Sharp decay for a family of solutions to the SU(2) Yang-Mills equations on a Schwarzschild black hole.
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Amphi Schwartz
The purely magnetic ansatz yields a family of spherically symmetric solutions to the SU(2) Yang-Mills equations parametrized by a single scalar function $W$. The equation on $W$ is a semilinear wave equation exhibiting rich dynamics while avoiding the algebraic complexity of the full Yang-Mills system. On the Schwarzschild spacetime, previous results include global existence, existence of an infinite family of unstable stationary solutions and stability of the trivial solutions $W =\pm 1$. In this talk, I will present a work in progress, joint with Cécile Huneau, adressing a conjecture of Bizoń-Chmaj-Rostworowski about the precise late-time asymptotics for solutions initially close to $W=1$. This sharpens the decay rate previously obtained by Ghanem-Häfner and provides a concrete example of how the nonlinearity and its structure influence the decay rate of the solution. In particular, the decay rate for the nonlinear problem differs from that of the corresponding linearized problem.