Page 398 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 398
MULTICOMPONENT DIFFUSION IN POROUS MEDIA (MS-42)
Gradient flow techniques for multicomponent diffusion-reaction
Martin Burger, martin.burger@fau.de
FAU Erlangen-Nürnberg, Germany
In this talk we will discuss some gradient flow approaches to reaction-diffusion systems with
nonlinear cross-diffusion, including the derivation from microscopic systems and the analysis
of the macroscopic systems in terms of renormalized solutions. We will also discuss the case of
coupled bulk-surface systems.
Well-posedness results for mixed-type PDE systems modelling
pressure-driven multicomponent flows
Pierre Etienne Druet, druet@wias-berlin.de
Weierstrass Institute Berlin, Germany
In this talk we consider the purely convective mass transport in such isothermal multicomponent
fluids for which the velocity field is negative proportional to the gradient of the thermodynamic
pressure. The equations of motion formally reduce to the Darcy law, and the main driving
mechanism is volume filling. Thus, this type of flow is mathematically related to the theory of
transport in porous media. We shall introduce a special equation of state, which allows to define
the thermodynamic pressure in a consistent way. This constitutive choice results into a system
of PDEs which, after an appropriate change of variables, consists of N-1 first-order transport
equations for the volume fractions, and one parabolic second-order equation of porous-medium-
type for the volume. We show that this system admits a unique classical or, in less smooth
geometrical settings, a unique weak solution. We shall also report on ongoing work concerning
the optimal control of the PDE–system. These results are obtained in the context of joined work
with A. Jüngel (TU Vienna) and J. Sprekels (WIAS Berlin).
Evolution nonlocal diffusion problems with Lipschitz-continuous diffusion
kernels
Gonzalo Galiano, galiano@uniovi.es
University of Oviedo, Spain
We present some recent results on the well-posedness of problems formulated as scalar or sys-
tems of scalar equations involving nonlocal (cross) diffusion and reaction terms. Lipschitz
continuous diffusion kernels are considered.
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Gradient flow techniques for multicomponent diffusion-reaction
Martin Burger, martin.burger@fau.de
FAU Erlangen-Nürnberg, Germany
In this talk we will discuss some gradient flow approaches to reaction-diffusion systems with
nonlinear cross-diffusion, including the derivation from microscopic systems and the analysis
of the macroscopic systems in terms of renormalized solutions. We will also discuss the case of
coupled bulk-surface systems.
Well-posedness results for mixed-type PDE systems modelling
pressure-driven multicomponent flows
Pierre Etienne Druet, druet@wias-berlin.de
Weierstrass Institute Berlin, Germany
In this talk we consider the purely convective mass transport in such isothermal multicomponent
fluids for which the velocity field is negative proportional to the gradient of the thermodynamic
pressure. The equations of motion formally reduce to the Darcy law, and the main driving
mechanism is volume filling. Thus, this type of flow is mathematically related to the theory of
transport in porous media. We shall introduce a special equation of state, which allows to define
the thermodynamic pressure in a consistent way. This constitutive choice results into a system
of PDEs which, after an appropriate change of variables, consists of N-1 first-order transport
equations for the volume fractions, and one parabolic second-order equation of porous-medium-
type for the volume. We show that this system admits a unique classical or, in less smooth
geometrical settings, a unique weak solution. We shall also report on ongoing work concerning
the optimal control of the PDE–system. These results are obtained in the context of joined work
with A. Jüngel (TU Vienna) and J. Sprekels (WIAS Berlin).
Evolution nonlocal diffusion problems with Lipschitz-continuous diffusion
kernels
Gonzalo Galiano, galiano@uniovi.es
University of Oviedo, Spain
We present some recent results on the well-posedness of problems formulated as scalar or sys-
tems of scalar equations involving nonlocal (cross) diffusion and reaction terms. Lipschitz
continuous diffusion kernels are considered.
396
