Journal of Prime Research in Mathematics

# A degenerate hydrodynamic dispersion model

Sergey Sazhenkov
Lavrentyev Institute of Hydrodynamics, Novosibirsk 630090, Russia.

$$^{1}$$Corresponding Author: sazhenkovs@yahoo.com

Copyright © 2007 Sergey Sazhenkov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

A Cauchy problem for a two-dimensional ultra-parabolic model of filtration through a porous ground of a viscous incompressible fluid containing a solute (tracer) is considered. The fluid is driven by the buoyancy force. The phenomenon of molecular diffusion of the tracer into the porous ground is taken into account. The porous ground consists of one dimensional filaments oriented along some smooth non-degenerate vector field. Two cases are distinguished depending on spatial orientation of the filaments, and existence of generalized entropy solutions is proved for the both. In the first case, all filaments are parallel to the buoyancy (gravitational) force and, except for this, the equations of the model have rather general forms. In the second case, the filaments can be nonparallel to the buoyancy force and to each other, in general, but their geometric structure must be genuinely nonlinear. The proofs rely on the method of kinetic equation and the theory of Young measures and H-measures.

#### Keywords:

Ultraparabolic Equation, Genuine Nonlinearity, Non-Isotropic Porous Medium, Nonlinear Diffusion-Convection.