On the Support of Anomalous Dissipation Measures

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 and by recent constructions of dissipative incompressible Euler solutions, as well as passive scalars. For $L^q_tL^r_x$ 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 soon as the solution lies in the Prodi-Serrin class. In the three-dimensional case, this matches with the Caffarelli-Kohn-Nirenberg partial regularity.