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Some recent results on the Muskat problem

Angel Castro, Diego Córdoba, Francisco Gancedo (2010)

Journées Équations aux dérivées partielles

We consider the dynamics of an interface given by two incompressible fluids with different characteristics evolving by Darcy’s law. This scenario is known as the Muskat problem, being in 2D mathematically analogous to the two-phase Hele-Shaw cell. The purpose of this paper is to outline recent results on local existence, weak solutions, maximum principles and global existence.

Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit

Giovanni Naldi, Lorenzo Pareschi, Giuseppe Toscani (2003)

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

In this paper we introduce numerical schemes for a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction equations in the so-called quasi elastic limit. To this aim we introduce a Fourier spectral method for the Boltzmann equation [25, 26] and show that the kernel modes that define the spectral method have the correct...

Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit

Giovanni Naldi, Lorenzo Pareschi, Giuseppe Toscani (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we introduce numerical schemes for a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction equations in the so-called quasi elastic limit. To this aim we introduce a Fourier spectral method for the Boltzmann equation [CITE] and show that the kernel modes that define the spectral method have the correct...

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