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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.

Global uniqueness in inverse boundary value problems for the Navier–Stokes equations and Lamé system in two dimensions