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A remark on the two dimensional water wave problem with surface tension
Authors:Shuanglin Shao  Hsi-Wei Shih
Institution:1. Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States of America;2. Department of Mathematics, National Cheng Kung University, No. 1 Dasyue Rd., Tainan City 70101, Taiwan
Abstract:We consider the motion of a two-dimensional interface between air (above) and an irrotational, incompressible, inviscid, infinitely deep water (below), with surface tension present. We propose a new way to reduce the original problem into an equivalent quasilinear system which is related to the interface's tangent angle and a quantity related to the difference of tangential velocities of the interface in the Lagrangian and the arc-length coordinates. The new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts. Then if assuming that waves are periodic, we establish a priori energy inequality.
Keywords:Water waves  Surface tension  The a priori energy inequality
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