Homogenization of a relaxed compressible viscous two-phase fluid model in a domain with very tiny holes

Abstract: We study an approximate system for the compressible Navier-Stokes-Korteweg equations in a bounded domain periodically perforated by small obstacles. As the size of the obstacles decrease much faster to zero than their mutual distances, we show in spatial dimension two and three that the limiting system remains unchanged. Our result applies for a large class of monotone and non-monotone pressure functions. In particular, it holds in the physically relevant case when the approximate system is used to describe the dynamics of a compressible viscous two-phase fluid extending the corresponding homogenization results known for the compressible Navier-Stokes equations in a single-phase setting.

• The paper establishes a homogenization result for a compressible two-phase fluid model (relaxed Navier-Stokes-Korteweg equations, rNSKE) with very tiny holes.
• When the hole radius shrinks significantly, the limiting system behaves as if no holes exist, indicating no macroscopic effects from the perforations.
• The study is significant for being the first to consider non-monotone pressure functions in the tiny-hole regime, relevant for modeling physical fluids with Van der Waals characteristics.

This paper by Oschmann and Wendt establishes a homogenization result for a parabolic relaxation system of the compressible Navier–Stokes–Korteweg equations (rNSKE) posed on a bounded domain periodically perforated by very small holes, in spatial dimensions two and three. The central finding is that when the hole radius $a_\varepsilon$ shrinks much faster than the mutual hole distance $\varepsilon$, the limiting system is identical to the rNSKE on the unperforated domain: the obstacles have no macroscopic effect (2608.17560). Notably, this is stated to be the first homogenization result in the tiny-hole regime that permits non-monotone, Van der Waals type pressure functions.

Background and positioning

For single-phase fluids, homogenization of Stokes, Navier–Stokes, and Navier–Stokes–Fourier systems in perforated domains is classified by the exponent $\alpha$ relating radius $a_\varepsilon = \varepsilon^\alpha$ to distance $\varepsilon$: in 3D, $\alpha < 3$ yields Darcy's law, $\alpha = 3$ Brinkman's law, and $\alpha > 3$ leaves the system unchanged. Prior to this work, rigorous homogenization for two-phase models existed only in Rohde and von Wolff's treatment of a nonlocal modification of the 3D NSKE with $\alpha = 1$ and a smooth convolution capillary term, producing a compressible Darcy law with inherited interaction kernel. The present paper departs from that setting along three axes: it treats tiny holes ($\alpha > d$), it replaces the Korteweg capillary force by a relaxation parameter $\varepsilon$0 governed by a linear parabolic equation with Neumann condition on the hole boundaries, and—most significantly—it admits pressure laws $\varepsilon$1 that are decreasing on a compact interval, i.e., of Van der Waals type.

The model

The rNSKE couples the continuity equation, a momentum balance with viscous stress $\varepsilon$2 and coupling force $\varepsilon$3 replacing the Korteweg tensor, and a parabolic relaxation equation $\varepsilon$4. As the relaxation coefficients satisfy $\varepsilon$5, $\varepsilon$6 appropriately, solutions formally converge to solutions of the NSKE; a rigorous convergence result via relative energy was obtained previously. The artificial pressure $\varepsilon$7 renders the first-order part hyperbolic for large enough $\varepsilon$8, which facilitates numerical treatment of two-phase flow. The pressure class allows the decomposition $\varepsilon$9 with $\alpha$0 growing like $\alpha$1 and $\alpha$2 smooth, compactly supported, and nonpositive—so $\alpha$3 may be non-monotone on a compact set while retaining coercivity of $\alpha$4.

Weak solutions are taken in the finite-energy class, with the energy functional comprising kinetic energy, pressure potential $\alpha$5, phase-mismatch term $\alpha$6, and capillary term $\alpha$7, plus dissipation from viscosity and $\alpha$8. Global-in-time existence of such solutions was established in companion work, so the homogenization theorem operates on an existing well-posedness theory.

Statement of the main result

Hole sizes are $\alpha$9 with $a_\varepsilon = \varepsilon^\alpha$0 in 3D and $a_\varepsilon = \varepsilon^\alpha$1 with $a_\varepsilon = \varepsilon^\alpha$2 in 2D, reflecting the logarithmic rather than polynomial scaling of Newtonian capacity in two dimensions. In dimension three, the adiabatic exponent must additionally obey

$a_\varepsilon = \varepsilon^\alpha$3

in dimension two, only $a_\varepsilon = \varepsilon^\alpha$4 and $a_\varepsilon = \varepsilon^\alpha$5 are required. Given initial data converging suitably as $a_\varepsilon = \varepsilon^\alpha$6 (zero extension of density in $a_\varepsilon = \varepsilon^\alpha$7, kinetic-energy density in $a_\varepsilon = \varepsilon^\alpha$8, extended relaxation parameter in $a_\varepsilon = \varepsilon^\alpha$9), any sequence of finite energy weak solutions has a subsequence whose zero extensions of $\varepsilon$0 and $\varepsilon$1 and whose extensions of $\varepsilon$2 converge strongly/weakly to a triple $\varepsilon$3 that is a finite energy weak solution of the rNSKE on all of $\varepsilon$4. Thus the effective dynamics are obstacle-free: no Darcy or Brinkman correction arises, consistent with the single-phase theory for subcritical hole size.

