# Improved duality estimates: time discrete case and applications to a class of cross-diffusion systems

## TL;DR

This paper develops a time-discrete version of improved duality estimates to establish new global existence results for a class of cross-diffusion systems with bounded pressures and superquadratic reactions.

## Contribution

It introduces a novel time-discrete adaptation of duality estimates and applies it to prove global existence for complex cross-diffusion systems.

## Key findings

- Established global existence for systems with bounded cross diffusion pressures.
- Extended duality estimates to a time-discrete framework.
- Applied results to systems with superquadratic reactions.

## Abstract

We adapt the improved duality estimates for bounded coefficients derived by Canizo et al. to the framework of cross diffusion. Since the estimates can not be directly applied we need to derive a time discrete version of their results and apply it to an implicit semi-discretization in time of the cross diffusion systems. This leads to new global existence results for cross diffusion systems with bounded cross diffusion pressures and potentially superquadratic reaction.