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Mathematical analysis of a two-phase parabolic free boundary problem derived from a Bingham-type model with visco-elastic core

A. Farina, L. Fusi (2005)

Bollettino dell'Unione Matematica Italiana

In this paper we consider a two-phase one-dimensional free boundary problem for the heat equation, arising from a mathematical model for a Bingham-like fluid with a visco-elastic core. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Existence of classical solution is proved using a fixed point argument based on Schauder's theorem. Uniqueness is proved using a technique based on a...

Mathematical model of tumour cord growth along the source of nutrient

S. Astanin, A. Tosin (2010)

Mathematical Modelling of Natural Phenomena

A mathematical model of the tumour growth along a blood vessel is proposed. The model employs the mixture theory approach to describe a tissue which consists of cells, extracellular matrix and liquid. The growing tumour tissue is supposed to be surrounded by the host tissue. Tumours where complete oxydation of glucose prevails are considered. Special attention is paid to consistent description of oxygen consumption and growth processes based on the energy balance. A finite difference numerical...

Maximal regularity and viscous incompressible flows with free interface

Senjo Shimizu (2008)

Banach Center Publications

We consider a free interface problem for the Navier-Stokes equations. We obtain local in time unique existence of solutions to this problem for any initial data and external forces, and global in time unique existence of solutions for sufficiently small initial data. Thanks to global in time L p - L q maximal regularity of the linearized problem, we can prove a global in time existence and uniqueness theorem by the contraction mapping principle.

Modeling Non-Stationary Processes of Diffusion of Solute Substances in the Near-Bottom Layer ofWater Reservoirs: Variation of the Direction of Flows and Assessment of Admissible Biogenic Load

V. V. Kozlov (2009)

Mathematical Modelling of Natural Phenomena

The paper is devoted to mathematical modelling and numerical computations of a nonstationary free boundary problem. The model is based on processes of molecular diffusion of some products of chemical decomposition of a solid organic substance concentrated in bottom sediments. It takes into account non-stationary multi-component and multi-stage chemical decomposition of organic substances and the processes of sorption desorption under aerobic and anaerobic conditions. Such a model allows one to...

Non linear phenomena in glaciology: ice-surging and streaming.

Emanuele Schiavi, Ana Isabel Muñoz, Ultano Kindelán (2002)

RACSAM

En estas notas presentamos algunos modelos físicos que han sido propuestos recientemente para tratar el problema de los movimientos repentinos y casi periódicos del hielo, así como la aparición de corrientes de hielo rápidas en los grandes mantos glaciares que se deslizan sobre lechos blandos y deformables. Estos fenómenos están relacionados con la transición de un régimen de flujo lento a uno rápido y pueden aparecer debido a una modificación del sistema de drenaje del glaciar. Los fenómenos en...

Nonlinear compressible vortex sheets in two space dimensions

Jean-François Coulombel, Paolo Secchi (2008)

Annales scientifiques de l'École Normale Supérieure

We consider supersonic compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. The problem is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. Nevertheless, we prove the local existence of such piecewise smooth solutions to the Euler equations. Since the a priori estimates for the linearized...

Numerical Approximations of the Relative Rearrangement: The piecewise linear case. Application to some Nonlocal Problems

Jean-Michel Rakotoson, Maria Luisa Seoane (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We first prove an abstract result for a class of nonlocal problems using fixed point method. We apply this result to equations revelant from plasma physic problems. These equations contain terms like monotone or relative rearrangement of functions. So, we start the approximation study by using finite element to discretize this nonstandard quantities. We end the paper by giving a numerical resolution of a model containing those terms.

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