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

TL;DR

This paper proves the global existence of solutions for the Euler-Korteweg equations in three dimensions with non-monotone pressure, using dispersive estimates and normal form methods to handle the zero sound speed case.

Contribution

It introduces a novel approach to establish global solutions for Euler-Korteweg equations with non-monotone pressure by reformulating the problem as a quasi-linear Schrödinger equation.

Findings

Constructed a class of global scattering solutions in 3D

Reformulated irrotational fluid perturbations as a quasi-linear Schrödinger equation

Applied dispersive estimates and normal form techniques successfully

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 $P'(\rho^{\ast})=0$, 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.