Local-in-space estimates near initial time for weak solutions of the   Navier-Stokes equations and forward self-similar solutions
  We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with $(-1)$-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are local-in-space regularity estimates near the initial time, which are of independent interest. 
Hao Jia, Vladim\'ir \v{S}ver\'ak
math.AP