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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.: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

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the 3-D incompressible Euler equations belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve kinetic energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which do not conserve kinetic energy. The first part, relating to conservation of kinetic energy, has since been confirmed (cf. Eyink 1994, Constantin-E-Titi 1994). The second part, relating to the existence of non-conservative solutions, remains an open conjecture and is the subject of this dissertation. In groundbreaking work of De Lellis and Székelyhidi Jr. (2012), the authors constructed the first examples of non-conservative Hölder continuous weak solutions to the Euler equations. The construction was subsequently improved by Isett (2012/2013), introducing many novel ideas in order to construct 1/5− Hölder continuous weak solutions with compact support in time. Adhering more closely to the original scheme of De Lellis and Székelyhidi Jr., we present a comparatively simpler construction of 1/5− Hölder continuous non-conservative weak solutions which may in addition be made to obey a prescribed kinetic energy profile. Furthermore, we extend this scheme in order to construct weak non-conservative solutions to the Euler equations whose Hölder 1/3− norm is Lebesgue integrable in time. The dissertation will be primarily based on three papers, two of which being in collaboration with De Lellis and Székelyhidi Jr.

In dieser Arbeit werden zwei Klassen von Evolutionsgleichungen in einem Matrixraum-Setting studiert: Hyperbolische Erhaltungsgleichungen und aktive skalare Gleichungen. Für erstere wird untersucht, wann man Oszillationen mit Hilfe polykonvexen Maßen ausschließen kann; für Zweitere wird mit Hilfe von Oszillationen gezeigt, dass es unendlich viele periodische schwache Lösungen gibt.

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the 3-D incompressible Euler equations belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve kinetic energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which do not conserve kinetic energy. The first part, relating to conservation of kinetic energy, has since been confirmed (cf. Eyink 1994, Constantin-E-Titi 1994). The second part, relating to the existence of non-conservative solutions, remains an open conjecture and is the subject of this dissertation. In groundbreaking work of De Lellis and Székelyhidi Jr. (2012), the authors constructed the first examples of non-conservative Hölder continuous weak solutions to the Euler equations. The construction was subsequently improved by Isett (2012/2013), introducing many novel ideas in order to construct 1/5− Hölder continuous weak solutions with compact support in time. Adhering more closely to the original scheme of De Lellis and Székelyhidi Jr., we present a comparatively simpler construction of 1/5− Hölder continuous non-conservative weak solutions which may in addition be made to obey a prescribed kinetic energy profile. Furthermore, we extend this scheme in order to construct weak non-conservative solutions to the Euler equations whose Hölder 1/3− norm is Lebesgue integrable in time. The dissertation will be primarily based on three papers, two of which being in collaboration with De Lellis and Székelyhidi Jr.

The aim of this work is to provide new results of global existence for dif- ferent evolution equations of fluid mechanics. We are in general interested in finding weak solutions without restrictions on the size of initial data. The proofs of existence are based on several different approaches including en- ergy methods, convergence analysis of finite numerical methods and convex integration. All these techniques significantly exploit results of mathematical analysis and other branches of mathematics. 1

In dieser Arbeit werden zwei Klassen von Evolutionsgleichungen in einem Matrixraum-Setting studiert: Hyperbolische Erhaltungsgleichungen und aktive skalare Gleichungen. Für erstere wird untersucht, wann man Oszillationen mit Hilfe polykonvexen Maßen ausschließen kann; für Zweitere wird mit Hilfe von Oszillationen gezeigt, dass es unendlich viele periodische schwache Lösungen gibt.