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On the steady-states of a two-species non-local cross-diffusion model

Rodríguez Nancy, Hu Yi

Journal of Applied Analysis

2020

Vol. 26, nr 1

1--19

We investigate the existence and properties of steady-state solutions to a degenerate, non-local system of partial differential equations that describe two-species segregation in homogeneous and heterogeneous environments. This is accomplished via the analysis of the existence and non-existence of global minimizers to the corresponding free energy functional. We prove that in the spatially homogeneous case global minimizers exist if and only if the mass of the potential governing the intra-species attraction is sufficiently large and the support of the potential governing the interspecies repulsion is bounded. Moreover, when they exist they are such that the two species have disjoint support, leading to complete segregation. For the heterogeneous environment we show that if a sub-additivity condition is satisfied then global minimizers exists. We provide an example of an environment that leads to the sub-additivity condition being satisfied. Finally, we explore the bounded domain case with periodic conditions through the use of numerical simulations.

Effect of cross diffusion on the onset of double - diffusive convection in a porous medium

Malashetty M. S., Gaikwad S. N.

International Journal of Applied Mechanics and Engineering

2001

Vol. 6, no 3

675-691

The effect of cross diffusion and the horizontal gradients of temperature and salinity on double diffusive convection in a fluid saturated densely packed porous medium is investigated using an analytical approach. The condition for the onset of convection through stationary and oscillatory modes has been reported. The effect of Soret, Dufour parameters, and horizontal gradients of temperature and salinity on the stability of the system is shown graphically. It is shown that the maximum growth rate of instability, the slope of the wave front depend on both Soret and Dufour coefficients. Some of the known results of the former problems are discussed as special cases.