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**Global well-posedness for the Yang-Mills equation in \(4+1\) dimensions. Small energy.**
*(English)*
Zbl 1377.58007

The hyperbolic Yang-Mills equations are critical in \(4+1\) dimensions. In this remarkable paper, global well-posedness of the system on the Minkowski space \({\mathbb R}^{4+1}\) is proven for initial data with sufficiently small Yang-Mills action. The proof modifies and extends methods that the authors used to establish a similar result for the related Maxwell-Klein-Gordon equation in [the authors with J. Sterbenz, Duke Math. J. 164, No. 6, 973–1040 (2015; Zbl 1329.35209)].

Since the equation is gauge invariant, a choice of gauge must be used to obtain another differential equation in order to make the system hyperbolic. The gauge imposed here is, maybe a bit surprisingly, the Coulomb gauge condition involving only the spatial components. It allows to view the equations as a nonlocal hyperbolic system for the spatial components. The nonlocality comes from the component \(A_0\) appearing in the system, which in turn solves an elliptic equation. The nonlinearity is split in two parts, a perturbative one and a non-perturbative paradifferential type component. The former cannot be estimated directly, but requires reiteration of the equality and the use of some null condition. The non-perturbative part is then eliminated via a paradifferential gauge renormalization. The choice of spaces is crucial and sophisticated.

Since the equation is gauge invariant, a choice of gauge must be used to obtain another differential equation in order to make the system hyperbolic. The gauge imposed here is, maybe a bit surprisingly, the Coulomb gauge condition involving only the spatial components. It allows to view the equations as a nonlocal hyperbolic system for the spatial components. The nonlocality comes from the component \(A_0\) appearing in the system, which in turn solves an elliptic equation. The nonlinearity is split in two parts, a perturbative one and a non-perturbative paradifferential type component. The former cannot be estimated directly, but requires reiteration of the equality and the use of some null condition. The non-perturbative part is then eliminated via a paradifferential gauge renormalization. The choice of spaces is crucial and sophisticated.

Reviewer: Andreas Gastel (Essen)

### MSC:

58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |

35L55 | Higher-order hyperbolic systems |

81T13 | Yang-Mills and other gauge theories in quantum field theory |