不可压理想流体在有角点区域边界上的速度估计

The velocity of a perfect incompressible fluid on the boundaryof a domain with corners

考虑有一个角被对称轴等分的对称有角点平面区域上的Euler方程.通过优化Kiselev和Zlatos的方法,在被等分角附近,垫一个有明确公式的正调和函数,在区域的格林函数下面,得到角点附近边界上流体速度的下界估计.当流体趋向角点时,下界估计趋于0,且角点处内角越大,下界估计越大.我们得到如下结论:第一,若角点处的内角大于π,则有光滑的初始涡量函数,使得没有全局光滑解以它为初值.第二,若内角不大于π,我们证明弱解的“涡量梯度”可以达到某些依赖于内角大小的增长率.类似的结果在非光滑区域上是稀缺的.

We consider the Euler equations on symmetric planar domains with corners.A quantitative refinement is used to obtain a lower bound 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 large corner,a larger lower bound is obtained.There are two consequences.Firstly,if the interior angle at the corner is large thanπ,there are smooth vorticities that can not be the initial value of a global smooth solution.Secondly,if the interior angle is not bigger thanπ,we prove some attainable boundary“vorticity gradient”growth rates of weak solutions that depend on the size of the angle.Such results are scarce for non-smooth domains.