A remark on very weak suitable solutions and Leray solutions of the Navier-Stokes equations

We introduce the notion of very weak suitable solutions for the Navier--Stokes equations. Here, the velocity and the pressure satisfy minimal conditions that make sense of the local energy inequality in the distributional setting. A well-known but still relevant question is to find sufficient conditions ensuring that very-weak solutions are in fact Leray solutions. Exploiting the \emph{local} energy inequality within the general framework of \emph{local} Morrey spaces, we establish such conditions. Local Morrey spaces provide a general framework that contains other useful functional settings in the theoretical analysis of the Navier--Stokes equations, such as Lebesgue, Lorentz, homogeneous Morrey, and parabolic Morrey spaces. As a by-product, we also derive some sufficient conditions to study the uniqueness and regularity of the resulting Leray solutions.