- Titel
- Geometric Analysis and Convex Integration
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-972306
- Datum der Einreichung
- 06.02.2025
- Datum der Verteidigung
- 21.05.2025
- Abstract (EN)
- The notion of well-posedness for mathematical problems describes three key properties: existence and uniqueness of solutions to the problem, and continuous dependence on parameters and initial data. Especially for non-linear partial differential equations, well-posedness of such is violated in the low regularity setting by means of non-uniqueness of solutions. However, for high enough regularity, uniqueness of solutions can be shown. It is then natural to ask about the regularity threshold of the PDE at hand which separates both regimes. The method of convex integration, originally developed by Gromov to construct solutions to differential inclusions, is a technique for studying ill-posedness of certain non-linear partial differential equations in the low regularity setting. This method exploits the flexibility of the PDE, induced by its non-linearity, and operates in certain regularity regimes where compactness mechanisms do not apply. In this thesis, we investigate the flexibility of the convex integration method itself on the basis of two examples. For the full Hall-Magnetohydrodynamics system, we obtain non-uniqueness of solutions in the class L^2 H^(beta) for sufficiently small beta > 0, closely following the work by Buckmaster and Vicol on the Navier-Stokes equations. Studying the stationary Hall equation, it turns out that the additional rigidity induced by the curl operator prevents us from applying the basic Tartar convex integration framework. In the setting of isometric embeddings between manifolds of low codimension, we show that it is possible to adapt the quantitative Nash scheme for Riemannian manifolds by Conti, De Lellis and Székelyhidi Jr. to the case of contact manifolds. This extends the previously known flexibility threshold of C^1 due to D'Ambra to the Hölder space C^(1,\alpha), with alpha being the contact equivalent to the regularity threshold in the Riemannian case.
- Freie Schlagwörter (EN)
- Geometric Analysis, Convex Integration, Partial Differential Equations
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- angenommene Version / Postprint / Autorenversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-972306
- Veröffentlichungsdatum Qucosa
- 23.05.2025
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY-NC-SA 4.0