# American Institute of Mathematical Sciences

July  2012, 11(4): 1453-1464. doi: 10.3934/cpaa.2012.11.1453

## On the regularity of steady periodic stratified water waves

 1 Faculty of Mathematics, University of Vienna, Nordbergstraße 15, 1090 Vienna, Austria 2 Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany

Received  June 2011 Revised  September 2011 Published  January 2012

In this paper we prove regularity results for steady periodic stratified water waves, where we allow for the effects of surface tension. Our results concern stratified water waves, without stagnation points, which exist in three distinct physical regimes, namely: capillary, capillary-gravity, and gravity water waves. We prove, for all three types of waves, that, when the Bernoulli function is Hölder continuous and the variable density function has a first derivative which is Hölder continuous, then the free-surface profile is the graph of a smooth function. Furthermore, we show that the streamlines are analytic a priori for capillary stratified waves, whereas for gravity and capillary-gravity stratified waves the streamlines are smooth in general, and analytic in an unstable regime. Moreover, if the Bernoulli function and the streamline density function are both real analytic functions then all of the streamlines, including the wave profile, are real analytic for all gravity, capillary, and capillary-gravity stratified waves.
Citation: David Henry, Bogdan--Vasile Matioc. On the regularity of steady periodic stratified water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1453-1464. doi: 10.3934/cpaa.2012.11.1453
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