On Nonperiodic Euler Flows with Hölder Regularity

Abstract

In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager’s conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/31/3 . This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L[∞ over t]B[1/3 over 3,∞] due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/51/5 . The main result of the present paper shows that any given smooth Euler flow can be perturbed in C[1/5−ϵ over t,x] on any pre-compact subset of R×R[superscript 3] to violate energy conservation. Furthermore, the perturbed solution is no smoother than C[1/5−ϵ over t,x]. As a corollary of this theorem, we show the existence of nonzero C[1/5−ϵ over t,x] solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.