The Velocity Of A Perfect Incompressible Fluid On The Boundary Of A Domain With Corners

We consider the Euler equations on symmetric planar domains with one corners bisected by the symmetry axis of the domain.Two results are obtained.First,if the interior angle at the corner is larger than ,there are smooth vorticities that cannot be the initial value of a global smooth solution.This is a partial extension of a result on domains with cusps by Kiselev and Zlato?s(J.Differential Equations,259,2015,pp.3490-3494).Second,if the interior angle is not bigger than ,we prove some attainable boundary‘vorticity gradient' growth rates of weak solutions that depend on the size of the angle.Such kind of results are scarce for non-smooth domains.The key ingradient is to use a quantitative refinement of an argument in Kiselev and Zlato?s to obtain a lower estimate of the velocity of a perfect fluid on the boundary near a corner bisected by the symmetry axis,which tends to zero as one approaches the corner.For a larger corner,a larger lower estimate is obtained.Also we find some explicit z-periodic axisymmetric solutions to the Euler equation on a cylinder.