Reacting multi-component fluids -- regular solutions in Lorentz spaces

TL;DR

This paper establishes the global existence of regular solutions for a multi-component incompressible fluid model with chemical reactions within a novel Lorentz space framework, under smallness and structural conditions.

Contribution

It introduces a new critical Lorentz space framework to analyze multi-component fluid flows with chemical reactions, enabling control over solution regularity and integral bounds.

Findings

Proves global existence of solutions under smallness conditions.

Develops a new analytical framework in Lorentz spaces.

Controls the integral of the gradient norm over time.

Abstract

The paper deals with analysis of a model of a multi-component fluid admitting chemical reactions. The flow is considered in the incompressible regime. The main result shows global existence of regular solutions under assumption of suitable smallness conditions. In order to control the solutions a special structure condition on the derivatives of chemical production functions determining the reactions is required. The existence is shown in a new critical functional framework of Lorentz spaces of type $L_{p,r}(0,T;L_q)$, which allows to control the integral $\int_0^\infty \|\nabla u(t)\|_{\infty} dt$.