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 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.

References in corpus (4)

• The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock
• A New Convex Integration Approach for the Compressible Euler Equations and Failure of the Local Maximal Dissipation Criterion
• The Least Action Admissibility Principle
• On the energy-constrained optimal mixing problem for one-dimensional initial configurations