Stable blowup for the supercritical Yang–Mills heat flow

Abstract

In this paper, we consider the heat flow for Yang-Mills connections on R 5 × SO(5). In the SO(5)−equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reactiondiffusion equation for which an explicit self-similar blowup solution was found by Weinkove [57]. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in L ∞ . 7 9

On the Stability of Type I Blow Up For the Energy Super Critical Heat Equation

“…Hence we avoid on purpose maximum principle like tools. In [13,14,15,16], this kind of question has also been addressed for the radially symmetric supercritical wave map problem, Yang-Mills, wave equation and Yang-Mills heat flow. In those works, the analysis requires a detailed description of the complex spectrum of the linearized operator in suitable spaces which is a delicate matter, and seems to rely heavily on the fact that in the cases under consideration, the self similar solution has an explicit formula.…”

“…Hence ϕ 0,n is not in the kernel of L n,0 . 16 Indeed, ϕn,0 would be an eigenvector for the eigenvalue 0, but 0 is not in the spectrum of An as seen above.…”

Construction of a spectrally stable self-similar blowup solution to the supercritical corotational harmonic map heat flow

“…The harmonic map heat flow bares a striking similarity to other parabolic equations, also displaying a blow-up scenario, such as Yang-Mills flow and semilinear heat equation. In fact, one of the authors and Schörkhuber have recently proved the nonlinear stability of a self-similar solution for the Yang-Mills flow [10]. The proof in [10] relies on a closed-form expression for the self-similar profile to solve the spectral stability problem.…”

“…In fact, one of the authors and Schörkhuber have recently proved the nonlinear stability of a self-similar solution for the Yang-Mills flow [10]. The proof in [10] relies on a closed-form expression for the self-similar profile to solve the spectral stability problem. Such a closedform expression is unavailable for the harmonic map flow but in this paper we show how to circumvent this issue.…”

On Strongly Anisotropic Type I Blowup

“…where r = √ 1 + r 2 . In particular, these solutions have infinite energy, but they can be shown to be the blow up profile for a finite codimensional class of finite energy smooth initial data, [7], see also [9]. The finite codimension of self similar blow up initial data is in one to one correspondance with the nonpositive eigenmodes of the linearized operator restricted to radial functions:…”