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Algebra i Analiz, 2018, Volume 30, Issue 3, Pages 112–128 (Mi aa1598)  

Research Papers

A comparison theorem for super- and subsolutions of $\nabla^2u+f(u)=0$ and its application to water waves with vorticity

V. Kozlova, N. G. Kuznetsovb

a Department of Mathematics, Linköping University, S-581 83, Linköping, Sweden
b Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr., 61, V.O., 199178, St. Petersburg, Russia

Abstract: A comparison theorem is proved for a pair of solutions that satisfy opposite nonlinear differential inequalities in a weak sense. The nonlinearity is of the form $f(u)$ with $f$ belonging to the class $L^p_\mathrm{loc}$ and the solutions are assumed to have nonvanishing gradients in the domain, where the inequalities are considered. The comparison theorem is applied to the problem describing steady, periodic water waves with vorticity in the case of arbitrary free-surface profiles including overhanging ones. Bounds for these profiles as well as streamfunctions and admissible values of the total head are obtained.

Keywords: comparison theorem, nonlinear differential inequality, partial hodograph transform in $n$ dimensions, periodic steady water waves with vorticity, streamfunction.

Funding Agency Grant Number
Swedish Research Council EO418401
Magnus Ehrnrooth Foundation
Linköping University
V. K. was supported by the Swedish Research Council (VR) through the grant [EO418401]. N. K. acknowledges the support from the G. S. Magnuson's Foundation of the Royal Swedish Academy of Sciences and Linköping University.


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Document Type: Article
Received: 10.10.2017
Language: English

Citation: V. Kozlov, N. G. Kuznetsov, “A comparison theorem for super- and subsolutions of $\nabla^2u+f(u)=0$ and its application to water waves with vorticity”, Algebra i Analiz, 30:3 (2018), 112–128

Citation in format AMSBIB
\Bibitem{KozKuz18}
\by V.~Kozlov, N.~G.~Kuznetsov
\paper A comparison theorem for super- and subsolutions of $\nabla^2u+f(u)=0$ and its application to water waves with vorticity
\jour Algebra i Analiz
\yr 2018
\vol 30
\issue 3
\pages 112--128
\mathnet{http://mi.mathnet.ru/aa1598}
\elib{http://elibrary.ru/item.asp?id=32855067}


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