Weak and Measure-Valued Solutions of the Incompressible Euler Equations

Abstract

This thesis is concerned with the existence problem for weak solutions of the incompressible Euler equations in arbitrary dimension, and with the relationship between weak solutions and other "very weak'' concepts of solution. In particular, measure-valued solutions as introduced by R. DiPerna and A. Majda (Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys., 108(4):667-689, 1987) are studied.<br /> There are three main results of this thesis: Theorem 1.1 asserts the global existence of weak solutions for the incompressible Euler equations. However, these solutions are physically not admissible since their kinetic energy increases at least at the initial time. Moreover, the solutions constructed are highly non-unique in the sense that there exist infinitely many weak solutions with the same initial data. Concerning admissible weak solutions (i.e. such whose energy never exceeds the initial energy), the second result, Theorem 1.2, shows that they exist globally in time at least for a dense subset of initial data. The last result, Theorem 1.3, elucidates the relationship between weak and measure-valued solutions: It is shown that every measure-valued solution is generated by a sequence of weak solutions and that therefore, surprisingly, weak solutions are as flexible as measure-valued solutions. <br /> A common feature of these results is their relying on methods recently developed by C. De Lellis and L. Székelyhidi Jr. (On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal., 195(1):225-260, 2010). This thesis includes a fairly detailed presentation of these methods.