半平面上不可压粘性流涡量的渐近行为

证明在上半平面Navier-Stokes方程具有紧支集初值的正涡量解在向上和向左的运动都有限制,就是说,其大部分都集中在一条宽度缓慢增长的水平条带区域里,其宽度为O(t1/2),也同时集中在一个以同阶速度向左缓慢扩展的1/4平面里.采用Marchioro在全平面上的讨论架构,先估计涡量单边矩的增长速度,但因半平面上的Biot-Savart核较复杂,故需要更仔细的分析.

Asymptotical Behavior of the Vorticity of Incompressible Viscous Flows in the Half- Plane

We show that the vorticity of Navier-Stokes flows in the upper half-plane with compactly supported nonnegative initial values is confined in the vertical and left directions.That is,the bulk of the vorticity is contained in a horizontal strip with slowly growing width,of order O(t1/2),and is also contained in a quarter-plane extending in the same order to the left.We adopt Marchioro's framework used in discussing the full-plane case,and first estimate the growth of the one-sided moments of the vorticity.However,more detail analysis is required as the Biot-Savart kernel on the half-plane is more complicate.