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Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

Juan Luis Vázquez (2014)

Journal of the European Mathematical Society

We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form u ( x , t ) = t α f ( | x | t β ) with suitable and β . As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov reflection...

Basic principles of mixed Virtual Element Methods

F. Brezzi, Richard S. Falk, L. Donatella Marini (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n − 1) − Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim...

BEM and FEM results of displacements in a poroelastic column

Bettina Albers, Stavros A. Savidis, H. Ercan Taşan, Otto von Estorff, Malte Gehlken (2012)

International Journal of Applied Mathematics and Computer Science

The dynamical investigation of two-component poroelastic media is important for practical applications. Analytic solution methods are often not available since they are too complicated for the complex governing sets of equations. For this reason, often some existing numerical methods are used. In this work results obtained with the finite element method are opposed to those obtained by Schanz using the boundary element method. Not only the influence of the number of elements and time steps on the...

Bipolar Barotropic Non-Newtonian Compressible Fluids

Šárka Matušu-Nečasová, Mária Medviďová-Lukáčová (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We are interested in a barotropic motion of the non-Newtonian bipolar fluids . We consider a special case where the stress tensor is expressed in the form of potentials depending on eii and ( e i j x k ) . We prove the asymptotic stability of the rest state under the assumption of the regularity of the potential forces.

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