By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier-Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler [14,36] and by recent constructions of dissipative incompressible Euler solutions [1, 2], as well as passive scalars [9,13]. For L q t L r x suitable Leray-Hopf solutions of the d−dimensional Navier-Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure P s , which gives s = 1 as soon as the solution lies in the Prodi-Serrin class. In the three-dimensional case, this matches with the Caffarelli-Kohn-Nirenberg partial regularity.

In the class of admissible weak solutions, we prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to, where L exp denotes the Orlicz space of exponentially integrable functions. Moreover, under the same assumptions on the limit solution to the Euler system, we obtain the convergence of vanishing-viscosity Leray-Hopf weak solutions of the Navier-Stokes equations.