Sharp Energy Regularity And Typicality Results For Holder Solutions Of Incompressible Euler Equations

We show a couple of typicality results for weak solutions v is an element of C-theta of the Euler equations, in the case theta < 1/3. It is known that convex integration schemes produce wild weak solutions that exhibit anomalous dissipation of the kinetic energy e(v). 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 similar to -Holder continuous weak solution v of the Euler equations satisfies e(v) not subset of C2 theta/(1-theta). As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space X-theta that is contained in the space of all C-theta weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions v is an element of X-theta, with e(v) not subset of C2 theta/(1-theta) but e(v) boolean OR(P >= 1 epsilon>0) W-2 theta/(1-theta)+epsilon,W-p(I) for any open I subset of [0, T], are a residual set in X-theta. 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 C-theta 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.