On well-posedness of energy supercritical NLS on product space R n × T 4 − n ${\mathbb{R}}^{n}{\times}{\mathbb{T}}^{4-n}$ ( n = 0, 1, 2, 3)

Abstract We establish the well-posedness theory for the quintic nonlinear Schrödinger equation (NLS) on four-dimensional tori (i.e., T 4 ${\mathbb{T}}^{4}$ ), which is an energy-supercritical model. Compared to the recent breakthrough work (B. Kwak and S. Kwon, Critical local well-posedness of the nonlinear Schrödinger equation on the torus , Ann. Inst. H. Poincaré C (2024)) on this topic, our approach provides a more concise and natural alternative. Moreover, the results we obtained naturally extend to the waveguide case, i.e. considering NLS on product spaces R m × T n ${\mathbb{R}}^{m}{\times}{\mathbb{T}}^{n}$ ( m , n ≥ 1, m + n = 4). The proof is based on Koch–Tataru functional framework and Killip–Visan scheme R. Killip and M. Vişan, Scale invariant Strichartz estimates on tori and applications , Math. Res. Lett. 23 (2016), no. 2, 445–472 with modifications. Moreover, motivated by recent work Y. Deng, H. Wang, Y. Wang, and Z. Zhao, On restricted-type strichartz estimates and the applications , preprint (2025), we establish improved local well-posedness results for the quintic NLS on R 2 × T ${\mathbb{R}}^{2}{\times}\mathbb{T}$ under a strip-type frequency restriction (i.e., the Fourier support is contained in a strip of bounded width) on the initial data, demonstrating that such geometric constraints can effectively mitigate the supercritical ill-posedness. Analogous results in this paper are expected to be obtained for other models with suitable modifications. Finally, we include the classical final state problems for NLS in the waveguide setting and the resonant Schrödinger system setting, respectively, which may have their own interests.