平面角域上Laplace-Leray交换子的最优估计
On Optimal Estimates for the Laplace-Leray Commutator in Planar Domains with Corners
摘要：
对于光滑域，Liu等人（Comm. Pure Appl.Math.60：1443-1487，2007）使用拉普拉斯算子和Leray投影算子的交换子的最优估计来建立扩展的Navier-Stokes动力学的适定性。在他们的工作中，压力不是由不可压缩性决定的，而是由涉及拉普拉斯-勒雷交换子的某个公式决定的。Liu等人的一个关键估计严格地控制了交换子在首阶能量范数下的拉普拉斯算子。在本文中，我们表明，这种严格的控制失败，在一个大家庭的有界平面域的角落。然而，当域是一个无限锥，我们发现，严格的控制可能会恢复在某些幂律加权范数。
英文摘要：
For smooth domains, Liu et al. (Comm. Pure Appl. Math. 60: 1443-1487, 2007) used optimal estimates for the commutator of the Laplacian and the Leray projection operator to establish well-posedness of an extended Navier-Stokes dynamics. In their work, the pressure is not determined by incompressibility, but rather by a certain formula involving the Laplace-Leray commutator. A key estimate of Liu et al. controls the commutator strictly by the Laplacian in energy norm at leading order. In this paper we show that this strict control fails in a large family of bounded planar domains with corners. However, when the domain is an infinite cone, we find that strict control may be recovered in certain power-law weighted norms.
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