Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Well-posedness of the Euler equations in a stably stratified ocean in isopycnal coordinates

Théo Fradin

TL;DR

This work establishes local well-posedness for the incompressible Euler equations describing a stably stratified ocean in isopycnal coordinates, with a perturbation of size $\varepsilon$ around a shear equilibrium and without any added regularization. The authors develop pressure estimates via two elliptic problems and prove energy bounds using Alinhac's good unknown to handle the semi-Lagrangian, quasi-2D structure of the reformulated system; crucially, the time of existence is uniform in $\varepsilon$ and, under additional smallness assumptions on $\epsilon$ and the background shear, uniform in the shallow-water parameter $\mu$. The analysis extends prior isopycnal-coordinate results to include shear flows without diffusion, bridging Desjardins–Lannes–Saut (2020) in Eulerian coordinates and Duchêne (2022) in isopycnal coordinates with regularization, and provides a robust framework for stratified ocean dynamics. The results have potential implications for rigorous understanding of internal-wave–shear interactions and for the development of stratified-flow models in oceanography that avoid artificial regularization terms.

Abstract

This article is concerned with the well-posedness of the incompressible Euler equations describing a stably stratified ocean, reformulated in isopycnal coordinates. Our motivation for using this reformulation is twofold: first, its quasi-2D structure renders some parts of the analysis easier. Second, it closes a gap between the analysis performed in the paper by Bianchini and Duch{ê}ne in 2022 in isopycnal coordinates, with shear velocity but with a regularizing term, and the analysis performed in the paper by Desjardins, Lannes, Saut in 2020 in Eulerian coordinates, without any regularizing term but without shear velocity. Our main result is a local well-posedness result in Sobolev spaces on the system in isopycnal coordinates, with shear velocity, without any regularizing term. The time of existence that we obtain is uniform with respect to the size $ε$ of the perturbation, and boils down to the large time $1/ε$ with the assumptions of the paper by Desjardins, Lannes, Saut in 2020. With additional assumptions, it is also uniform in the shallow-water parameter. The main difficulty consists in transposing to the isopycnal reformulation the symmetric structure of the system which is more straightforward in Eulerian coordinates.

Well-posedness of the Euler equations in a stably stratified ocean in isopycnal coordinates

TL;DR

This work establishes local well-posedness for the incompressible Euler equations describing a stably stratified ocean in isopycnal coordinates, with a perturbation of size around a shear equilibrium and without any added regularization. The authors develop pressure estimates via two elliptic problems and prove energy bounds using Alinhac's good unknown to handle the semi-Lagrangian, quasi-2D structure of the reformulated system; crucially, the time of existence is uniform in and, under additional smallness assumptions on and the background shear, uniform in the shallow-water parameter . The analysis extends prior isopycnal-coordinate results to include shear flows without diffusion, bridging Desjardins–Lannes–Saut (2020) in Eulerian coordinates and Duchêne (2022) in isopycnal coordinates with regularization, and provides a robust framework for stratified ocean dynamics. The results have potential implications for rigorous understanding of internal-wave–shear interactions and for the development of stratified-flow models in oceanography that avoid artificial regularization terms.

Abstract

This article is concerned with the well-posedness of the incompressible Euler equations describing a stably stratified ocean, reformulated in isopycnal coordinates. Our motivation for using this reformulation is twofold: first, its quasi-2D structure renders some parts of the analysis easier. Second, it closes a gap between the analysis performed in the paper by Bianchini and Duch{ê}ne in 2022 in isopycnal coordinates, with shear velocity but with a regularizing term, and the analysis performed in the paper by Desjardins, Lannes, Saut in 2020 in Eulerian coordinates, without any regularizing term but without shear velocity. Our main result is a local well-posedness result in Sobolev spaces on the system in isopycnal coordinates, with shear velocity, without any regularizing term. The time of existence that we obtain is uniform with respect to the size of the perturbation, and boils down to the large time with the assumptions of the paper by Desjardins, Lannes, Saut in 2020. With additional assumptions, it is also uniform in the shallow-water parameter. The main difficulty consists in transposing to the isopycnal reformulation the symmetric structure of the system which is more straightforward in Eulerian coordinates.
Paper Structure (25 sections, 26 theorems, 306 equations, 2 figures)

This paper contains 25 sections, 26 theorems, 306 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General setting
  3. Ansatz
  4. Non-dimensionalization
  5. Main result
  6. Perspectives
  7. Structure of the proof and plan of the paper
  8. Notations
  9. Pressure estimates
  10. Statement of the results
  11. Proofs of the results
  12. A first elliptic equation
  13. A second elliptic equation
  14. Proof of Proposition \ref{['lemma:elliptic_high_reg']}
  15. Proof of Proposition \ref{['lemma:elliptic:high_reg:perte']}
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $d \in \mathbb{N}^*$, $s_0>\frac{d}{2}$, $s \in \mathbb{N}$ with $s \geq s_0+\frac{5}{2}$. Let $\epsilon \in [0,1]$, $\mu \in ]0,1]$. Let $\underline{V}, \varrho \in W^{s+1,\infty}([0,1])$ , $(V_{\mathrm{in}},w_{\mathrm{in}},\eta_{\mathrm{in}}) \in H^{s}(S_r)^{d+2}$ such that assumptions hyp:non Then there exists $T > 0$ independent of $\epsilon,\mu$ such that there exists a unique solution $(

Figures (2)

  • Figure 1: Setting in Eulerian coordinates. Isopycnals are sets where the density is constant.
  • Figure 2: Setting in isopycnal coordinates. Isopycnals are flattened.

Theorems & Definitions (65)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • ...and 55 more