Micropolar fluids with initial angular velocities in non-homogeneous Sobolev spaces of order $-1/2$

TL;DR

The paper develops a fractional-energy framework to prove local existence for micropolar fluids starting from an initial angular velocity with negative Sobolev regularity, specifically in a non-homogeneous setting that captures the critical scaling of the associated simplified system. By constructing mollified approximations and employing energy balances with nonlocal fractional derivatives, the authors derive a priori bounds in $H^\tau$ for the velocity and $H^\sigma$ for the angular velocity, then pass to the limit via Aubin–Lions compactness to obtain solutions of the micropolar system $(M)$ with pressure given by the Riesz transform. The results cover ranges $\frac12<\tau<\frac32$, $\tau-1<\sigma<\frac32$, and also include a small-data regime at $\tau=\tfrac12$, $-\tfrac12<\sigma<\tfrac32$, as well as a $(M^{\mu,\nu})$-extension under large viscosity product, thereby providing a non-mild-solution route that complements classical approaches and connects to CK-N-type singularity analyses for coupled fluids.

Abstract

In this paper, we investigate fractional energy methods for Micropolar fluids, starting with an initial angular velocity of negative Sobolev regularity. For the initial angular velocity assumption, we consider a non-homogeneous Sobolev norm of negative order. The regularity -1/2 studied here corresponds to the critical scaling of a simplified associated system, and the general framework can also be applied to the Boussinesq system with viscosity. Since our approach differs from those based on mild solutions and does not rely on a projected system, this work provides new tools for studying the Caffarelli-Kohn-Nirenberg theory of singularities in coupled variables within the Navier-Stokes equations.

Theorems & Definitions (7)

• Theorem 2.1
• Theorem 2.2
• Remark 2.1
• Definition 3.1
• Remark 3.1
• Lemma 3.1
• Lemma 3.2: Aubin--Lions compactness theorem