- Titel
- Intermittent Convex Integration for Partial Differential Equations describing Fluid Flows
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-960739
- Datum der Einreichung
- 30.07.2024
- Datum der Verteidigung
- 05.02.2025
- Abstract (EN)
- Intermittent Convex Integration is a technique for constructing weak solutions to non-linear partial differential equations. It originates from Buckmaster and Vicol's celebrated result about non-uniqueness of distributional solutions to the three-dimensional Navier-Stokes equation. Their construction uses highly concentrated functions as building blocks in a recursively defined infinite series. The recursive definition stems from the method De Lellis and Székelyhidi developed for the Euler equation which later led to the proof of Onsager's conjecture by Isett. Using methods from but other building blocks, Modena and Székelyhidi proved the non-uniqueness of solutions to the transport equation with incompressible velocity fields with Sobolev regularity, relying on a much simpler construction than in the first instance of Intermittent Convex Integration mentioned above. In a similar manner Luo proved the existence of stationary solutions to the Navier-Stokes equation in dimension 4. Another step in the development was the introduction of temporal intermittency by Cheskidov and Luo - earlier constructions were highly concentrated in the spatial variable but homogeneous in time. This innovation admitted results on the two-dimensional Navier-Stokes equation as well as the transport equation with almost Lipschitz velocity field and almost smooth density. In a series of works which combine the iterative ansatz from the proof of Onsager's conjecture with methods and building blocks from Intermittent Convex Integration Novack et al. were able to prove an intermittent analog of the conjecture. The contrary approach, in some sense, was taken by the author of this thesis and Székelyhidi by showing that in most results which use Intermittent Convex Integration, iterations are unnecessary and can be replaced by a simple perturbation argument and applying the Baire category theorem. This allows for stronger results since not only existence but also genericity (in the Baire category sense) of solutions can be concluded. In this work all these developments are presented in an accessible and transparent manner; to this end we will not follow the historically correct order (which is outlined above) but the didactically optimal one: starting from the proof of Onsager's conjecture (which can be considered classical by now) we introduce 'concentrated Mikado flows' and show how they can be applied to the transport equation and the Navier-Stokes equation. In the next step we present building blocks which are entirely localised in space and therefore feature optimal concentration properties, and showcase their use in the transport equation. Then we introduce temporal intermittency as described in and show that it can be used in convex integration independently from spatial intermittency in order to give an elementary proof for non-uniqueness of solutions to the hypodissipative Navier-Stokes equation. The final step is the introduction of the 'Baire category method' and its application to transport and Navier-Stokes equations.
- Freie Schlagwörter (EN)
- Fluid mechanics, Intermittency in Turbulence, Convex Integration, Non-Uniqueness
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-960739
- Veröffentlichungsdatum Qucosa
- 06.03.2025
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY 4.0- Inhaltsverzeichnis
Contents Chapter I. Introduction Chapter II. Turbulent Energy Cascade and Onsager’s Conjecture 1. Observations and Heuristics: Richardson and Kolmogorov 2. Onsager’s conjecture on dissipation of energy 3. Proof of Conservation of energy and why it fails for low regularity 4. A proof of Onsager’s Conjecture by Convex Integration Chapter III. Intermittency in Turbulence and Intermittent Onsager Conjecture 5. Deviation from Homogeneity in Experiments and Modelling 6. Excursion into dyadic energy cascade models 7. Intermittent Energy Cascade and Onsager’s Conjecture Chapter IV. Concentrated Mikado Flows and Applications 8. Technical Prerequisites 9. Transport equation with Sobolev fields 10. Navier Stokes equation in dimension four and higher 11. Convex Integration for the Intermittent Onsager Conjecture Chapter V. Full Dimensional Concentration 12. Building blocks and methods 13. Transport equation with Sobolev fields 14. Three-dimensional Navier-Stokes equation Chapter VI. Temporal Intermittency 15. Hypodissipative Navier-Stokes equations 16. Two-dimensional Navier-Stokes equation and sharp non-uniqueness 17. Transport with almost Lipschitz fields and almost smooth density Chapter VII. Baire Category Method for Intermittent Convex Integration 18. Outline of the Baire category method 19. Genericity of three-dimensional Navier-Stokes solutions 20. Genericity of solutions to the transport equation with Sobolev fields Bibliography