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.

In the first part of our paper we discuss linear 2nd order differential equations in complex domain, especially Heun class equations, that is, equation and its confluent cases. The second is devoted to Painlev\'e I-VI equations. Our philosophy treat these families a unified way. This works well for We classification into 5 supertypes, subdivided 10 types (not counting trivial cases). also intro...