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Global existence of Euler-Korteweg equations with the non-monotone pressure

Zihao Song

TL;DR

This work analyzes the 3D compressible Euler-Korteweg system with zero sound speed ($P'\!(\rho^{*})=0$) near a constant equilibrium, reformulating the perturbation into a quasi-linear Schrödinger equation for the irrotational case. By combining dispersive decay, normal-form transformations, and a space-time resonance analysis, the authors construct global scattering solutions for small initial data, providing a detailed final-data framework and a bootstrap energy method to control derivative losses. The main results establish global existence and uniqueness for irrotational perturbations and show precise scattering behavior: the density and velocity decompose into a leading Schrödinger evolution and a decaying remainder with $t^{-3/2}$ decay. This advances understanding of inviscid Korteweg flows with non-monotone pressure, demonstrating global stability and asymptotic freedom in 3D for a physically relevant, non-dissipative regime.

Abstract

We are concerned with the global solution of the compressible Euler-Korteweg equations in $\mathbb{R}^{3}$. In the case of zero sound speed $P'(ρ^{\ast})=0$, it is found that the perturbation problem of irrotational fluids could be reformulated into a quasi-linear Schr$\ddot{o}$dinger equation. Based on techniques of dispersive estimates and methods of normal form, we construct a class of global scattering solutions for 3D case.

Global existence of Euler-Korteweg equations with the non-monotone pressure

TL;DR

This work analyzes the 3D compressible Euler-Korteweg system with zero sound speed () near a constant equilibrium, reformulating the perturbation into a quasi-linear Schrödinger equation for the irrotational case. By combining dispersive decay, normal-form transformations, and a space-time resonance analysis, the authors construct global scattering solutions for small initial data, providing a detailed final-data framework and a bootstrap energy method to control derivative losses. The main results establish global existence and uniqueness for irrotational perturbations and show precise scattering behavior: the density and velocity decompose into a leading Schrödinger evolution and a decaying remainder with decay. This advances understanding of inviscid Korteweg flows with non-monotone pressure, demonstrating global stability and asymptotic freedom in 3D for a physically relevant, non-dissipative regime.

Abstract

We are concerned with the global solution of the compressible Euler-Korteweg equations in . In the case of zero sound speed , it is found that the perturbation problem of irrotational fluids could be reformulated into a quasi-linear Schrdinger equation. Based on techniques of dispersive estimates and methods of normal form, we construct a class of global scattering solutions for 3D case.
Paper Structure (10 sections, 7 theorems, 120 equations)

This paper contains 10 sections, 7 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Main results and strategy
  3. Preliminaries
  4. Littlewood-Paley theory and Besov spaces
  5. Linearization and bootstrap argument
  6. Estimates for the second approximation
  7. Proof of Lemma \ref{['final priori2']}
  8. $L^2$ estimates
  9. High order Sobolev estimates
  10. Proof of Theorem \ref{['thm2.3']}

Key Result

Theorem 1.1

Let $P'(\rho^{\ast})=0$. Assume $\varphi$ irrotational and $\varphi\in H^{3s+7}\cap\dot{H}^{-2}\cap B^{1}_{1,1}$, $x^{2}\varphi\in H^{1}$ for $s>\frac{5}{2}$. If $\varphi$ fulfills then there exists a unique global-in-time solution $(\rho-\rho^*,u)$ for (1.1) on $[0,\infty]$satisfying while where $\delta$ is some positive constant while $\mathcal{Z}$ fulfills for any $t>0$

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.1
  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Lemma 3.1
  • Proposition 3.1
  • proof
  • Lemma 5.1
  • ...and 1 more