Determining the viscosity from the boundary information for incompressible fluid

For the Stokes equations in a compact connected Riemannian $n$-manifold $(\Omega,g)$ with smooth boundary $\partial \Omega$, we give an equivalent new system of elliptic equations with $(n+1)$ independent unknown functions on $\Omega$. We show that the Dirichlet-to-Neumann map ${\tilde{\Lambda}}_{\tilde{\epsilon}, \mu,g}$ associated with this new system is also equivalent to the original Dirichlet-to-Neumann map $\Lambda_{\mu,g}$ associated with the Stokes equations. We explicitly give the full symbol expression for the ${\tilde{\Lambda}}_{\tilde{\epsilon},\mu,g}$ by a method of factorization, and prove that Dirichlet-to-Neumann map ${\tilde{\Lambda}}_{\tilde{\epsilon},\mu,g}$ (or equivalently, $\Lambda_{\mu, g}$) uniquely determines viscosity $\mu$ and all tangential and normal derivatives of $\mu$ on $\partial \Omega$. 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 ${\mathbb{R}}^n$, ($n=2,3$).