  In [Isett,13], the first author proposed a strengthening of Onsager's
conjecture on the failure of energy conservation for incompressible Euler flows
with H\"{o}lder 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 $L_t^\infty
B_{3,\infty}^{1/3}$ due to low regularity of the energy profile. In this paper,
we establish two theorems that may be viewed as first steps towards
establishing the conjectured failure of energy regularity for generic solutions
with H\"{o}lder exponent less than $1/5$.
  Our first result shows that any non-negative function with compact support
and H\"older regularity $1/2$ can be prescribed as the energy profile of an
Euler flow in the class $C^{1/5-\epsilon}_{t,x}$. The exponent $1/2$ is sharp
in view of a regularity result of [Isett,13]. The proof employs an improved
greedy algorithm scheme that builds upon that in [Buckmaster-De
Lellis-Sz\'ekelyhidi, 13]. Our second result shows that any given smooth Euler
flow can be perturbed in $C^{1/5-\epsilon}_{t,x}$ on any pre-compact subset of
${\mathbb R}\times {\mathbb R}^3$ to violate energy conservation. In
particular, there exist nonzero $C^{1/5-\epsilon}_{t,x}$ solutions to Euler
with compact space-time support, generalizing previous work of the first author
[Isett,12] to the nonperiodic setting.
