Local well-posedness for the \(H^2\)-critical nonlinear Schrödinger equation (Q2821677)

This article establishes local well-posedness for the \(H^2\)-critical NLS in dimensions \(N\geq 5\): NEWLINE\[NEWLINE iu_t+\Delta u=\lambda |u|^\alpha u, \quad \alpha=\frac{4}{N-4}. NEWLINE\]NEWLINE The main result (Theorem 1.1) includes local existence and uniqueness, regularity, global character of solutions for small data in \(H^2\), the blow-up alternative and continuous dependence on initial data. This concludes the discussion of local well-posedness in \(H^2/\dot{H}^2\) for this equation, some elements of which were scattered among earlier papers carefully referenced in the introduction, notably the seminal work [the first author and \textit{F. B. Weissler}, Nonlinear Anal. 14, No. 10, 807--836 (1990; Zbl 0706.35127)]. The proof is based on a fixed point argument in a suitably constructed metric space. An interesting feature of the problem is the doubly `critical' nature of the equation: (i) the \(\dot{H}^2\) norm is scaling invariant; (ii) the \(s=2\) regularity level typically requires estimating the size of two space derivatives in order to carry out the fixed point argument. Point (ii) is delicate, as the nonlinearity considered is only \(C^{1,\alpha}\), and not \(C^2\) if \(N\geq8\). This difficulty is overcome via an argument originating from [\textit{T. Kato}, Ann. Inst. H. Poincaré Phys. Théor. 46, No. 1, 113--129 (1987; Zbl 0632.35038)], which makes use of one time derivative instead of two space derivatives. The regularity exponent \(s=2\) is the highest for which this approach works.