On the Support of Anomalous Dissipation Measures

Abstract 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 Majda in Am Math Soc 43 (281): 93, 1983. https: //doi. org/10. 1090/memo/0281) and by recent constructions of dissipative incompressible Euler solutions (Brue and De Lellis in Commun Math Phys 400 (3): 1507–1533, 2023. https: //doi. org/10. 1007/s00220-022-04626-0 624 ; Brue et al. in Commun Pure App Anal, 2023), as well as passive scalars (Colombo et al. in Ann PDE 9 (2): 21–48, 2023. https: //doi. org/10. 1007/s40818-023-00162-9 ; Drivas et al. in Arch Ration Mech Anal 243 (3): 1151–1180, 2022. https: //doi. org/10. 1007/s00205-021-01736-2). For LqₜLʳₓ L t q L x r suitable Leray–Hopf solutions of the d- d - dimensional Navier–Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure P^s P s, which gives s=d-2 s = d - 2 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.