Onsager's Conjecture

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. [Eyi94, CWT94]). 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. [DLSJ12a, DLSJ12b], the authors constructed the first examples of non-conservative Hölder continuous weak solutions to the Euler equations. The construction was subsequently improved by Isett [Ise12, Ise13a], introducing many novel ideas in order to construct 1/5 − ε 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 − ε Hölder continuous nonconservative weak solutions which may in addition be made to obey a prescribed kinetic energy profile.1 Furthermore, we extend this scheme in order to construct weak nonconservative solutions to the Euler equations whose Hölder 1/3 − ε norm is Lebesgue integrable in time. The dissertation will be primarily based on three papers: [BDLSJ13], [Buc13] and [BDLS14] – the first and third paper being in collaboration with De Lellis and Székelyhidi Jr. Introduction The Euler Equation The Onsager Conjecture References and Remarks Outline of Convex Integration Scheme Convex Integration and the Approach of De Lellis and Székelyhidi Jr. to Onsager's Conjecture The Convex Integration Scheme of Isett An Examination of Scales Convergence of the Energy References and Remarks Cancellation of low frequency error Beltrami Flows The Operator R References and Remarks Minimisation of Transport Error The Principal Transport Error Transport Error of Previous Reynolds Stress Transport Estimates References and Remarks Perturbation estimates Additional Notation and Parameter Orderings Estimates on Components of Perturbation References and Remarks Reynolds Stress Estimates Reynolds Stress Estimates References and Remarks Proof of Theorem 1.2.2 Estimates on the Energy Main Proposition and Choice of Parameters Conclusion of Proof of Theorem 1.2.2 References and Remarks Proof of Theorem 1.2.3 Bookkeeping, Partitioning and Parameter Choice Main Proposition and Parameter Inequalities Conclusion of the Proof of Theorem 1.2.3 References and Remarks Appendix Hölder spaces Linear Partial Differential Equation Theory References