Theoretical and numerical analysis for an interspecies competition model with chemoattraction-consumption in fluids

TL;DR

This paper analyzes a two-species chemotaxis-Navier-Stokes system with competitive kinetics, establishing existence, regularity, and optimal numerical schemes with error estimates, supported by numerical simulations.

Contribution

It provides the first combined theoretical and numerical analysis of a chemotaxis-Navier-Stokes system with Lotka-Volterra competition, including regularity criteria and an efficient finite element scheme.

Findings

Existence of global weak solutions established.

A finite element scheme with proven optimal error estimates.

Numerical simulations confirm scheme stability and accuracy.

Abstract

This work is devoted to the theoretical and numerical analysis of a two-species chemotaxis- Navier-Stokes system with Lotka-Volterra competitive kinetics in a bounded domain of Rd, d = 2, 3. First, we study the existence of global weak solutions and establish a regularity criterion which provides sufficient conditions to ensure the strong regularity of the weak solutions. After, we propose a finite element numerical scheme in which we use a splitting technique obtained by introducing an auxiliary variable given by the gradient of the chemical concentration and applying an inductive strategy, in order to deal with the chemoattraction terms in the two-species equations and prove optimal error estimates. For this scheme, we study the well-posedness and derive some uniform estimates for the discrete variables required in the convergence analysis. Finally, we present some numerical simulations oriented to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis.