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The Rayleigh-Taylor instability with local energy dissipation

Björn Gebhard, József J. Kolumbán

TL;DR

The paper develops a rigorous framework to construct turbulently mixing solutions of the inhomogeneous incompressible Euler equations with local energy dissipation, by embedding gravity-driven Rayleigh–Taylor dynamics into a differential inclusion and applying convex integration. It introduces accelerated coordinates to absorb gravity, derives a full \\Lambda-convex hull description, and builds one-dimensional subsolutions governed by hyperbolic conservation laws for the coarse-grained density; these subsolutions generate infinitely many weak solutions with a mixing zone growing as $gt^2$ while respecting the local energy inequality. A key finding is that, among a family of one-dimensional constitutive laws $m_n=gt( ho+F( ho))$, the choice $F( ho)=F_{ rac13}( ho)$ maximizes total energy dissipation, yielding the strongest turbulent mixing, whereas the conservative choice $F$ corresponding to $ ext{$ rac12$}$ produces a background profile that attains sharp scale-invariant bounds. The results extend prior work on local energy admissibility (notably GK-EE) to the inhomogeneous Euler setting, provide a clear subsolution selection criterion, and offer explicit expressions for the mixing zone and energy budgets, with potential implications for modeling RT-driven turbulence in variable-density flows.

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.

The Rayleigh-Taylor instability with local energy dissipation

TL;DR

The paper develops a rigorous framework to construct turbulently mixing solutions of the inhomogeneous incompressible Euler equations with local energy dissipation, by embedding gravity-driven Rayleigh–Taylor dynamics into a differential inclusion and applying convex integration. It introduces accelerated coordinates to absorb gravity, derives a full \\Lambda-convex hull description, and builds one-dimensional subsolutions governed by hyperbolic conservation laws for the coarse-grained density; these subsolutions generate infinitely many weak solutions with a mixing zone growing as while respecting the local energy inequality. A key finding is that, among a family of one-dimensional constitutive laws , the choice maximizes total energy dissipation, yielding the strongest turbulent mixing, whereas the conservative choice corresponding to rac12 produces a background profile that attains sharp scale-invariant bounds. The results extend prior work on local energy admissibility (notably GK-EE) to the inhomogeneous Euler setting, provide a clear subsolution selection criterion, and offer explicit expressions for the mixing zone and energy budgets, with potential implications for modeling RT-driven turbulence in variable-density flows.

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.
Paper Structure (27 sections, 26 theorems, 232 equations)

This paper contains 27 sections, 26 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Change of coordinates
  3. The associated differential inclusion
  4. Statement of results 1 -- Convex integration
  5. Statement of results 2 -- Subsolutions
  6. Related results
  7. Local energy inequality
  8. Inhomogeneous Euler and Boussinesq
  9. Selection of subsolutions
  10. Connection to scale invariant bounds
  11. Convex integration via the Tartar framework
  12. Localized plane waves
  13. The unbounded and bounded convex hulls
  14. Step 0. - Preliminaries
  15. Computing the hulls
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\rho_+>\rho_->0$, $\Omega=\mathbb{T}^{n-1}\times(-L,L)$, $L>0$, $\left|\mathbb{T}\right|=1$, $n=2,3$ and $\lambda\in (0,1/2)$. There exist infinitely many weak solutions $(\rho,v)\in L^\infty(\Omega\times(0,T);\mathbb{R}\times \mathbb{R}^n)$ of eq:flat_initial_data, eq:euler_only, eq:euler_loca The solutions satisfy $\rho\in\left\{\,\rho_-,\rho_+\,\right\}$ almost everywhere and are induced b

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Proposition 1.7
  • Definition 1.8
  • Definition 1.9
  • Proposition 1.10
  • ...and 42 more