Global oscillatory solutions for the Yang-Mills heat flow

We investigate the long-time dynamics for the global solution of the $SO(4)$-equivariant Yang-Mills heat flow (YMHF) with structure group $SU(2)$ in space dimension $4$. For a class of initial data with specific decay at spatial infinity, we prove that the long-time dynamics of YMHF can be described by the initial data in a unified manner. As a consequence, the global solutions can exhibit blow-up, blow-down, and more exotically, {\it oscillatory} asymptotic behavior at time infinity. This seems to be the first example of Yang-Mills heat flows with oscillatory behavior as $t\to \infty$.

💡 Research Summary

The paper studies the long‑time dynamics of the Yang–Mills heat flow (YMHF) on $\mathbb R^4$ under an $SO(4)$‑equivariant ansatz with structure group $SU(2)$. By imposing $SO(4)$ symmetry the Yang–Mills connection reduces to a single scalar function $\psi(t,r)$ satisfying a nonlinear radial heat equation (1.3). After the change of variables $\phi=r^{-2}\psi$, the problem becomes a six‑dimensional radial heat equation (1.4) with a critical cubic nonlinearity.

A one‑parameter family of static solutions (instantons) is given by
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