Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows

Abstract

In this paper, we consider the Yang-Mills heat flow on R d × SO(d) with d ≥ 11. Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to :We are interested in describing the singularity formation of this parabolic equation. We construct non-self-similar blowup solutions for d ≥ 11 and prove that the asymptotic of the solution is of the formwhere Q is the ground state with boundary conditions Q(0) = −1, Q ′ (0) = 0 and the blowup speedIn particular, when ℓ = 1, this asymptotic is stable whereas for ℓ ≥ 2 it becomes stable on a space of codimension ℓ − 1. Our approach here is not based on energy estimates but on a careful construction of time dependent eigenvectors and eigenvalues combined with maximum principle and semigroup pointwise estimates.