Weak solutions of the Euler equations: non-uniqueness and dissipation

Abstract

These notes are based on a series of lectures given at the meeting Journées EDP in Roscoff in June 2015 on recent developments concerning weak solutions of the Euler equations and in particular recent progress concerning the construction of Hölder continuous weak solutions and Onsager's conjecture.

Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations

“…Thus, in analogy with the Nash-Kuiper result, Theorem 1.2 says that any smooth strict subsolution can be weakly approximated by C 0 weak solutions with prescribed energy. For more information on the connection between the Nash-Kuiper iteration and the Euler equations we refer to the surveys [9,6,21] and the lecture notes [22]. 1.3.…”

On Turbulence and Geometry: from Nash to Onsager

“…In this way we can push the error to high frequencies by successively "undoing" the averaging process leading to Reynolds stresses in (8)- (9). As explained in [29,62], starting from the Ansatz above it is possible to write down a family of conditions that would have to satisfy, ideally, so to give a "clean" convex integration iteration leading to a proof of Theorem 1(b). Although this family of conditions is somewhat naive and unfortunately no such exists (indeed, the scaling of time in (15) is clearly "wrong"), approximations based on a special family of stationary solutions of Euler called Beltrami flows can be used.…”

High dimensionality and h-principle in PDE

“…The paradox of Scheffer. There are various notions of weak solutions (see for instance the survey articles [36] and [80]), and despite the fact that uniqueness in general fails for such notions (see Theorem 6.1 below and [30,31,35] for further results), weak solutions have been studied because of their possible relevance to homogeneous 3-dimensional turbulence [18,24,42,68]. In particular we will consider pairs (v, p) :…”

“…Similar considerations (see for instance [80]) lead to the following set of conditions that we would like to impose on W :…”

“…Here we merely state Scheffer's amazing result from 1993 and refer to previous surveys [36,80] for further results on distributional solutions. This theorem was proved first by V. Scheffer [72] …”