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Existence of multi-travelling waves in capillary fluids

Part of: Representations of solutions Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  19 August 2019

Corentin Audiard
Corentin Audiard*
Affiliation:
Laboratoire Jacques-Louis Lions (LJLL), Sorbonne université, CNRS, Université de Paris, F75005, Paris, France ([email protected])
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Abstract

We prove the existence of multi-soliton and kink-multi-soliton solutions of the Euler–Korteweg system in dimension one. Such solutions behave asymptotically in time like several travelling waves far away from each other. A kink is a travelling wave with different limits at ±∞. The main assumption is the linear stability of the solitons, and we prove that this assumption is satisfied at least in the transonic limit. The proof relies on a classical approach based on energy estimates and a compactness argument.

Keywords

traveling wavesEuler–Kortewegstability

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35C07: Traveling wave solutions 35Q53: KdV-like equations (Korteweg-de Vries) 35Q31: Euler equations 35B35: Stability
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2905 - 2936
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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