A note on the uniqueness of weak solutions to a class of cross-diffusion systems

Abstract

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.

From Nonlocal to Classical Shigesada--Kawasaki--Teramoto Systems: Triangular Case with Bounded Coefficients

“…The duality lemma is an a priori estimate inspired by the papers [32,37] (see [6] for improved versions), which allows in one go to justifiy the integrability of each of the nonlinearities of the system (1) (and is thus useful to handle concentration issues in the approximation process); together with the entropy estimate (allowing for gradient estimates and thus strong compactness) they form the cornerstone of the global existence results obtained in [15,16,27]. Let us finally mention a recent result of Chen and Jüngel concerning the uniqueness of weak solutions: [10].…”

Weak–strong uniqueness of renormalized solutions to reaction–cross-diffusion systems

“…The reaction part does not obey any growth condition which makes it necessary to use the concept of renormalized solutions like in [17]. The uniqueness of weak solutions to cross-diffusion systems is a very delicate topic, and there are very few results only for special problems; we refer to [7] and references therein. In this work, we show a weak-strong uniqueness result for the population cross-diffusion system.…”

“…which can be seen as a generalized distance between a renormalized solution u and a strong solution v. There is a relation between Gajewski's semimetric and the relative entropy; see the discussion in [7,Remark 4]. To simplify the following formal arguments (which are made rigourous in section 3), we set b i = 0, λ i = 0, and π i = 1.…”

Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

“…The difficulty here is that such an effect may lead to unexpected behaviour, see e.g. [15]. Finally, the study of a problem with cross-diffusion is motivated by the fact that parabolic problems with cross-diffusion play a crucial role in biological applications like epidemic diseases, chemotaxis phenomena, cancer growth and population development, cf.…”