论文标题
弯曲的薄域中的Navier-Stokes方程，第三部分：薄膜极限
Navier-Stokes equations in a curved thin domain, Part III: thin-film limit
论文作者
论文摘要
我们考虑在给定的闭合表面周围的三维弯曲薄域中，带有Navier-Stokes方程。在适当的假设下，我们表明，在较大的溶液的薄方向上，散装Navier-Stokes方程在极限表面的适当函数空间中弱收敛，因为薄域的厚度往往为零。此外，我们将极限表征为限制方程的弱解决方案，后者是极限表面上的阻尼和加权的Navier-Stokes方程。我们还证明，强大解决方案对散装方程的平均值通过显示对它们之间的差异的估计值的较弱解决方案的强烈收敛性。在某些特殊情况下，我们的极限方程与Riemannian歧管上的Navier-Stokes方程一致，其中粘性术语包含RICCI曲率。这是对薄膜限制在一般闭合表面上严格推导表面Navier-Stokes方程的第一个结果。
We consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a given closed surface. Under suitable assumptions we show that the average in the thin direction of a strong solution to the bulk Navier-Stokes equations converges weakly in appropriate function spaces on the limit surface as the thickness of the thin domain tends to zero. Moreover, we characterize the limit as a weak solution to limit equations, which are the damped and weighted Navier-Stokes equations on the limit surface. We also prove the strong convergence of the average of a strong solution to the bulk equations towards a weak solution to the limit equations by showing estimates for the difference between them. In some special case our limit equations agree with the Navier-Stokes equations on a Riemannian manifold in which the viscous term contains the Ricci curvature. This is the first result on a rigorous derivation of the surface Navier-Stokes equations on a general closed surface by the thin-film limit.
