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The zero capillarity limit for the Euler-Korteweg system with no-flux boundary conditions

Paolo Antonelli, Yuri Cacchiò

TL;DR

This work analyzes the vanishing capillarity limit for the Euler-Korteweg system with no-flux boundary conditions on bounded domains. It develops a relative-entropy framework to prove convergence of finite-energy weak solutions to strong Euler solutions, while addressing boundary-layer effects through a high-order augmented entropy and a boundary-layer correction. The authors extend prior relative-entropy results from periodic settings to nontrivial boundaries and provide detailed remainder estimates that accommodate gradient information, yielding convergence for both the basic variables and their first-order derivatives under suitable scaling of the boundary layer. The results offer a robust approach to singular limits with nontrivial boundaries and connect to quantum hydrodynamics and semiclassical limits via augmented energy formulations.

Abstract

In this article, we study the small dispersion limit of the Euler-Korteweg system in a bounded domain with no-flux boundary conditions. We exploit a relative entropy approach to study the convergence of finite energy weak solutions towards strong solutions to the compressible Euler system. Since we consider non-trivial boundary conditions, our approach needs a correction for the limiting particle density, due to the appearance of a (weak) boundary layer. We believe this approach can be adapted to study similar singular limits involving non-trivial boundary conditions.

The zero capillarity limit for the Euler-Korteweg system with no-flux boundary conditions

TL;DR

This work analyzes the vanishing capillarity limit for the Euler-Korteweg system with no-flux boundary conditions on bounded domains. It develops a relative-entropy framework to prove convergence of finite-energy weak solutions to strong Euler solutions, while addressing boundary-layer effects through a high-order augmented entropy and a boundary-layer correction. The authors extend prior relative-entropy results from periodic settings to nontrivial boundaries and provide detailed remainder estimates that accommodate gradient information, yielding convergence for both the basic variables and their first-order derivatives under suitable scaling of the boundary layer. The results offer a robust approach to singular limits with nontrivial boundaries and connect to quantum hydrodynamics and semiclassical limits via augmented energy formulations.

Abstract

In this article, we study the small dispersion limit of the Euler-Korteweg system in a bounded domain with no-flux boundary conditions. We exploit a relative entropy approach to study the convergence of finite energy weak solutions towards strong solutions to the compressible Euler system. Since we consider non-trivial boundary conditions, our approach needs a correction for the limiting particle density, due to the appearance of a (weak) boundary layer. We believe this approach can be adapted to study similar singular limits involving non-trivial boundary conditions.

Paper Structure

This paper contains 14 sections, 11 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Survey of related results
  3. Main contributions of the paper
  4. Preliminaries and statement of the main results
  5. Relative entropy functionals
  6. Proof of general relative entropy result
  7. Preliminary results
  8. Energy inequality
  9. Estimate on $R(\rho,\Lambda, \rho^E , u^E )$
  10. Proof of Theorem \ref{['main']}
  11. Proof of high order relative entropy result
  12. Boundary layer correction
  13. Estimate on $R_h$
  14. Proof of Theorem \ref{['main high']}

Key Result

Theorem 2.2

Let $\Omega\subset \mathbb{R}^d$ such that $\partial \Omega \in C^{n+1}$ with $n=2,3$. Let $( \rho^E _0, u^E _0)\in C^{n+\delta}(\Omega)$, with $\delta>0$, be such that $0<\inf_\Omega \rho^E _0$. Then, there exists $T^*>0$ and a unique solution of Euler,BC_E,initial data E such that

Theorems & Definitions (27)

  • Definition 2.1: Weak solutions
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • ...and 17 more