The theory of local energy solutions for the incompressible Navier-Stokes equations, pioneered by Lemarie-Rieusset, considers those solutions satisfying the local energy inequality but not be in the global energy class. The original global existence theorem for initial data in uniform L2 loc with decay is extended in two directions, one for data without decay but with oscillation decay or suitable growth bounds, the other for data in the Wiener amalgam spaces between L2 and uniform L2 loc. The latter is motivated in identifying the smallest functional spaces that still contain type one singularity and self-similar solutions. We will also present regularity criteria for local energy solutions and use them to estimate their regular sets, extending the eventual regularity result of Leray, and Theorems C and D of Caffarelli, Kohn and Nirenberg for initial data in weighted spaces.

This talk summarizes joint work with Bradshaw, Kang, Kukavica, Kwon, and Miura.