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.