论文标题
liouville型定理，用于在平板中稳定解决Navier-Stokes系统的解决方案
Liouville-type theorems for steady solutions to the Navier-Stokes system in a slab
论文作者
论文摘要
在具有无滑动边界条件或周期性边界条件的三维平板中，研究了用于稳定不可压缩的Navier-Stokes系统的Liouville型定理。当规定了无滑移边界条件时，我们证明，如果轴对称或$ ru^r $有界，则任何有限的解决方案都是微不足道的，并且当速度在$ l^\ infty $空间中不大时，一般的三维解决方案必须是Poiseuille的流动。当施加周期性边界条件在平板边界上时，如果旋转或径向速度独立于角变量，或者$ ru^r $衰减至零，则有限的解决方案必须是恒定向量。证明基于方程和能量估计的基本结构。关键技术是建立一个圣人类型的估计，该估计是特征了非平凡溶液组成部分的增长。
Liouville-type theorems for the steady incompressible Navier-Stokes system are investigated for solutions in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or $ru^r$ is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big in $L^\infty$ space. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vectors if either the swirl or radial velocity is independent of the angular variable, or $ru^r$ decays to zero as $r$ tends to infinity. The proofs are based on the fundamental structure of the equations and energy estimates. The key technique is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions.
