# Scale invariant solutions to Navier Stokes equation and implications to Leray-Hopf weak solutions

# Scale invariant solutions to Navier Stokes equation and implications to Leray-Hopf weak solutions

In this talk, I will first discuss the existence of scale invariant solutions to Navier Stokes equation with arbitrary $-1$ homogeneous initial data. Since these solutions may not be small, linearized analysis seem to suggest nontrivial bifurcations. Under a quite plausible spectral assumption, we show rigorously that such bifurcations do occur and they imply non-uniqueness of scale-invariant solutions. By appropriately localizing such solutions, we then obtain non-uniqueness of Leray-Hopf weak solutions with initial data which are compactly supported, smooth away from origin, and having a singularity at the origin of the type $O(\frac{1}{|x|})$, which will be sharp. The verification of the spectral assumption involves only smooth and decaying functions, and seems to be doable numerically. This is joint work with V.Sverak.