Proof strategy in 3D

The argument follows the now-standard architecture for tiny-hole homogenization of compressible Navier–Stokes systems. Uniform bounds come first from the energy inequality, then an improved density bound $\varepsilon$5 is derived by testing the momentum equation with a Bogovskiĭ-based test function built from $\varepsilon$6, with $\varepsilon$7. This step requires the strict inequality on $\varepsilon$8: the constraint $\varepsilon$9 ensures the Bogovskiĭ norm factor $\alpha < 3$0 decays. The relaxation force term $\alpha < 3$1 enters these estimates through Sobolev embedding and is controlled precisely because $\alpha < 3$2 was chosen so that the relevant exponents stay below $\alpha < 3$3.

Extensions to $\alpha < 3$4 use zero extension for $\alpha < 3$5 and $\alpha < 3$6 and a uniform extension operator $\alpha < 3$7 for $\alpha < 3$8—the latter because the Neumann condition prevents zero extension from preserving $\alpha < 3$9 regularity, exactly as with temperature in Navier–Stokes–Fourier homogenization. The residual error between $\alpha = 3$0 and $\alpha = 3$1 vanishes at rate $\alpha = 3$2, exploiting the exponentially small total hole measure.

A solenoidal cut-off matrix $\alpha = 3$3 (equal to the identity away from $\alpha = 3$4-neighborhoods of the holes, vanishing inside them) converts test functions into admissible ones on the perforated domain. All commutator terms collect into a remainder $\alpha = 3$5 satisfying $\alpha = 3$6 for some explicit $\alpha = 3$7 determined by the interplay among $\alpha = 3$8 and $\alpha = 3$9. Compactness arguments (Banach–Alaoglu, Aubin–Lions, Arzelà–Ascoli) yield strong convergence of $\alpha > 3$0 in $\alpha > 3$1 and identify products like $\alpha > 3$2 and $\alpha > 3$3. Strong convergence of the density—in turn identifying $\alpha > 3$4 and $\alpha > 3$5—is obtained via weak compactness of the effective viscous flux and Feireisl's oscillation-kill technique applied to truncated densities $\alpha > 3$6. Finally, the energy inequality passes to the limit; the capillary contribution requires testing the limit parabolic equation with $\alpha > 3$7 itself to show $\alpha > 3$8 converges to its expected limit.

The two-dimensional case

The 2D proof parallels the 3D one but requires substantially finer Bogovskiĭ operators. Because the harmonic capacity of 2D holes is logarithmic, an additional parameter $\alpha > 3$9 interpolates between hole distance and exponentially small radius, and the resulting operator bound involves terms like $\alpha = 1$0. A technical contribution of independent value is the appendix, where the extension of this 2D Bogovskiĭ operator to negative Sobolev spaces is proved rigorously; the authors note that this extension was used but not proved in prior work. The improved density estimate becomes $\alpha = 1$1, and the cut-off remainder again vanishes at rate $\alpha = 1$2 with $\alpha = 1$3 guaranteed by $\alpha = 1$4. The rest of the argument mirrors 3D.

Limitations and open questions

Two structural limitations deserve emphasis. First, both the 3D and 2D results require strict inequalities on $\alpha = 1$5 ($\alpha = 1$6 and $\alpha = 1$7, respectively). At criticality the remainder exponent $\alpha = 1$8 degenerates to zero and the proof fails; the authors state plainly that the critical case should produce a Brinkman-type limiting term—as known in the incompressible setting—but "in the fully compressible case, this question is completely open," in both dimensions. Second, the result concerns the relaxed system rather than the NSKE itself; the combined limit of homogenization followed by relaxation, and whether the diagram "homogenization of rNSKE → NSKE" commutes with taking the relaxation limit directly on the perforated domain, is left unaddressed. The authors also announce as planned work a quantitative generalization using relative energy methods, extending recent convergence-rate results for large-hole homogenization of the compressible NSE.

Conclusion

The paper extends the homogenization theory of compressible viscous flows to the two-phase setting by proving that, for sufficiently small holes, the rNSKE with Van der Waals type pressure converges to the same system on the unperforated domain, in dimensions two and three. Its principal contributions are the first treatment of non-monotone pressures in the tiny-hole regime, the handling of a parabolic relaxation variable subject to Neumann conditions on hole boundaries, and a self-contained construction of the 2D negative-Sobolev Bogovskiǐ operator. The critical hole-size regime remains open for compressible two-phase models.