论文标题
具有障碍的域中的螺线管扩展：明确的界限和应用到Navier-Stokes方程
Solenoidal extensions in domains with obstacles: explicit bounds and applications to Navier-Stokes equations
论文作者
论文摘要
我们引入了一种新方法，用于在包含障碍物的（2D或3D）立方体中构造相当通用边界数据的螺线管扩展。这种方法使我们能够为扩展程序的Dirichlet规范提供明确的界限。它的运行如下：通过反转痕量运算符，我们首先确定数据的合适扩展（不一定是螺线管）。然后，我们分析了Bogovskii问题，从而获得了产生的差异，以获得螺线管扩展。最后，通过解决涉及无限拉普拉斯式的变分问题并使用临时截止函数，我们根据障碍物的几何参数找到明确的界限。我们的结果的自然应用在于分析流出流问题，其中需要明确绑定流入速度，以估计固定的Navier-Stokes方程中唯一性的阈值，并且在对称性的情况下，障碍物的稳定性是浸入流体流中的障碍物。
We introduce a new method for constructing solenoidal extensions of fairly general boundary data in (2d or 3d) cubes that contain an obstacle. This method allows us to provide explicit bounds for the Dirichlet norm of the extensions. It runs as follows: by inverting the trace operator, we first determine suitable extensions, not necessarily solenoidal, of the data; then we analyze the Bogovskii problem with the resulting divergence to obtain a solenoidal extension; finally, by solving a variational problem involving the infinity-Laplacian and using ad hoc cutoff functions, we find explicit bounds in terms of the geometric parameters of the obstacle. The natural applications of our results lie in the analysis of inflow-outflow problems, in which an explicit bound on the inflow velocity is needed to estimate the threshold for uniqueness in the stationary Navier-Stokes equations and, in case of symmetry, the stability of the obstacle immersed in the fluid flow.
