[Paper Review] Global renormalized solutions to reaction-cross-diffusion systems

Xiuqing Chen, Ansgar Jüngel|arXiv (Cornell University)|Nov 4, 2017

Mathematical and Theoretical Epidemiology and Ecology Models12 references3 citations

TL;DR

This paper establishes the global-in-time existence of renormalized solutions for reaction-cross-diffusion systems with non-diagonal, non-symmetric diffusion matrices and arbitrary reaction kinetics, leveraging an entropy structure and a generalized boundedness-by-entropy approach. The key contribution is extending J. Fischer's renormalized solution framework to strongly coupled, population-model-type systems without growth restrictions on reaction terms.

ABSTRACT

The global-in-time existence of renormalized solutions to reaction-cross-diffu-sion systems for an arbitrary number of variables in bounded domains with no-flux boundary conditions is proved. The cross-diffusion part describes the segregation of population species and is a generalization of the Shigesada-Kawasaki-Teramoto model. The diffusion matrix is not diagonal and generally neither symmetric nor positive semi-definite, but the system possesses a formal gradient-flow or entropy structure. The reaction part includes reversible reactions of mass-action kinetics and does not obey any growth condition. The existence result generalizes both the condition on the reaction part required in the boundedness-by-entropy method and the proof of J. Fischer for reaction-diffusion systems with diagonal diffusion matrices.

Motivation & Objective

• To establish the existence of global-in-time renormalized solutions for strongly coupled reaction-cross-diffusion systems with non-diagonal diffusion matrices.
• To remove the standard growth restrictions on reaction terms that typically prevent the definition of weak solutions in such systems.
• To extend the boundedness-by-entropy method to systems with non-symmetric, non-positive semi-definite diffusion matrices.
• To generalize Fischer's renormalized solution framework from diagonal to cross-diffusion systems with complex coupling.
• To ensure nonnegativity and stability of solutions through entropy structure and quasi-positivity conditions on reaction terms.

Proposed method

• Introduces a renormalized formulation of the reaction-cross-diffusion system using a truncation function $\varphi^L(u)$ to control large densities.
• Employs a Lyapunov functional based on entropy density $h(u) = \sum \pi_i h_i(u_i)$, where $h_i(s) = s(\log s - 1 + \lambda_i) + e^{-\lambda_i}$, to control the system's long-time behavior.
• Applies the entropy structure via the condition $\sum \pi_i f_i(u)(\log u_i + \lambda_i) \leq 0$, ensuring quasi-positivity and entropy dissipation.
• Imposes either the weak cross-diffusion condition $\alpha > 0$ or the detailed-balance condition $\pi_i a_{ij} = \pi_j a_{ji}$ to ensure the entropy structure holds.
• Uses a sequence of truncated solutions $u^L$ with bounded entropy and applies the compensated compactness method to pass to the limit $L \to \infty$.
• Applies Lebesgue's dominated convergence theorem to pass to the limit in the renormalized formulation, proving convergence to a renormalized solution.

Experimental results

Research questions

• RQ1 Can global renormalized solutions be constructed for reaction-cross-diffusion systems with non-diagonal, non-symmetric diffusion matrices and arbitrary reaction kinetics?
• RQ2 Does the boundedness-by-entropy method extend to systems where the diffusion matrix is neither symmetric nor positive semi-definite?
• RQ3 Can the renormalized solution framework of Fischer be generalized to strongly coupled, population-type models with cross-diffusion?
• RQ4 Is it possible to remove growth restrictions on reaction terms while still ensuring existence and nonnegativity of solutions?
• RQ5 What conditions on the diffusion coefficients and reaction terms ensure the existence of a Lyapunov functional (entropy) for the full system?

Key findings

• Global-in-time renormalized solutions exist for reaction-cross-diffusion systems with arbitrary $n \geq 2$ species, bounded domains, and no-flux boundary conditions.
• The reaction terms $f_i(u)$ are only required to be continuous on $[0,\infty)^n$, with no growth condition, extending beyond the classical linear-growth restriction.
• The entropy structure is preserved under the weak formulation, with the entropy density $h(u)$ serving as a Lyapunov functional via the condition $\sum \pi_i f_i(u)(\log u_i + \lambda_i) \leq 0$.
• The diffusion matrix $A(u)$ is not required to be symmetric or positive semi-definite, but the weak cross-diffusion condition $\alpha > 0$ or detailed-balance $\pi_i a_{ij} = \pi_j a_{ji}$ ensures entropy dissipation.
• The limit $L \to \infty$ in the truncated problem yields a solution satisfying the renormalized formulation, with convergence of all terms in the weak formulation.
• The method successfully handles the lack of control on reaction terms by using renormalization and entropy-based a priori estimates, avoiding the need for strong compactness on $f_i(u)$.