Abstract. We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution.As a byproduct we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.

We propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in ‫ޒ‬ n with n 2. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy

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We prove that given any β < 1/3, a time interval [0, T ], and given any smooth energy profile e : [0, T ] → (0, ∞), there exists a weak solution v of the three-dimensional Euler equations such that v ∈ C β ([0, T ] × T 3 ), with e(t) = ´T3 |v(x, t)| 2 dx for all t ∈ [0, T ]. Moreover, we show that a suitable h-principle holds in the regularity class C β t,x , for any β < 1/3. The implication of this is that the dissipative solutions we construct are in a sense typical in the appropriate space of subsolutions as opposed to just isolated examples.Date: January 31, 2017. 1 The smallest constant C satisfying (1.2) will be denoted by [v] β , cf. Appendix A. We will write v ∈ C β (T 3 ×[0, T ]) when v is Hölder continuous in the whole space-time.