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.