Abstract
The uniqueness of bounded weak solutions to strongly coupled parabolic equations in a bounded domain with no-flux boundary conditions is shown. The equations include cross-diffusion and drift terms and are coupled self-consistently to the Poisson equation. The model class contains special cases of the Maxwell–Stefan equations for gas mixtures, generalized Shigesada–Kawasaki–Teramoto equations for population dynamics, and volume-filling models for ion transport. The uniqueness proof is based on a combination of the H^{-1} technique and the entropy method of Gajewski.
Highlights
Several techniques have been developed for the analysis of nonlinear parabolic systems, including sufficient conditions for the global existence of weak or strong solutions [3,18,22,29]
We prove the uniqueness of bounded weak solutions to a class of cross-diffusion systems
The proof is based on a combination of the H −1 technique and the method of Gajewski [14], where a certain semimetric measures the distance between two solutions
Summary
Several techniques have been developed for the analysis of nonlinear parabolic systems, including sufficient conditions for the global existence of weak or strong solutions [3,18,22,29]. The proof of uniqueness of weak solutions is generally much more delicate, in particular for strongly coupled systems. We prove the uniqueness of bounded weak solutions to a class of cross-diffusion systems. The proof is based on a combination of the H −1 technique and the method of Gajewski [14], where a certain semimetric measures the distance between two solutions. It is shown that the semimetric is related to relative entropies
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