The model is treated without simplified assumptions on the gas density. This model is degenerated due to vanishing terms. We establish an existence result for the nonlinear degenerate parabolic system based on new energy estimate on pressures.
From particles scale to anomalous or classical convection-diffusion models with path integrals
Catherine Choquet and  Marie-Christine Néel
2014, 7(2): 207-238 doi: 10.3934/dcdss.2014.7.207 +[Abstract](135) +[PDF](709.3KB)
The present paper is devoted to the rigorous upscaling of some particles displacement model with trapping events, to the continuum scale. It focuses especially on the transitions between sub-diffusive and diffusive models. The work gives emphasis to the following points: 1. The distribution of waiting times in passing to the continuum limit. The common idea is that the distributions with slowly decaying long tails produce anomalous diffusion while the classical diffusion model corresponds to distributions with short tails. This is shown to be not always true by introducing a simple model of geometrical heterogeneity leading to trapping events without characteristic time scale. 2. The extension of the Feynman-Kac theory to some non-Brownian setting. 3. Constructing a microscopic random walk model that, thought based on a MIM approach, gives both fMIM and FFPE at the mesoscopic limit.
Shallow water asymptotic models for the propagation of internal waves
Vincent Duchêne , Samer Israwi and  Raafat Talhouk
2014, 7(2): 239-269 doi: 10.3934/dcdss.2014.7.239 +[Abstract](39) +[PDF](698.0KB)
We are interested in asymptotic models for the propagation of internal waves at the interface between two shallow layers of immiscible fluid, under the rigid-lid assumption. We review and complete existing works in the literature, in order to offer a unified and comprehensive exposition. Anterior models such as the shallow water and Boussinesq systems, as well as unidirectional models of Camassa-Holm type, are shown to descend from a broad Green-Naghdi model, that we introduce and justify in the sense of consistency. Contrarily to earlier works, our Green-Naghdi model allows a non-flat topography, and horizontal dimension $d=2$. Its derivation follows directly from classical results concerning the one-layer case, and we believe such strategy may be used to construct interesting models in different regimes than the shallow-water/shallow-water studied in the present work.
Some mathematical models in phase transition
Alain Miranville
2014, 7(2): 271-306 doi: 10.3934/dcdss.2014.7.271 +[Abstract](89) +[PDF](551.5KB)
Our aim in these notes is to discuss the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, of models in phase transition. In particular, we focus on the Caginalp phase field model.
Two-dimensional individual clustering model
Elissar Nasreddine
2014, 7(2): 307-316 doi: 10.3934/dcdss.2014.7.307 +[Abstract](30) +[PDF](364.0KB)
This paper is devoted to study a model of individual clustering with two specific reproduction rates in two space dimensions. Given $q>2$ and an initial condition in $W^{1,q}(\Omega)$, the local existence and uniqueness of solution have been shown in [6]. In this paper we give a detailed proof of existence of global solution.
Numerical analysis of a non equilibrium two-component two-compressible flow in porous media
Bilal Saad and  Mazen Saad
2014, 7(2): 317-346 doi: 10.3934/dcdss.2014.7.317 +[Abstract](28) +[PDF](545.2KB)
We propose and analyze a finite volume scheme to simulate a non equilibrium two components (water and hydrogen) two phase flow (liquid and gas) model. In this model, the assumption of local mass non equilibrium is ensured and thus the velocity of the mass exchange between dissolved hydrogen and hydrogen in the gas phase is supposed finite.
    The proposed finite volume scheme is fully implicit in time together with a phase-by-phase upwind approach in space and it is discretize the equations in their general form with gravity and capillary terms We show that the proposed scheme satisfies the maximum principle for the saturation and the concentration of the dissolved hydrogen. We establish stability results on the velocity of each phase and on the discrete gradient of the concentration. We show the convergence of a subsequence to a weak solution of the continuous equations as the size of the discretization tends to zero. At our knowledge, this is the first convergence result of finite volume scheme in the case of two component two phase compressible flow in several space dimensions.
Asymptotics of wave models for non star-shaped geometries
Farah Abou Shakra
2014, 7(2): 347-362 doi: 10.3934/dcdss.2014.7.347 +[Abstract](26) +[PDF](474.7KB)
In this paper, we provide a detailed study and interpretation of various non star-shaped geometries linking them to recent results for the 3D critical wave equation and the 2D Schrödinger equation. These geometries date back to the 1960's and 1970's and they were previously studied only in the setting of the linear wave equation.

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