The Rayleigh–Taylor instability with local energy dissipation

Abstract We consider the inhomogeneous incompressible Euler equations including their local energy inequality as a differential inclusion. Providing a corresponding convex integration theorem and constructing subsolutions, we show the existence of locally dissipative Euler flows emanating from the horizontally flat Rayleigh–Taylor configuration and having a mixing zone which grows quadratically in time. For the Rayleigh–Taylor instability these are the first turbulently mixing solutions known to respect local energy dissipation, and outside the range of Atwood numbers considered in Gebhard et al. (Arch Ration Mech Anal 241(3):1243–1280, 2021), the first weakly admissible solutions in general. In the coarse grained picture the existence relies on one-dimensional subsolutions described by a family of hyperbolic conservation laws, among which one can find the optimal background profile appearing in the scale invariant bounds from Kalinin et al. (SIAM J Math Anal 56(6):7846–7865, 2024), and as we show, the optimal conservation law with respect to maximization of the total energy dissipation. Furthermore, we also show that the least action admissibility criteria from Gimperlein et al. (Arch Ration Mech Anal 249(2):22, 2025; arXiv:2503.03491 , 2025) selects rather the stationary solution within our family of conservation laws.