On the uniqueness of a suitable weak solution to the Navier–Stokes Cauchy problem

The paper is concerned with the Navier–Stokes Cauchy problem. We investigate on some results of regularity and uniqueness related to suitable weak solutions corresponding to a special set of initial data. The suitable weak solution notion is meant in the sense introduced by Caffarelli–Kohn–Nirenberg. As further result we discuss the uniqueness of a set of suitable weak solutions (wider than the previous one) enjoying a “Prodi–Serrin” condition which is “relaxed” in space.

Estimate on the Dimension of the Singular Set of the Supercritical Surface Quasigeostrophic Equation

We consider the SQG equation with dissipation given by a fractional Laplacian of order $$\alpha <\frac{1}{2}$$ α < 1 2 . We introduce a notion of suitable weak solution, which exists for every $$L^2$$ L 2 initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most $$\frac{1}{2\alpha } \left( \frac{1+\alpha }{\alpha } (1-2\alpha ) + 2\right) $$ 1 2 α 1 + α α ( 1 - 2 α ) + 2 .

On regularity properties of a surface growth model

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local smoothness of solutions.