On Nonperiodic Euler Flows with Hölder Regularity

By Philip Isett, Sung-Jin Oh. 2014

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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 $${1/3}$$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 $${L_t^\infty B_{3,\infty}^{1/3}}$$Lt∞B3,∞1/3 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 $${1/5}$$1/5. The main result of the present paper shows that any given smooth Euler flow can be perturbed in $${C^{1/5-\epsilon}_{t,x}}$$Ct,x1/5-ϵ on any pre-compact subset of $${\mathbb{R}\times \mathbb{R}^3}$$R×R3 to violate energy conservation. Furthermore, the perturbed solution is no smoother than $${C^{1/5-\epsilon}_{t,x}}$$Ct,x1/5-ϵ. As a corollary of this theorem, we show the existence of nonzero $${C^{1/5-\epsilon}_{t,x}}$$Ct,x1/5-ϵ 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.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/383628cbdfe4ed929a1af9497c6933814791e4bd

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations (2016, background): “1 in [14], the energy obtained will be very large compared with the energy of v̄.”
Convex integration and phenomenologies in turbulence (2019, methodology): “In earlier papers [48, 49, 16, 14, 103, 17], Beltrami waves were used as the building blocks of the convex”