An estimate on the parabolic fractal dimension of the singular set for solutions of the Navier–Stokes system

Abstract

We estimate the parabolic fractal (or parabolic box-counting) dimension of the singular set for suitable weak solutions of the Navier–Stokes equations in a bounded domain D. We prove that the parabolic fractal dimension is bounded by 45/29 improving an earlier result from (Kukavica 2009 Nonlinearity 22 2889–900). Also, we introduce the new (parabolic) λ-fractal dimension, where λ is a parameter, which for λ = 1 agrees with the parabolic fractal and for λ = ∞ with the parabolic Hausdorff dimension. We prove that for a certain range of λ, the dimension of the singular set is bounded by 3/2.