Critical hole-size scaling in three-dimensional compressible homogenization

Determine whether the fully compressible relaxed Navier–Stokes–Korteweg equations develop an additional Brinkman term at the critical scaling \(\alpha = \max\{3,(2\gamma-3)/(\gamma-3)\}\), or otherwise characterize the homogenized limit in this regime.

Background

The paper proves that, in three spatial dimensions, the homogenization remainder vanishes when the hole-size exponent satisfies the strict condition $\alpha > \max\{3,(2\gamma-3)/(\gamma-3)\}$. At equality, the proof yields no positive decay exponent for the remainder, so the authors cannot establish convergence to the unmodified relaxed Navier–Stokes–Korteweg system.

The authors compare this critical regime with the incompressible case, where the analogous critical scaling produces a Brinkman term. They explicitly state that the corresponding question for the fully compressible setting is unresolved.