On the global well-posedness of the quadratic NLS on $L^2(\mathbb{R})+H^1(\mathbb{R})$
Chaichenets, Leonid
; Hundertmark, Dirk; Kunstmann, Peer; Pattakos, Nikolaos
; Hundertmark, Dirk; Kunstmann, Peer; Pattakos, NikolaosAbstract:
We study the one dimensional nonlinear Schrödinger equation with power nonlinearity $|u|^{\alpha-1}$ for $\alpha \in [2, 5]$ and initial data $u_0 ∈ L^2(\mathbb{R})+H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity $(\alpha = 2)$ we obtain unconditional global well-posedness in the space $C(\mathbb{R}, L^2(\mathbb{R})+H^1(\mathbb{T}))$ via Gronwall’s inequality.
| Zugehörige Institution(en) am KIT | Institut für Analysis (IANA) Sonderforschungsbereich 1173 (SFB 1173) |
| Publikationstyp | Forschungsbericht/Preprint |
| Publikationsjahr | 2019 |
| Sprache | Englisch |
| Identifikator | ISSN: 2365-662X KITopen-ID: 1000093752 |
| Verlag | Karlsruher Institut für Technologie (KIT) |
| Umfang | 23 S. |
| Serie | CRC 1173 ; 2019/9 |
| Projektinformation | SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015) |
| Schlagwörter | nonlinear Schrödinger equation, local well-posedness, global well-posedness, Gronwall’s inequality, Strichartz estimates |