# On the time of existence of solutions of the Euler-Korteweg system

## TL;DR

This paper investigates the existence time of solutions to the Euler-Korteweg system, establishing bounds based on initial rotational data and providing a finite-time blow-up example.

## Contribution

It extends previous global existence results to cases with rotational initial data and offers a finite-time blow-up example in a special case.

## Key findings

- Lower bounds on solution existence time depending on initial vorticity
- Recovery of global well-posedness in the zero vorticity limit
- Existence of finite-time blow-up solutions in specific scenarios

## Abstract

Under a natural stability condition on the pressure, it is known that for small irrotational initial data, the solutions of the Euler-Korteweg system are global in time. When the initial velocity has a small rotational part, we obtain a lower bound on the time of existence that depends only on the rotational part. In the zero vorticity limit we recover the previous global well-posedness result. Independently of this analysis, we also provide (in a special case) a simple example of solution that blows up in finite time.