SHARP ENERGY REGULARITY AND TYPICALITY RESULTS FOR HÖLDER SOLUTIONS OF INCOMPRESSIBLE EULER EQUATIONS

We show a couple of typicality results for weak solutions v ∈ Cθ of the Euler equations, in the case (Formula Presented). It is known that convex integration schemes produce wild weak solutions that exhibit anomalous dissipation of the kinetic energy ev. We show that those solutions are typical in the Baire category sense. From work of Isett (2013, arXiv:1307.0565), it is know that the kinetic energy ev of a θ-Hölder continuous weak solution v of the Euler equations satisfies ev ∈ C2θ/(1-θ). As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space Xθ that is contained in the space of all Cθ weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions v ∈ Xθ, with (Formula Presented) but (Formula Presented)for any open I ⊂ T [0,T], are a residual set in Xθ. This, in particular, partially solves Conjecture 1 of Isett and Oh (Arch. Ration. Mech. Anal. 221:2 (2016), 725–804). We also show that smooth solutions form a nowhere dense set in the space of all the Cθ weak solutions. The technique is the same and what really distinguishes the two cases is that in the latter there is no need to introduce a different complete metric space with respect to the natural one