On the Support of Anomalous Dissipation Measures

• Published in: Journal of mathematical fluid mechanics Vol. 26; no. 4
• Main Authors: De Rosa, Luigi, Drivas, Theodore D., Inversi, Marco
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2024; Springer Nature B.V
• Subjects: Classical and Continuum Physics; Compressibility; Dissipation; Fluid flow; Fluid- and Aerodynamics; Incompressible flow; Lower bounds; Mathematical Methods in Physics; Navier-Stokes equations; Physics; Physics and Astronomy; Scalars; Conservation laws; Dimension lower bounds; Dissipation; 76F02; 35Q31; 76D05; 35L65; Incompressible fluids
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-024-00894-z

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 (Drivas and Eyink in Commun Math Phys 359(2):733–763, 2018. https://doi.org/10.1007/s00220-017-3078-4 ; 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 L t q L x r 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 = 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.