The Stokes conjecture for waves with vorticity
Based on the author's abstract: Stagnation points of two-dimensional steady gravity free-surface water waves with vorticity are considered. The authors obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation points is given either by the Stokes corner flow where the free surface has a corner of \(120\degree\), 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. The authors exclude horizontally flat singularities in the case that the vorticity is 0 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.
scientific article; zbMATH DE number 6156243
scientific article