Large time existence of Euler-Korteweg equations and two-fluid Euler-Maxwell equations with vorticity

TL;DR

This paper investigates how vorticity affects the lifespan of solutions in Euler-Korteweg and two-fluid Euler-Maxwell systems, showing that vorticity can significantly influence the time of existence.

Contribution

It establishes a lower bound on the lifespan of solutions based on the initial vorticity's $H^s$ norm, extending previous results from irrotational to rotational initial data.

Findings

Vorticity reduces the lifespan bound proportionally to its initial $H^s$ norm.

Energy estimates and decay results are used to derive lifespan bounds.

The results apply to systems with small irrotational initial data, now extended to rotational cases.

Abstract

The aim of this manuscript is to study the influence of the vorticity on the existence time in fluid systems for which global smoothness and decay is known in the case of small irrotational data. We focus on two examples: the Euler-Korteweg system and the two-fluid Euler Maxwell system. We prove that the lower bound of the lifespan of these systems is no less than the inverse of the $H^s$ $(s>5/2)$ norm of the rotational part of the initial velocity. Our approach is based on energy estimates and the fast time decay results of global solutions to these systems with small irrotational initial data.