Dokument: Local and global well-posedness for the higher-order NLS & dNLS hierarchy equations
| Titel: | Local and global well-posedness for the higher-order NLS & dNLS hierarchy equations | |||||||
| URL für Lesezeichen: | https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=69188 | |||||||
| URN (NBN): | urn:nbn:de:hbz:061-20250402-152143-8 | |||||||
| Kollektion: | Dissertationen | |||||||
| Sprache: | Englisch | |||||||
| Dokumententyp: | Wissenschaftliche Abschlussarbeiten » Dissertation | |||||||
| Medientyp: | Text | |||||||
| Autor: | Adams, Joseph [Autor] | |||||||
| Dateien: |
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| Beitragende: | apl. Prof. Dr. Grünrock, Axel [Gutachter] Prof.Dr. Saal, Jürgen [Gutachter] | |||||||
| Dewey Dezimal-Klassifikation: | 500 Naturwissenschaften und Mathematik » 510 Mathematik | |||||||
| Beschreibung: | The subject of this thesis is the well-posedness theory of two hierarchies of
higher-order nonlinear dispersive partial differential equations (PDEs). The first of these hierarchies is the nonlinear Schrödinger (NLS) hierarchy, which is anchored in the classical cubic NLS equation $$i\partial_t u + \partial_x^2 u \pm |u|^2u = 0, \qquad u(x, 0) = u_0(x).$$ It is well known that, being completely integrable, it possesses an infinite number of conservation laws. We start by deriving a general representation of the structure of the higher-order Hamiltonian PDEs associated with these conservation laws. Using tools from Fourier analysis, bi- and tri-linear refinements of Strichartz estimates are derived. These can subsequently be used to prove local well-posedness of the Cauchy problems associated with all higher-order equations in the NLS hierarchy in the non-periodic setting. As data spaces we cover the classical $L^2$-based Sobolev spaces $H^s(\mathbb{R})$, which, despite being a natural choice of data space, turn out not to be well suited for achieving well-posedness close to critical regularity. We therefore also consider alternative classes of data spaces: the Fourier-Lebesgue spaces $\hat H^s_r(\mathbb{R})$, $s\in\mathbb{R}$, $2 \ge r > 1$, and the modulation spaces $M_{2,p}^s(\mathbb{R})$, $s \in \mathbb{R}$, $2 \le p < \infty$. Combined with a-priori estimates derived from the complete integrability of the hierarchy equations taken from the literature we are able to extend our $L^2$-based local solutions to global in time solutions. Concluding this first part of the thesis we also prove that, within the framework of tools we are using (deriving well-posedness with fixed-point methods), we have obtained optimal results up to the endpoint, the critical regularity, in our respective classes of data spaces. Furthermore we show that fixed-point methods are not applicable in the periodic setting. In the second part of the thesis we consider a different hierarchy of PDEs based on the infinite number of conservation laws of the derivative nonlinear Schrödinger (dNLS) equation $$i\partial_t u + \partial_x^2 u - i\partial_x (|u|^2u) = 0, \qquad u(x, 0) = u_0(x),$$ another completely integrable model. We again derive a workable representation of the higher-order PDEs in this dNLS hierarchy, and even go one step further by determining, for a subset of nonlinear terms appearing in the higher-order equations, the coefficients appearing in front of them. This is necessary to establish the effectiveness of a gauge-transformation that will rid the equation of what we call `bad' cubic terms in the nonlinearity. A similar detailed analysis of the coefficients of a hierarchy of integrable PDEs has not yet been undertaken. The local well-posedness theory for the dNLS hierarchy equations follows similar arguments as for the NLS hierarchy equations, reusing the smoothing estimates from earlier. However, an additional difficulty arises due to an extra derivative in nonlinear terms of the equations. The smoothing estimates alone are not sufficient to derive well-posedness. We resort to additional smoothing resulting from the structure of the nonlinear terms, expressed through their resonance relation. For the extension of the local $L^2$-based well-posedness results to global in time ones, no equivalent to the NLS hierarchy a-priori estimates is available. We therefore have to resort to proving that it is possible to derive sufficient a-priori estimates from the conservation laws associated with the dNLS hierarchy equations. Finally we again establish optimality of our results within the framework we are using and rule out the possibility of developing a similar theory for the periodic setting (using fixed-point methods). | |||||||
| Lizenz: |  Dieses Werk ist lizenziert unter einer Creative Commons Namensnennung 4.0 International Lizenz | |||||||
| Fachbereich / Einrichtung: | Mathematisch- Naturwissenschaftliche Fakultät » WE Mathematik » Angewandte Analysis | |||||||
| Dokument erstellt am: | 02.04.2025 | |||||||
| Dateien geändert am: | 02.04.2025 | |||||||
| Promotionsantrag am: | 25.02.2025 | |||||||
| Datum der Promotion: | 26.03.2025 |
