Stable self-similar blowup in energy supercritical Yang–Mills theory

Abstract

Abstract. We consider the Cauchy problem for an energy supercritical nonlinear wave equation that arises in (1 + 5)-dimensional Yang-Mills theory. A certain self-similar solution W0 of this model is conjectured to act as an attractor for generic large data evolutions. Assuming mode stability of W0, we prove a weak version of this conjecture, namely that the self-similar solution W0 is (nonlinearly) stable. Phrased differently, we prove that mode stability of W0 implies its nonlinear stability. The fact that this statement is not vacuous follows from careful numerical work by Bizoń and Chmaj that verifies the mode stability of W0 beyond reasonable doubt.

Stable blowup for the supercritical Yang–Mills heat flow

“…For d = 5, singularity formation for the resulting YM wave equation is known and the existence of infinitely many selfsimilar blowup solutions has been proved by Bizoń [3], who also found a closed form expression for the ground state solution. The stability of this object has been established by the first author in [14] under a spectral assumption which, however, was recently removed in a joint paper of the first author with Costin, Glogić and Huang [11]. The proof in [14] is based on a canonical method that was developed by the authors to investigate stable self-similar blowup for wave equations, cf.…”

“…The stability of this object has been established by the first author in [14] under a spectral assumption which, however, was recently removed in a joint paper of the first author with Costin, Glogić and Huang [11]. The proof in [14] is based on a canonical method that was developed by the authors to investigate stable self-similar blowup for wave equations, cf. similar results for supercritical wave maps [21] and wave equations with focusing power-nonlinearities, [17], [20], [19].…”

Stable self-similar blowup in the supercritical heat flow of harmonic maps

“…• From now on we follow the argument introduced in our earlier works [11][12][13][14][15][16] on selfsimilar blowup for wave-type equations. We first show that the nonlinearity is locally Lipschitz on X .…”

On blowup of co-rotational wave maps in odd space dimensions

“…Finally, concerning the method, we remark that our proof relies on the techniques developed in the series of papers [13][14][15][16][17][18][19]. However, we would like to emphasize that the present paper is not just a straightforward continuation of these works.…”