Existence of Suitable Weak Solutions to the Navier–Stokes Equations for Intermittent Data

Local in time weak solutions to the 3D Navier–Stokes are constructed for a class of initial data in $$L^2_\mathrm {loc}$$. In contrast to other constructions (e.g. Lemarie-Rieusset in Recent developments in the Navier–Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol 431. Chapman & Hall/CRC, Boca Raton, 2002; Kikuchi and Seregin in Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality. Nonlinear equations and spectral theory, American Mathematical Society translations: series 2, vol 220. American Mathematical Society, Providence, pp 141–164, 2007; Kwon and Tsai in Global Navier–Stokes flows for non-decaying initial data with slowly decaying oscillation. arXiv:1811.03249 ), the initial data is not required to be uniformly locally square integrable and, in particular, can exhibit growth in a local $$L^2$$ sense. This class of initial data includes vector fields in the critical Morrey space and discretely self-similar vector fields in $$L^2_\mathrm {loc}$$.