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Planar Gradient Diffusion System to Investigate Chemotaxis in a 3D Collagen Matrix

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Published on: June 12, 2015

Gradient structures and geodesic convexity for reaction-diffusion systems.

Matthias Liero¹, Alexander Mielke

¹Weierstraß-Institut für Angewandte Analysis und Stochastik, , Mohrenstraße 39, 10117 Berlin, Germany.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|November 20, 2013

Summary

This study treats reaction-diffusion systems as gradient systems using an entropy functional and Onsager operator. Novel differential methods establish geodesic λ-convexity, applicable to drift-diffusion systems.

Keywords:

Onsager operator Wasserstein metric geodesic convexity gradient structures reaction–diffusion system relative entropy

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Area of Science:

Mathematical Physics
Chemical Kinetics
Nonlinear Dynamics

Background:

Reaction-diffusion systems are fundamental in modeling various phenomena.
Understanding their behavior often involves thermodynamic principles and gradient flows.
Existing methods for analyzing these systems can be mathematically intensive.

Purpose of the Study:

To frame reaction-diffusion systems as gradient systems.
To develop new methods for analyzing the entropy functional's properties.
To demonstrate the theory's applicability to practical systems.

Main Methods:

Utilizing an entropy functional and a dissipation metric defined by an Onsager operator.
Employing purely differential methods to establish geodesic λ-convexity.
Analyzing the interplay between diffusion (Wasserstein type) and reaction terms.

Main Results:

The entropy functional is shown to be geodesically λ-convex.
Differential methods successfully circumvent the need for mass transportation arguments.
The developed theory is applicable to a range of systems.

Conclusions:

The gradient system approach provides a powerful framework for reaction-diffusion equations.
The novel differential methods offer an alternative to existing analytical techniques.
The theory's utility is confirmed through examples like drift-diffusion systems.