Onsager's energy conservation for inhomogeneous Euler equations

TL;DR

This paper establishes conditions under which energy conservation holds for inhomogeneous Euler equations, extending classical results to include rough density profiles and inhomogeneous flows.

Contribution

It introduces new regularity conditions on density and velocity that guarantee energy conservation for inhomogeneous Euler equations, generalizing previous homogeneous results.

Findings

Energy conservation holds under integrability of density gradient.

Energy conservation holds with additional time Besov regularity on velocity.

Results apply to a broad class of rough density profiles.

Abstract

This paper addresses the problem of energy conservation for the two- and three-dimensional density-dependent Euler equations. Two types of sufficient conditions on the regularity of solutions are provided to ensure the conservation of total kinetic energy on the entire time interval including the initial time. The first class of data assumes integrability on the spatial gradient of the density, and hence covers the classical result of Constantin-E-Titi for the homogeneous Euler equations. The other type of data imposes extra time Besov regularity on the velocity profile, and the corresponding result can be applied to deal with a wide class of rough density profiles.