Steady Periodic Water Waves with Unbounded Vorticity: Equivalent Formulations and Existence Results

被引：0
作者：
Calin Iulian Martin
Bogdan-Vasile Matioc
机构：
[1] Institut für Mathematik,
[2] Universität Wien,undefined
来源：
Journal of Nonlinear Science | 2014年 / 24卷
关键词：
Equivalent formulations; Local bifurcation; Unbounded vorticity; Gravity waves; Capillary-gravity waves; Capillary waves; 35J60; 76B03; 76B15; 76B45; 47J15;
D O I：
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学科分类号：
摘要：
In this paper we consider the steady water wave problem for waves that possess a merely Lr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_r$$\end{document}-integrable vorticity, with r∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in (1,\infty )$$\end{document} being arbitrary. We first establish the equivalence of the three formulations – the velocity formulation, the stream function formulation, and the height function formulation – in the setting of strong solutions, regardless of the value of r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document}. Based upon this result and using a suitable notion of weak solution for the height function formulation, we then establish, by means of local bifurcation theory, the existence of small-amplitude capillary and capillary–gravity water waves with an Lr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_r$$\end{document}-integrable vorticity.
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页码：633 / 659
页数：26
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