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On the zero capillarity limit for the Euler-Korteweg system

Corentin Audiard, Marc-Antoine Vassenet

TL;DR

This work analyzes the zero-capillarity limit of the Euler-Korteweg system, proving rigorous convergence to the Euler equations in $\mathbb{R}^d$ and introducing a high-order BKW framework to capture the limit with explicit rates. It also initiates a half-space study, deriving a priori estimates that degenerate as the capillarity vanishes and revealing boundary-layer structures that explain the degeneracy. The full-space results connect to Grenier’s semi-classical analysis for nonlinear Schrödinger equations by providing a fluid-dynamic, symmetrizable-hyperbolic perspective and arbitrarily accurate approximate solutions. In the half-space, the analysis treats quantum fluids with irrotational velocity under $K(\rho)=1/\rho$ and constructs a two-scale BKW expansion featuring boundary layers, with solvability and decay properties established for the interior and boundary-layer components. The paper thus clarifies the interplay between capillarity, boundary effects, and the Euler limit, and sets groundwork for more general boundary conditions and extensions in quantum-fluid models.

Abstract

We study the Euler-Korteweg equations with a weak capillarity tensor. It formally converges to the Euler equations in the zero capillarity limit. Our aim is two-fold : first we prove rigorously this limit in R d , d $\ge$ 1, and obtain a more precise BKW expansion of the solution, second we initiate the study of the problem on the half space. In this case we obtain a priori estimates for the solutions that degenerate as the capillary coefficient converges to zero, and we explain this degeneracy with the construction of a (formal) BKW expansion that exhibits boundary layers. The results on the full space extend and improve a classical result of Grenier (1998) on the semi-classical limit of nonlinear Schr{ö}dinger equations. The analysis on the half space is restricted to the case of quantum fluids with irrotational velocity.

On the zero capillarity limit for the Euler-Korteweg system

TL;DR

This work analyzes the zero-capillarity limit of the Euler-Korteweg system, proving rigorous convergence to the Euler equations in and introducing a high-order BKW framework to capture the limit with explicit rates. It also initiates a half-space study, deriving a priori estimates that degenerate as the capillarity vanishes and revealing boundary-layer structures that explain the degeneracy. The full-space results connect to Grenier’s semi-classical analysis for nonlinear Schrödinger equations by providing a fluid-dynamic, symmetrizable-hyperbolic perspective and arbitrarily accurate approximate solutions. In the half-space, the analysis treats quantum fluids with irrotational velocity under and constructs a two-scale BKW expansion featuring boundary layers, with solvability and decay properties established for the interior and boundary-layer components. The paper thus clarifies the interplay between capillarity, boundary effects, and the Euler limit, and sets groundwork for more general boundary conditions and extensions in quantum-fluid models.

Abstract

We study the Euler-Korteweg equations with a weak capillarity tensor. It formally converges to the Euler equations in the zero capillarity limit. Our aim is two-fold : first we prove rigorously this limit in R d , d 1, and obtain a more precise BKW expansion of the solution, second we initiate the study of the problem on the half space. In this case we obtain a priori estimates for the solutions that degenerate as the capillary coefficient converges to zero, and we explain this degeneracy with the construction of a (formal) BKW expansion that exhibits boundary layers. The results on the full space extend and improve a classical result of Grenier (1998) on the semi-classical limit of nonlinear Schr{ö}dinger equations. The analysis on the half space is restricted to the case of quantum fluids with irrotational velocity.

Paper Structure

This paper contains 28 sections, 15 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Link with the Schrödinger equation
  3. The Schrödinger equation on the full space
  4. The Schrödinger equation on a domain
  5. The general case of the Euler-Korteweg system
  6. Main results
  7. Remark
  8. Plan of the paper
  9. Notations, functional settings
  10. Differential calculus
  11. Irrotational and solenoidal vector fields
  12. Functional spaces
  13. Analysis on the whole space
  14. Energy estimates and the time of existence
  15. Computation of $D_1:=\frac{d}{2dt}\int_{\mathbb{R}^d}a^{2n}\rho |\mathbb{Q}\Delta^nz|^2dx$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\psi_\varepsilon$ solution of schrodinger with, for some $J\in \mathbb{N}$, $\psi_\varepsilon|_{t=0}= a_0(x,\varepsilon)e^{i\varphi_0(x,\varepsilon)/\varepsilon)}$, $a_0=\sum_0^J \varepsilon^ja^j_0(x)+\varepsilon^Jr^J_\varepsilon(x)$ , $\varphi_0=\sum_{j=0}^J\varepsilon^j\varphi_0^j+\varepsilon Then there exists $T>0$ such that $\psi_\varepsilon$ has the form $\psi_\varepsilon=a_\varepsilon e

Theorems & Definitions (26)

  • Theorem 1.1: Grenier '98
  • Proposition 1.2: Existence of an approximate solution
  • Theorem 1.3: Convergence of the approximate solution
  • Proposition 1.4: Approximate solution as a two scale expansion
  • Proposition 2.1
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Remark 1
  • Lemma 3.3
  • ...and 16 more