Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations

TL;DR

This paper proves that for entropy-dissipating reaction-diffusion equations, any weak or renormalized solution must match the strong solution if it exists, under minimal assumptions on reaction rates, extending the understanding of solution uniqueness.

Contribution

It establishes a weak-strong uniqueness principle for reaction-diffusion equations with minimal assumptions, applicable to systems satisfying detailed balance and mass-action kinetics.

Findings

Weak solutions coincide with strong solutions when the latter exist.

The result applies without growth restrictions on reaction rates.

Renormalized solutions are globally existent, but weak solutions' global existence remains open.

Abstract

We establish a weak-strong uniqueness principle for solutions to entropy-dissipating reaction-diffusion equations: As long as a strong solution to the reaction-diffusion equation exists, any weak solution and even any renormalized solution must coincide with this strong solution. Our assumptions on the reaction rates are just the entropy condition and local Lipschitz continuity; in particular, we do not impose any growth restrictions on the reaction rates. Therefore, our result applies to any single reversible reaction with mass-action kinetics as well as to systems of reversible reactions with mass-action kinetics satisfying the detailed balance condition. Renormalized solutions are known to exist globally in time for reaction-diffusion equations with entropy-dissipating reaction rates; in contrast, the global-in-time existence of weak solutions is in general still an open problem - even for smooth data - , thereby motivating the study of renormalized solutions. The key ingredient of our result is a careful adjustment of the usual relative entropy functional, whose evolution cannot be controlled properly for weak solutions or renormalized solutions.