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 a...
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Published in | Journal of mathematical fluid mechanics Vol. 26; no. 4 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Cham
Springer International Publishing
2024
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 1422-6928 1422-6952 |
DOI: | 10.1007/s00021-024-00894-z |