Blowup Criterion for the Navier–Stokes Equations via the Middle Eigenvalue of the Strain Tensor in the Anisotropic Besov Spaces With Negative Indices

ABSTRACT In this paper, we study blowup criteria for the three‐dimensional incompressible Navier–Stokes equations in terms of the middle eigenvalue of the strain tensor. It is known that regularity criteria involving eigenvalues of the strain tensor were first investigated by Neustupa and Penel, and later developed by Miller and others. First, we extend the known blowup criterion in anisotropic Lebesgue spaces by removing the restriction on the admissible exponent range. More precisely, the condition appearing in previous results is relaxed to . Furthermore, we establish several new blowup criteria in anisotropic Besov spaces with negative indices. These results extend the eigenvalue regularity criteria to a broader class of scale‐invariant anisotropic spaces.