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Weak-strong uniqueness for general cross-diffusion systems with volume filling

Maria Heitzinger, Ansgar Jüngel

TL;DR

The paper establishes a general weak-strong uniqueness result for cross-diffusion systems with volume filling by exploiting a Boltzmann entropy structure to guarantee bounded weak solutions and a unifying relative entropy framework. A central advancement is augmenting the mobility matrix to an enlarged system and projecting onto a subspace where the augmented mobility is positive definite, enabling a clean relative entropy estimate that yields Gronwall-type stability between a weak and a strong solution. Under hypotheses (H1)–(H5), any weak solution coinciding with a positive strong solution at initial time must coincide for all later times, and the authors verify the framework across several models including scalar diffusion, multiphase tissue dynamics, tumor growth, modified Busenberg–Travis systems, Maxwell–Stefan diffusion, and thin-film solar cells; extensions to reaction terms and certain ion-channel settings are discussed. The results provide a broad, robust tool for proving uniqueness in degenerate, non-symmetric cross-diffusion systems with volume-filling constraints, with implications for diverse physical and biological applications.

Abstract

The weak-strong uniqueness of solutions to a broad class of cross-diffusion systems with volume filling is established. In general, the diffusion matrices are neither symmetric nor positive definite. This issue is overcome by supposing that the equations possess a Boltzmann-type entropy structure, which ensures the existence of bounded weak solutions. In this framework, general conditions on the mobility matrix are identified that allow for the proof of the weak-strong uniqueness property by means of the relative entropy method. The core idea consists in analyzing an augmented mobility matrix that is positive definite only on a specific subspace. Several examples that meet the required assumptions are provided, together with a discussion on possible extensions.

Weak-strong uniqueness for general cross-diffusion systems with volume filling

TL;DR

The paper establishes a general weak-strong uniqueness result for cross-diffusion systems with volume filling by exploiting a Boltzmann entropy structure to guarantee bounded weak solutions and a unifying relative entropy framework. A central advancement is augmenting the mobility matrix to an enlarged system and projecting onto a subspace where the augmented mobility is positive definite, enabling a clean relative entropy estimate that yields Gronwall-type stability between a weak and a strong solution. Under hypotheses (H1)–(H5), any weak solution coinciding with a positive strong solution at initial time must coincide for all later times, and the authors verify the framework across several models including scalar diffusion, multiphase tissue dynamics, tumor growth, modified Busenberg–Travis systems, Maxwell–Stefan diffusion, and thin-film solar cells; extensions to reaction terms and certain ion-channel settings are discussed. The results provide a broad, robust tool for proving uniqueness in degenerate, non-symmetric cross-diffusion systems with volume-filling constraints, with implications for diverse physical and biological applications.

Abstract

The weak-strong uniqueness of solutions to a broad class of cross-diffusion systems with volume filling is established. In general, the diffusion matrices are neither symmetric nor positive definite. This issue is overcome by supposing that the equations possess a Boltzmann-type entropy structure, which ensures the existence of bounded weak solutions. In this framework, general conditions on the mobility matrix are identified that allow for the proof of the weak-strong uniqueness property by means of the relative entropy method. The core idea consists in analyzing an augmented mobility matrix that is positive definite only on a specific subspace. Several examples that meet the required assumptions are provided, together with a discussion on possible extensions.

Paper Structure

This paper contains 18 sections, 5 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. State of the art
  4. Key ideas
  5. Main result
  6. Preliminaries
  7. Proof of Theorem \ref{['thm.wsu']}
  8. Examples
  9. Scalar diffusion model
  10. Multiphase model for biological tissues
  11. Tumor-growth model
  12. Modified Busenberg--Travis model
  13. Two-species Maxwell--Stefan system
  14. Thin-film solar-cell model
  15. Extensions
  16. ...and 3 more sections

Key Result

Theorem 1

Let Hypotheses (H1)--(H5) hold, let $u$ be a weak solution to 1.eq--1.bic satisfying 1.ei and let $v$ be a positive strong solution. If the initial data coincide then $u(t)=v(t)$ in $\Omega$ for $t>0$.

Theorems & Definitions (10)

  • Definition 1: Weak solution
  • Definition 2: Strong solution
  • Theorem 1: Weak--strong uniqueness
  • Lemma 2
  • proof
  • Lemma 3: Relative entropy inequality
  • proof
  • Lemma 4
  • Lemma 5: Entropy inequality
  • proof