The Stokes conjecture for gravity water waves with vorticity

image086 The paper “The Stokes conjecture for waves with vorticity“, co-authored by Eugen Varvaruca and Georg Weiss, has been published in the Annales de l’Institut Henri Poincare (C) Non Linear Analysis.  The paper studies stagnation points of two-dimensional steady gravity free-surface water waves with vorticity. In the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a corner of 120°, or the free surface ends in a horizontal cusp, or the free surface is horizontally flat at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity.  In a second main result, they exclude horizontally flat singularities in the case that the vorticity is zero on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far. A link to the paper is here.

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