Solvability of the initial value problem to a model system for water waves

Yuuta Murakami, Tatsuo Iguchi

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We consider the initial value problem to a model system for water waves. The model system is the Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The model are nonlinear dispersive equations and the hypersurface t = 0 is characteristic for the model equations. Therefore, the initial data have to be restricted in an infinite dimensional manifold in order to the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh–Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces.

Original languageEnglish
Pages (from-to)470-491
Number of pages22
JournalKodai Mathematical Journal
Volume38
Issue number2
DOIs
Publication statusPublished - 2015 Jul 11

Fingerprint

Water Waves
Initial Value Problem
Solvability
Nonlinear Dispersive Equations
Euler-Lagrange Equations
Model
Sobolev Spaces
Hypersurface
Necessary Conditions

Keywords

  • Luke’s lagrangian
  • Shallow water
  • Water waves

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Solvability of the initial value problem to a model system for water waves. / Murakami, Yuuta; Iguchi, Tatsuo.

In: Kodai Mathematical Journal, Vol. 38, No. 2, 11.07.2015, p. 470-491.

Research output: Contribution to journalArticle

@article{bfb844f832cb410e88e5c76d91686c5e,
title = "Solvability of the initial value problem to a model system for water waves",
abstract = "We consider the initial value problem to a model system for water waves. The model system is the Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The model are nonlinear dispersive equations and the hypersurface t = 0 is characteristic for the model equations. Therefore, the initial data have to be restricted in an infinite dimensional manifold in order to the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh–Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces.",
keywords = "Luke’s lagrangian, Shallow water, Water waves",
author = "Yuuta Murakami and Tatsuo Iguchi",
year = "2015",
month = "7",
day = "11",
doi = "10.2996/kmj/1436403901",
language = "English",
volume = "38",
pages = "470--491",
journal = "Kodai Mathematical Journal",
issn = "0386-5991",
publisher = "Tokyo Institute of Technology",
number = "2",

}

TY - JOUR

T1 - Solvability of the initial value problem to a model system for water waves

AU - Murakami, Yuuta

AU - Iguchi, Tatsuo

PY - 2015/7/11

Y1 - 2015/7/11

N2 - We consider the initial value problem to a model system for water waves. The model system is the Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The model are nonlinear dispersive equations and the hypersurface t = 0 is characteristic for the model equations. Therefore, the initial data have to be restricted in an infinite dimensional manifold in order to the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh–Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces.

AB - We consider the initial value problem to a model system for water waves. The model system is the Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The model are nonlinear dispersive equations and the hypersurface t = 0 is characteristic for the model equations. Therefore, the initial data have to be restricted in an infinite dimensional manifold in order to the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh–Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces.

KW - Luke’s lagrangian

KW - Shallow water

KW - Water waves

UR - http://www.scopus.com/inward/record.url?scp=84937578547&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937578547&partnerID=8YFLogxK

U2 - 10.2996/kmj/1436403901

DO - 10.2996/kmj/1436403901

M3 - Article

VL - 38

SP - 470

EP - 491

JO - Kodai Mathematical Journal

JF - Kodai Mathematical Journal

SN - 0386-5991

IS - 2

ER -