Large homogeneous initial data for the 3D Navier-Stokes equations

Due to the scaling properties of the Navier-Stokes equations,

homogeneous initial data may lead to forward self-similar solutions.

When the initial data is small enough, it is well known that the

formalism of mild solutions (through the Picard-Duhamel formula) give

such self-similar solutions. We shall discuss the issue of large initial

data, where we can only prove the existence of weak solutions; those

solutions may lack self-similarity, due to the fact that we have no

results about uniqueness for such weak solutions. We study some tools

which may be useful to get a better understanding of those weak solutions.