Confinement of vorticity in two dimensional ideal incompressible exterior flow
Confinement of vorticity in two dimensional ideal incompressible exterior flow
复制标题
二维理想不可压缩外流中涡度的限制
DOI：
10.1090/s0033-569x-07-01059-4
复制
发表时间：
2006
影响因子：
0.8
通讯作者：
H. N. Lopes
中科院分区：
文献类型：
--
作者：
D. Iftimie;M. L. Filho;H. N. Lopes
在[数学。应用科学19（1996）53-62]，C.马尔基奥罗研究了在光滑有界域外部的涡量限制问题。Marchioro文章的主要结果是：在连通有界区域的外部具有紧支集的非负初始涡量的不可压缩二维Euler方程的解，对于任何“> 0，具有直径增长至多为O（t（1/2）+”）的涡量支集.此外，如果区域是圆盘的外部，那么涡度支集包含在半径为O（t1/3）的圆盘中。本文的目的是改进Marchioro的结果。我们将证明，如果初始涡度相对于原点是偶数，则盘外部的指数可以改进为1/4。对于流动的外部的一个光滑的，连接的，有界域，我们证明了一个限制估计与指数1/2（即，我们删除“），并在某些情况下，根据谐波部分的流量，我们建立一个对数改善指数1/2。我们方法中的主要新成分是：（1）通过使用Riemann映射获得的无穷远附近外部Poisson问题解的详细渐进描述;（2）重整化能量估计和涡度对数矩的界限以及（3）惯性矩对数扰动时间导数的新先验估计。
In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the problem of vorticity confinement in the exterior of a smooth bounded domain. The main result in Marchioro’s paper is that solutions of the incompressible 2D Euler equations with compactly supported nonnegative initial vorticity in the exterior of a connected bounded region have vorticity support with diameter growing at most like O(t (1/2)+" ), for any " > 0. In addition, if the domain is the exterior of a disk, then the vorticity support is contained in a disk of radius O(t 1/3 ). The purpose of the present article is to refine Marchioro’s results. We will prove that, if the initial vorticity is even with respect to the origin, then the exponent for the exterior of the disk may be improved to 1/4. For flows in the exterior of a smooth, connected, bounded domain we prove a confinement estimate with exponent 1/2 (i.e. we remove the ") and in certain cases, depending on the harmonic part of the flow, we establish a logarithmic improvement over the exponent 1/2. The main new ingredients in our approach are: (1) a detailed asymptotic description of solutions to the exterior Poisson problem near infinity, obtained by the use of Riemann mappings; (2) renormalized energy estimates and bounds on logarithmic moments of vorticity and (3) a new a priori estimate on time derivatives of logarithmic perturbations of the moment of inertia.