Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Iterative method for mass diffusion model with density dependent viscosity

Pages: 823 - 841, Volume 10, Issue 4, November 2008      doi:10.3934/dcdsb.2008.10.823

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Francisco Guillén-González - Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain (email)
Mamadou Sy - Laboratoire d'Analyse Numérique et d'Informatique (LANI), Université Gaston Berger, BP 234, Saint-Louis, Senegal (email)

Abstract: The aim of this work is to study the existence of strong solutions for $3D$ fluids models with mass diffusion (also called Kazhikhov-Smagulov type system) assuming density dependent viscosity. The considered system represents a pollutant model.
    We use an iterative method to approach regular solutions. Moreover, some convergence rates are obtained, depending on weak, strong and more regular norms. This work extend to [1], where this technique has been used for the model with constant viscosity.
    The model has a diffusive operator $-\lambda$div$(\rho (\nabla v +\nabla v^t))$ with $v$ the velocity field, which not allows us to use direct Stokes regularity (as has been done in [1]. Thus, it becomes more difficult to obtain the $H^2\times H^1$ and $H^3\times H^2$ regularity for the velocity-pressure pair $(v,p)$. The key is to use a new regularity result for a Stokes type problem with $\rho\Delta v$ as diffusion term.

Keywords:  Iterative method, Strong solutions, Stokes regularity, Convergence rates.
Mathematics Subject Classification:  Primary: 35Q30, 35Q35 ; Secondary: 76D05, 76R50.

Received: February 2008;      Revised: June 2008;      Available Online: August 2008.