Title: The Energy Measure for the Euler and Navier-Stokes Equations,
Abstract: The potential failure of energy equality for a solution $u$ of the Euler or
Navier-Stokes equations can be quantified using a so-called `energy measure':
the weak-$*$ limit of the measures $|u(t)|^2\,\mbox{d}x$ as $t$ approaches the
first possible blowup time. We show that membership of $u$ in certain (weak or
strong) $L^q L^p$ classes gives a uniform lower bound on the lower local
dimension of $\mathcal{E}$; more precisely, it implies uniform boundedness of a
certain upper $s$-density of $\mathcal{E}$. We also define and give lower
bounds on the `concentration dimension' associated to $\mathcal{E}$, which is
the Hausdorff dimension of the smallest set on which energy can concentrate.
Both the lower local dimension and the concentration dimension of $\mathcal{E}$
measure the departure from energy equality. As an application of our estimates,
we prove that any solution to the $3$-dimensional Navier-Stokes Equations which
is Type-I in time must satisfy the energy equality at the first blowup time.