The Rayleigh-Taylor instability with local energy dissipation

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 arXiv:2002.08843, 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 arXiv:2303.01889, and as we show, the optimal conservation law with respect to maximization of the total energy dissipation.