We consider spherically symmetric Yang-Mills equations with gauge group $SO(d)$ in $d+1$-dimensional Minkowski spacetime. For any given odd $d\ge 11$, we establish existence and uniqueness (modulo reflection symmetry) of exactly $N$ smooth self-similar solutions, where $N$ is the number of zeros of an explicit polynomial $P_m(z)$ of degree $m=(d-5)/2$ in the interval $0<z<1$. The number $N$ can be determined algorithmically by an explicit computation. Our extensive computations for large odd dimensions suggest that $N=3$ for all odd $d\ge 11$. Two of these self-similar solutions admit closed-form expressions: one has been known previously, while the other appears to be new. Our result points toward a relatively simple landscape of possible blowup scenarios for high-dimensional Yang-Mills equations. Beyond its purely mathematical interest, this rigidity of self-similar blowup may also be relevant from a physical perspective, as it constrains the possible ultraviolet dynamics of non-Abelian gauge fields in higher-dimensional Yang–Mills theories arising in string-inspired extra-dimensional setups and in holographic models.

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• Classical black holes
• Classical solutions in field theory
• General relativity
• Gravity in dimensions other than four
• Higher-dimensional field theories