Fine dissipative properties of Euler solutions with measure first derivatives

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 yield nontrivial conclusions when only the vorticity is assumed to be a measure.