  We consider inverse boundary value problems for the Navier-Stokes equations
and the isotropic Lam\'e system in two dimensions. The uniqueness without any
smallness assumptions on unknown coefficients, which is called global
uniqueness, was longstanding open problems for the Navier-Stokes equations and
the isotropic Lam\'e system in two dimensions. We prove the global uniqueness
for both inverse boundary value problems. Our methodology are common for both
systems and the key is the construction of complex geometric optics solutions
after decoupling the systems into weakly coupling systems.
