Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient
- Druet, Pierre-Étienne
- Jüngel, Ansgar
2010 Mathematics Subject Classification
- 35K45 35L65 35Q79 35M31 35Q92 92C17
- Parabolic-hyperbolic system, cross diffusion, fluid mixture, existence of solutions, transport equation
The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy?s law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.