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Subject: positive solutions × WGL Institute: WIAS ×

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    Modeling of drift-diffusion systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Stephan, Holger
    We derive drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, non-degenerate diffusivity matrices turn out to be diagonal, or
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    Positivity and time behavior of a general linear evolution system, non-local in space and time
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Khrabustovskyi, Andrii; Stephan, Holger
    We consider a general linear reaction-diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle, and positivity of the solution, and investigate its asymptotic behavior. Moreover, we give an explicite expression of the limit of the solution for large times. In order to obtain these results we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution of a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction-diffusion system what allows us to investigate its properties.