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 (27)

• On admissibility criteria for weak solutions of the Euler equations
• Global ill-posedness of the isentropic system of gas dynamics
• Regularity and Energy Conservation for the Compressible Euler Equations
• A counterexample to well-posedness of entropy solutions to the compressible Euler system
• Lack of uniqueness for weak solutions of the incompressible porous media equation
• On the energy dissipation rate of solutions to the compressible isentropic Euler system
• Maximal dissipation and well-posedness for the compressible Euler system
• Bounded solutions of ideal MHD with compact support in space-time
• The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock
• Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations
• Piecewise constant subsolutions for the Muskat problem
• On Non-uniqueness of Hölder continuous globally dissipative Euler flows
• A new approach to the Rayleigh-Taylor instability
• Existence and non-uniqueness of global weak solutions to inviscid primitive and Boussinesq equations
• The 2-d isentropic compressible Euler equations may have infinitely many solutions which conserve energy
• Weak stablity and closure in turbulence
• Relaxation of the Boussinesq system and applications to the Rayleigh-Taylor instability
• Semiclassical estimates for pseudodifferential operators and the Muskat problem in the unstable regime
• On Non-uniqueness of continuous entropy solutions to the isentropic compressible Euler equations
• H-principle for the 2D incompressible porous media equation with viscosity jump
• On bounded two-dimensional globally dissipative Euler flows
• The Least Action Admissibility Principle
• A New Convex Integration Approach for the Compressible Euler Equations and Failure of the Local Maximal Dissipation Criterion
• On a degenerate elliptic problem arising in the least action principle for Rayleigh-Taylor subsolutions
• Scale invariant bounds for mixing in the Rayleigh-Taylor instability
• Entropy solutions to macroscopic IPM
• On the energy-constrained optimal mixing problem for one-dimensional initial configurations