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 1/3. 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^∞_t(B^(1/3)_(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 1/5 . The main result of the present paper shows that any given smooth Euler flow can be perturbed in C^(1/5 − ϵ)_(t,x) on any pre-compact subset of ℝ×ℝ^3 to violate energy conservation. Furthermore, the perturbed solution is no smoother than C^(1/5 − ϵ)_(t,x). As a corollary of this theorem, we show the existence of nonzero C^(1/5 − ϵ)_(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.

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