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 Rd×SO(d) with d≥11. Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to the following nonlinear equation:∂tu=∂r2u+d+1r∂ru−3(d−2)u2−(d−2)r2u3, and (r,t)∈R+×R+. We are interested in describing the singularity formation of this parabolic equation. More precisely, we aim to construct non self-similar blowup solutions in higher dimensions d≥11, and prove that the asymptotic behavior of the solution is of the formu(r,t)∼1λℓ(t)Q(rλℓ(t)), as t→T, where Q is the steady state corresponding to the boundary conditions Q(0)=−1,Q′(0)=0 and the blowup speed λℓ satisfiesλℓ(t)=(C(u0)+ot→T(1))(T−t)2ℓα as t→T,ℓ∈N+⁎,α>1. In particular, the case ℓ=1 corresponds to the stable type II blowup regime, whereas for the cases ℓ≥2 corresponds to a finite co-dimensional stable regime. 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.