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We prove local higher-order asymptotics for extreme water waves withvorticity near stagnation points. We obtain that the behaviour of solutions andtheir regularity depend substantially on the vorticity. In particular, we showthat extreme waves with a negative vorticity distribution have concave profilesnear the crest. Our approach is based on new regularity results and asymptoticanalysis of the corresponding nonlinear problem in a half-strip. Our mainresult is local and therefore is valid for a broad range of problems, such asfor waves with a piecewise constant vorticity, stratified waves, flows withcounter-currents or waves on infinite depth.

An asymptotic behaviour near the crest of waves of extreme form on water of finite depth