Onsager's conjecture for admissible weak solutions

2017/01/30 by Buckmaster, Tristan, De Lellis, Camillo, Székelyhidi, László +1 · 8 citations
#35D30 #35Q31 (Primary) #76B03 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1701.08678

Abstract

We prove that given any β<1/3, a time interval [0,T], and given any smooth energy profile e \colon [0,T] → (0,∞), there exists a weak solution v of the three-dimensional Euler equations such that v ∈ C^β([0,T]× \mathbbT³), with e(t) = ∫_\mathbbT³ |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.

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