论文标题
关于Leray的问题，在无限长的管道中具有Navier-Slip边界条件
On Leray's problem in an infinite-long pipe with the Navier-slip boundary condition
论文作者
论文摘要
原始的Leray的问题涉及弱解决方案对扭曲的管道中稳定不可压缩的Navier-Stokes方程的适当性，该方程可用于poiseuille流动，但在空间无穷大处处于无滑动边界状态。在本文中，解决了Navier-Slip边界条件而不是防滑边界条件的同样问题。由于边界条件的复杂性，引入了一些新想法，以应对边界项造成的额外困难。 首先，将引入半无限直管中的Poiseuille流，将引入带有Navier-Slip边界条件的条件，这将用作溶液的渐近型，以实现leray在Gentai Infinity上的广义Leray问题。其次，将仔细构建整个管道中定义的螺线管向量函数，满足带有指定通量并等于在较大距离的Poiseuille流量的Navier-Slip边界条件。这在重新制定我们的问题中起着重要作用。第三，能量估计取决于梯度的$ l^2 $估计值和速度的应力张量。
The original Leray's problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe, which approach to the Poiseuille flow subject to the no-slip boundary condition at spacial infinity. In this paper, the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition, is addressed. Due to the complexity of the boundary condition, some new ideas, presented as follows, are introduced to handle the extra difficulties caused by boundary terms. First, the Poiseuille flow in the semi-infinite straight pipe with the Navier-slip boundary condition will be introduced, which will be served as the asymptotic profile of the solution to the generalized Leray's problem at spacial infinity. Second, a solenoidal vector function defined in the whole pipe, satisfying the Navier-slip boundary condition, having the designated flux and equalling to the Poiseuille flow at large distance, will be carefully constructed. This plays an important role in reformulating our problem. Third, the energy estimates depend on a combined $L^2$ estimate of the gradient and the stress tensor of the velocity.
