Determine whether the viscosity-error coefficient can vanish

Determine whether the coefficient C in the local relation E(\hat\nu)=C(\hat\nu-\nu)^2 for the continuous-data-assimilation viscosity-recovery objective can equal zero for cases other than the isolated or degenerate situations suggested by the authors, such as w=0.

Background

The viscosity-recovery analysis defines E(\hat\nu)=\tfrac12|I_H e(\hat\nu)|^2, where e(\hat\nu)=u-v(\hat\nu) is the discrepancy between the true steady Navier–Stokes velocity and the continuous-data-assimilation solution computed with trial viscosity \hat\nu. Under a Taylor expansion, the authors derive the local quadratic form E(\hat\nu)=C(\hat\nu-\nu)^2, with C determined by the sensitivity w of the assimilated solution with respect to viscosity.

The proposed modified Newton method relies on this quadratic behavior and on a nonzero coefficient C. The authors state that they cannot exclude C=0, although they believe this can occur only in isolated or exceptional cases and report no such issue in their numerical experiments. Establishing whether C is always nonzero under identifiable conditions would clarify the theoretical validity and robustness of the viscosity-recovery method.