Navier–Stokes equations regularity criteria via strain sufficiently close to being eigenfunctions of the Laplacian in Vishik spaces

Abstract This paper develops new regularity criteria for the 3D Navier–Stokes equations. Motivated by the observation that the strain tensor can be approximated by eigenfunctions of the Laplacian, we extend the framework from classical Lebesgue spaces to Vishik spaces with negative regularity indices. We prove a generalized Vishik-type trilinear estimate of independent analytic interest and, combined with the strain-vorticity orthogonality identity, derive a new continuation criterion. Our result extends Miller’s earlier work and shows that Vishik spaces offer a natural and flexible setting for critical regularity and blow-up problems in fluid dynamics.