Critical hole-size scaling in two-dimensional compressible homogenization

Determine whether the two-dimensional relaxed Navier–Stokes–Korteweg equations acquire an additional Brinkman term at the critical hole-size exponent \(\alpha=2\), or otherwise characterize the homogenized limit in that regime.

Background

The paper establishes homogenization without an obstacle-induced term in two dimensions for exponentially small holes with exponent $\alpha>2$. The strict inequality is needed because the estimates for the momentum-equation remainder produce a positive decay exponent only when $\alpha>2$.

At the borderline value $\alpha=2$, the authors expect a Brinkman-type correction, by analogy with related critical homogenization regimes, but do not derive the limiting equation. They explicitly identify this two-dimensional critical question as completely open.