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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 & 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$$L^q_tL^r_x$$ $L_t^q{L}_x^r$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$$\mathcal {P}^{s}$$ $P^s$, 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. 

Abstract In the context of incompressible fluids, the observation that turbulent singular structures fail to be space filling is known as “intermittency”, and it has strong experimental foundations. Consequently, as first pointed out by Landau, real turbulent flows do not satisfy the central assumptions of homogeneity and self-similarity in the K41 theory, and the K41 prediction of structure function exponents$$\zeta _p={p}/{3}$$ ${\zeta }_p=p/3$might be inaccurate. In this work we prove that, in the inviscid case, energy dissipation that is lower-dimensional in an appropriate sense implies deviations from the K41 prediction in everyp-th order structure function for$$p>3$$ $p>3$. By exploiting a Lagrangian-type Minkowski dimension that is very reminiscent of the Taylor’sfrozen turbulencehypothesis, our strongest upper bound on$$\zeta _p$$ ${\zeta }_p$coincides with the$$\beta $$ $\beta$-model proposed by Frisch, Sulem and Nelkin in the late 70s, adding some rigorous analytical foundations to the model. More generally, we explore the relationship between dimensionality assumptions on the dissipation support and restrictions on thep-th order absolute structure functions. This approach differs from the current mathematical works on intermittency by its focus on geometrical rather than purely analytical assumptions. The proof is based on a new local variant of the celebrated Constantin-E-Titi argument that features the use of a third order commutator estimate, the special double regularity of the pressure, and mollification along the flow of a vector field.