Weak Solutions to the Navier–Stokes Equations with Data in $ $\mathbb {L}(3,\infty) $ $ L (3,∞)

The paper concerns the existence of weak solutions to the 3d-Navier–Stokes initial boundary value problem in exterior domains. The problem is considered with an initial data belonging to (3,∞) which is a special subspace of the Lorentz’s space L(3,∞) . The nature of the domain and the initial data in L(3,∞) make the result of existence not comparable with the usual Leray-Hopf theory of weak solutions. However, we are able to prove both that the weak solutions enjoy the partial regularity in the sense of Leray’s structure theorem and the asymptotic limit of |u(t)|3∞ .