Local well posedness of the Euler-Korteweg equations on $ \mathbb T^d $

TL;DR

This paper proves local existence of classical solutions for the Euler-Korteweg equations on periodic domains, under minimal regularity assumptions, focusing on irrotational velocity fields.

Contribution

It establishes the local well-posedness of the Euler-Korteweg system with minimal regularity assumptions on initial data for irrotational flows.

Findings

Existence of classical solutions under minimal regularity

Results applicable to periodic boundary conditions

Focus on irrotational velocity fields

Abstract

We consider the Euler-Korteweg system with space periodic boundary conditions $ x \in \mathbb T^d $. We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.