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A Weighted Regularity Criterion for Suitable Weak Solutions of Incompressible Non-Newtonian Fluids

Jae-Myoung Kim

TL;DR

This work studies regularity for a 3D incompressible non-Newtonian fluid with power-law stress $S(Du)=(\mu_0+\mu_1|Du|^{q-2})Du$ in $\mathbb{R}^3$ and develops a weighted regularity criterion based on the weighted gradient of the velocity and leverages the Caffarelli–Kohn–Nirenberg and Stein inequalities to obtain weighted a priori bounds. An $\varepsilon$-regularity framework for suitable weak solutions is established, leading to semi-regularity under finite weighted norms and appropriate initial data. The results extend existing weighted criteria for shear-thinning flows to the non-Newtonian setting and clarify how spatial weights govern regularity, with implications for singularity formation.

Abstract

It establishes a regularity criterion for non-Newtonian fluids in $\mathbb{R}^3$ in terms of the weighted gradient of the velocity field, based on the Caffarelli--Kohn--Nirenberg inequality.

A Weighted Regularity Criterion for Suitable Weak Solutions of Incompressible Non-Newtonian Fluids

TL;DR

This work studies regularity for a 3D incompressible non-Newtonian fluid with power-law stress in and develops a weighted regularity criterion based on the weighted gradient of the velocity and leverages the Caffarelli–Kohn–Nirenberg and Stein inequalities to obtain weighted a priori bounds. An -regularity framework for suitable weak solutions is established, leading to semi-regularity under finite weighted norms and appropriate initial data. The results extend existing weighted criteria for shear-thinning flows to the non-Newtonian setting and clarify how spatial weights govern regularity, with implications for singularity formation.

Abstract

It establishes a regularity criterion for non-Newtonian fluids in in terms of the weighted gradient of the velocity field, based on the Caffarelli--Kohn--Nirenberg inequality.
Paper Structure (3 sections, 3 theorems, 53 equations)

This paper contains 3 sections, 3 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and proofs
  3. Preliminaries

Key Result

Theorem 1.1

Let $q>2$. Assume that $u_0 \in H^1({ \mathbb{R} }^3)$ with $\nabla \cdot u_0 = 0$ satisfies and if a weak solution $u$ satisfies with Then the weak solution $u$ is semi-regular up to time $T>0$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.1
  • Lemma 2.1