The critical Yang–Mills global well-posedness conjecture

Let $\mathcal G$ be a compact Lie group with Lie algebra $\mathbf g$, let $s_0=\frac{n-2}{2}$ be the scaling index, and consider the Yang–Mills equation

$$D^jF_{ij}=0$$

for a connection with curvature $F_{ij}$. Yang–Mills global well-posedness conjecture. (i) The Yang–Mills equation is globally well-posed for small data in $H^{s_0}\times H^{s_0-1}$ for $n\geq4$. (ii) The Yang–Mills equation is globally well-posed for large data in $H^{s_0}\times H^{s_0-1}$ for $n=4$. The source notes that local well-posedness above the scaling index was known for $n\geq4$, whereas the critical global questions, especially the four-dimensional large-data case, remained open there.