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.

我们介绍了一种新方法，用于在包含障碍物的（2d或3d）多维数据集中构造相当通用的边界数据的螺线扩展。这种方法可以让我们提供明确的扩展的Dirichlet范数的界。它的运行过程如下：通过反转跟踪运算符，我们首先确定数据的合适扩展名（不一定是螺旋的）。然后我们分析Bogovskii问题，并由此得出发散以获得螺线管扩展；最后，通过解决涉及无穷拉普拉斯算式的变分问题并使用临时截断函数，我们在障碍物的几何参数方面找到了明确的界限。我们的结果的自然应用在于对流入-流出问题的分析，其中需要对流入速度有一个明确的界限，以估计平稳的Navier-Stokes方程中唯一性的阈值，并且在对称的情况下，需要确定其稳定性。障碍物浸没在流体中。