Determining the viscosity of the Navier–Stokes equations from observations of finitely many modes

The paper proves that the viscosity-update map Γ(γ) defined from the determining map W(γ,PNu) is a contraction under explicit conditions on N and μ0, and gives a residual-based stopping rule. The candidate solution derives the same weighted identity for ν−Γ(γ), decomposes the nonlinear remainder identically (up to relabeling w=u−v vs. v−u), uses the same Ladyzhenskaya/Poincaré and spectral inequalities, obtains the same M1 constant and 1/N factor, and concludes the same contraction factor ε1/(ν1−ν0+ε1) and stopping test. The steps, constants, and hypotheses match Theorem 7.1 and Algorithm 1 in the paper, with only presentational differences and standard estimates filled in. See Algorithm 1 and Theorem 7.1 for the update and convergence statement, and the proof sketch around β(t) identities and bounds of the trilinear terms that yield the M1/N control.