Global Well-Posedness of the Energy-Critical Nonlinear Schrödinger Equations on $$\mathbb {T}^{d}$$

Abstract In this paper, we prove the global well-posedness of the energy-critical nonlinear Schrödinger equations on the torus $$\mathbb {T}^{d}$$ T d for general dimensions. This result is new for dimensions $$d\ge 5$$ d ≥ 5 , extending previous results for $$d=3,4$$ d = 3 , 4 [11, 23]. Compared to the cases $$d=3,4$$ d = 3 , 4 , the regularity theory for higher d , developed in the underlying local well-posedness result [18], is less understood. In particular, stability theory and inverse inequalities, which are ingredients in [11, 23] and more generally in the widely used concentration compactness framework since [14], are too weak to be applied to higher dimensions. Our proof introduces a new strategy for addressing global well-posedness problems. Without relying on perturbation theory, we develop tools to analyze the concentration dynamics of the nonlinear flow. On the way, we show the formation of a nontrivial concentration.