The Energy Measure for the Euler and Navier-Stokes Equations

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.