On the Support of Anomalous Dissipation Measures

TL;DR

This paper establishes measure-theoretic bounds on the support of anomalous dissipation in fluid equations, linking it to the Hausdorff dimension and providing sharp results for various fluid models and boundary conditions.

Contribution

It introduces a unifying measure-theoretic framework to bound the Hausdorff dimension of dissipation support, extending previous results to boundary cases and zero viscosity limits.

Findings

Lower bounds on dissipation support dimension for weak solutions.

Sharpness demonstrated via shock solutions and dissipative Euler solutions.

Bound on dissipation in terms of Parabolic Hausdorff measure for Navier-Stokes solutions.

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 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.