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Regularity criteria of 3D generalized magneto-micropolar fluid system in terms of the pressure

Jae-Myoung Kim

TL;DR

The paper addresses the regularity problem for a 3D generalized magneto-micropolar fluid system by deriving pressure-based regularity criteria in Lorentz spaces. The authors reformulate the system using $z^{+}$ and $z^{-}$ and develop energy estimates that connect small Lorentz-norms of the total pressure $\pi$ (and its gradient) to the regularity of the weak solution $(u,b,w)$, employing a nonlinear Gronwall bootstrap to close the estimates. In the isotropic case $\alpha=\beta=\gamma=1$, they obtain Serrin-type criteria for $\nabla\pi$ and extended conditions involving $\nabla\mathcal{P}$ and $b$, including joint gradient/magnetic-field criteria. These results extend pressure-based regularity theory to the generalized magneto-micropolar system and provide a Lorentz-space framework for future analyses in similar PDEs with fractional diffusion.

Abstract

This work focuses on regularity criteria of 3D generalized magneto-micropolar fluid system in terms of the pressure in Lorentz spaces inspired by the recent works in \cite{FS22} and \cite{LN22}.

Regularity criteria of 3D generalized magneto-micropolar fluid system in terms of the pressure

TL;DR

The paper addresses the regularity problem for a 3D generalized magneto-micropolar fluid system by deriving pressure-based regularity criteria in Lorentz spaces. The authors reformulate the system using and and develop energy estimates that connect small Lorentz-norms of the total pressure (and its gradient) to the regularity of the weak solution , employing a nonlinear Gronwall bootstrap to close the estimates. In the isotropic case , they obtain Serrin-type criteria for and extended conditions involving and , including joint gradient/magnetic-field criteria. These results extend pressure-based regularity theory to the generalized magneto-micropolar system and provide a Lorentz-space framework for future analyses in similar PDEs with fractional diffusion.

Abstract

This work focuses on regularity criteria of 3D generalized magneto-micropolar fluid system in terms of the pressure in Lorentz spaces inspired by the recent works in \cite{FS22} and \cite{LN22}.
Paper Structure (3 sections, 5 theorems, 94 equations)

This paper contains 3 sections, 5 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['the1.1']}
  3. Proof of Theorem \ref{['the1.2']}

Key Result

Theorem 1.1

Let $\quad 0 < T < \infty$ and $u_0, b_0, w_0 \in H^m(\mathbb R^3)$ with $m > \frac{5}{2}$ and $1\leq \alpha, \gamma \leq \frac{5}{4}$. There exists a sufficient constant $\epsilon > 0$ such that if $\pi$ or $\nabla \pi$ satisfy then a weak $\left( u,b\right)$ is regular on $(0,T].$

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2