On Type I blowup and $\varepsilon$-regularity criteria of suitable weak solutions to the 3D incompressible MHD equations

TL;DR

This work develops a unified ε-regularity and Type I blow-up framework for suitable weak solutions to the 3D incompressible MHD equations. By a direct iteration of the Caffarelli–Kohn–Nirenberg energy quantities and through careful dimensionless estimates, it establishes ε-regularity criteria under independent smallness/boundedness conditions for the velocity and magnetic fields, and proves that finite limsup of a single scaling-invariant quantity per field enforces Type I blow-up. The analysis further extends to a mixed Lebesgue-norm formulation, showing that ε-regularity and Type I characterisations can be expressed in terms of scale-invariant mixed norms for $u$, $b$, and their derivatives. These results generalise Seregin’s Type I Navier–Stokes criteria to MHD and provide a natural route toward investigating Type II blow-up within a robust, scale-aware framework.

Abstract

We study interior $\varepsilon$-regularity and Type I blowup criteria for suitable weak solutions to the three-dimensional incompressible MHD equations. Our starting point is a direct iteration scheme for the classical Caffarelli--Kohn--Nirenberg scaled energy quantities $A,E,C$ and $D$, which yields $\varepsilon$-regularity criteria under smallness assumptions on the velocity field $u$ and boundedness assumptions on the magnetic field $b$, with the underlying scaling-invariant quantities chosen independently. As an intermediate step, we prove that finiteness of one such scaling-invariant quantity for each of $u$ and $b$ allows only Type I blowup, in the sense that $A(u,b;r)+E(u,b;r)+C(u,b;r)+D(p;r)<\infty$ for small $r$. This extends Seregin's Type I criteria for the Navier--Stokes equations to the MHD setting and provides a natural point of departure for the analysis of Type II blowup. By interpolation and embedding, we further obtain $\varepsilon$-regularity criteria and Type I characterisations in terms of general scaled mixed Lebesgue norms for $u$ and $b$, with independent exponent choices. While we do not aim to sharpen existing mixed-norm $\varepsilon$-regularity criteria, the present formulation offers a unified and comparatively direct route that is naturally compatible with the Type I framework; in particular, the mixed-norm Type I description does not follow from earlier mixed-norm $\varepsilon$-regularity proofs by a formal replacement of the smallness parameter.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof