Fine dissipative properties of Euler solutions with measure first derivatives
Marco Inversi
TL;DR
The paper studies fine dissipation properties for bounded Euler solutions whose first derivatives are Radon measures, establishing energy conservation ($D\equiv0$) in several critical regimes: BV, BD, and measure-vorticity settings. The main strategy leverages the specific structure of the Euler nonlinearity and incompressibility, deriving direct BV-chain-rule proofs and BD-based arguments that avoid reliance on kernel-optimization or linear renormalization tactics. In the BV case, a direct chain-rule argument shows the Duchon–Robert defect vanishes; in the BD case, blow-up analysis of divergence-free fields yields the same conclusion; for measure vorticity, a refined momentum-equation manipulation demonstrates no energy dissipation under mild singular-set assumptions. These results extend Onsager-type energy conservation into critical, measure-valued regularity classes and provide robust mechanisms to conclude energy conservation without resorting to heavy mollification dependencies. The findings have implications for understanding energy transfer and dissipation mechanisms in turbulent Euler flows at sharp regularity thresholds, including vortex-sheet configurations.
Abstract
We study fine properties of bounded weak solutions to the incompressible Euler equations whose first derivatives, or only some combinations of them, are Radon measures. As consequences we obtain elementary proofs of the local energy conservation for solutions in BV and BD, without relying on the freedom in choosing the convolution kernel appearing in the approximation of the dissipation. The argument heavily exploits the form of the Euler nonlinearity and it does not apply to the linear transport equations, where the renormalization property for BD vector fields is an open problem. The methods also yields to nontrivial conclusions when only the vorticity is assumed to be a measure.