Partial suitable solutions for the micropolar equations and regularity properties

TL;DR

This paper introduces the concept of partial suitable solutions for the micropolar equations, demonstrating regularity results and Hölder continuity for velocity and microrotation fields under certain conditions.

Contribution

It proposes a new notion of partial suitable solutions focusing on the velocity field and establishes regularity and Hölder continuity for both variables.

Findings

Hölderian gain for velocity and microrotation fields

Regularity results for weak solutions of micropolar equations

Introduction of partial suitable solutions concept

Abstract

The incompressible Micropolar system is given by two coupled equations: the first equation gives the evolution of the velocity field u while the second equation gives the evolution of the microrotation field $\omega$. In this article we will consider regularity problems for weak solutions of this system. For this we will introduce the new notion of partial suitable solutions, which imposes a specific behavior for the velocity field u only, and under some classical hypotheses over the pressure, we will obtain a h{\"o}lderian gain for both variables u and $\omega$.