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Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit

Roberta Bianchini, Michele Coti Zelati, Lucas Ertzbischoff

Abstract

We investigate the hydrostatic approximation for inviscid stratified fluids, described by the two-dimensional Euler-Boussinesq equations in a periodic channel. Through a perturbative analysis of the hydrostatic homogeneous setting, we exhibit a stratified steady state violating the Miles-Howard criterion and generating a growing mode, both for the linearized hydrostatic and non-hydrostatic equations. By leveraging long-wave nonlinear instability for the original Euler-Boussinesq system, we demonstrate the breakdown of the hydrostatic limit around such unstable profiles. Finally, we establish the generic nonlinear ill-posedness of the limiting hydrostatic system in Sobolev spaces.

Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit

Abstract

We investigate the hydrostatic approximation for inviscid stratified fluids, described by the two-dimensional Euler-Boussinesq equations in a periodic channel. Through a perturbative analysis of the hydrostatic homogeneous setting, we exhibit a stratified steady state violating the Miles-Howard criterion and generating a growing mode, both for the linearized hydrostatic and non-hydrostatic equations. By leveraging long-wave nonlinear instability for the original Euler-Boussinesq system, we demonstrate the breakdown of the hydrostatic limit around such unstable profiles. Finally, we establish the generic nonlinear ill-posedness of the limiting hydrostatic system in Sobolev spaces.
Paper Structure (10 sections, 21 theorems, 279 equations)

This paper contains 10 sections, 21 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Main difficulties and context
  4. Strategy of the proof and organization of the paper
  5. Construction of linear instabilities
  6. Unstable unbounded spectrum for the hydrostatic Euler-Boussinesq equations
  7. Long-wave instability for the Euler-Boussinesq system
  8. Failure of the hydrostatic limit
  9. Nonlinear ill-posedness for the hydrostatic Euler-Boussinesq system
  10. Unstable profile for the homogeneous hydrostatic equations

Key Result

Theorem 1.1

There exists an analytic stationary profile $(\rho_s(z), U_s(z))$, with $\rho_s'<0$, which does not satisfy the Miles-Howard condition eq:Miles-Howard, such that the following holds. For all $p,k \in \mathbb{N}$, there exist $m>0$, a family of smooth solutions $(\varrho_\varepsilon, u_\varepsilon, v while

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 37 more