Stokes waves with vorticity

2009/11/30 by Vera Mikyoung Hur, Hur, Vera Mikyoung
Earth and Planetary Sciences · #35J60 #47J15 #76B03 #76B15 #Analysis of PDEs (math.AP) #Coastal and Marine Dynamics #FOS: Mathematics #Ocean Waves and Remote Sensing #Oceanographic and Atmospheric Processes

paper · pdf · doi:10.48550/arxiv.0911.5542

openalex publication_date 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep water under gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a fixed semi-infinite strip with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global connected set of nontrivial solutions is obtained via singular theory of bifurcation. Each solution on the continuum has a symmetric and monotone wave profile. The proof uses a generalized degree theory, global bifurcation theory and Wyburn's lemma in topology, combined with the Schauder theory for elliptic problems and the maximum principle.

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