On the regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space

TL;DR

This paper establishes regularity criteria for weak solutions of the micropolar fluid equations in Lorentz spaces, identifying conditions under which solutions become smooth based on integrability properties of velocity, pressure, and their gradients.

Contribution

It provides new regularity criteria in Lorentz spaces for weak solutions to micropolar fluid equations, extending previous results in Lebesgue spaces.

Findings

Weak solutions become smooth if velocity belongs to certain Lorentz space integrability conditions.

Regularity is guaranteed when pressure or its gradient satisfy specific Lorentz space bounds.

The criteria depend on the relation between time integrability and spatial Lorentz space parameters.

Abstract

In this paper the regularity of weak solutions and the blow-up criteria of smooth solutions to the micropolar fluid equations on three dimension space are studied in the Lorentz space $L^{p,\infty}(\mathbb{R}^3)$. We obtain that if $u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 1$ with $3<p\le \infty$; or $\nabla u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 2$ with $\frac32<p\le \infty$; or the pressure $P\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 2$ with $\frac32<p\le \infty$; or $\nabla P\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 3$ with $1<p\le \infty$, then the weak solution $(u,\omega)$ satisfying the energy inequality is a smooth solution on $[0,T)$.