  We consider the Yangs-Mills equations in 4+1 dimensions. This is the energy
critical case and we show that it admits a family of solutions which blow up in
finite time. They are obtained by the spherically symmetric ansatz in the SO(4)
gauge group and result by rescaling of the instanton solution. The rescaling is
done via a prescribed rate which in this case is a modification of the
self-similar rate by a power of |log t|. The powers themselves take any value
exceeding 3/2 and thus form a continuum of distinct rates leading to blow-up.
The methods are related to the authors' previous work on wave maps and the
energy critical semi-linear equation. However, in contrast to these equations,
the linearized Yang-Mills operator (around an instanton) exhibits a zero energy
eigenvalue rather than a resonance. This turns out to have far-reaching
consequences, amongst which are a completely different family of rates leading
to blow-up (logarithmic rather than polynomial corrections to the self-similar
rate).
