论文标题
从边界信息中确定不可压缩流体的粘度
Determining the viscosity from the boundary information for incompressible fluid
论文作者
论文摘要
对于紧凑型连接的Riemannian $ n $ -Manifold $（ω，g）$，带有光滑边界$ \ partialω$的Stokes方程，我们在$ω$上给出了具有$（n+1）$独立不知情功能的椭圆方程式的同等新系统。我们表明，与此新系统相关的dirichlet到neumann地图$ {\tildeλ} _ {\tildeε，μ，g} $也等同于与Stokes方程相关的原始dirichlet to-neumann map $λ_{μ，g} $。 We explicitly give the full symbol expression for the ${\tildeΛ}_{\tildeε,μ,g}$ by a method of factorization, and prove that Dirichlet-to-Neumann map ${\tildeΛ}_{\tildeε,μ,g}$ (or equivalently, $Λ_{μ, g}$) uniquely determines viscosity $μ$以及$ \ partialω$上的$μ$的所有切向和普通导数。特别是，结合了这个结果，Lai-uhlmann-Wang的定理和Heck-Li-Wang的定理，我们完全解决了一个长期存在的开放问题，该问题询问是否可以通过$ {\ Mathbbbb {$ n = n Navier-Stokes方程的粘度和Navier-Stokes方程来确定navier-Stokes方程的粘度。
For the Stokes equations in a compact connected Riemannian $n$-manifold $(Ω,g)$ with smooth boundary $\partial Ω$, we give an equivalent new system of elliptic equations with $(n+1)$ independent unknown functions on $Ω$. We show that the Dirichlet-to-Neumann map ${\tildeΛ}_{\tildeε, μ,g}$ associated with this new system is also equivalent to the original Dirichlet-to-Neumann map $Λ_{μ,g}$ associated with the Stokes equations. We explicitly give the full symbol expression for the ${\tildeΛ}_{\tildeε,μ,g}$ by a method of factorization, and prove that Dirichlet-to-Neumann map ${\tildeΛ}_{\tildeε,μ,g}$ (or equivalently, $Λ_{μ, g}$) uniquely determines viscosity $μ$ and all tangential and normal derivatives of $μ$ on $\partial Ω$. In particular, combining this result, Lai-Uhlmann-Wang's theorem and Heck-Li-Wang's theorem, we completely solve a long-standing open problem that asks whether one can determine the viscosity for the Stokes equations and for the Navier-Stokes equations by boundary measurements on an arbitrary bounded domain in ${\mathbb{R}}^n$, ($n=2,3$).
