Extension to generalized density-dependent viscosity coefficients

Develop a global existence theory for compressible Navier–Stokes–Korteweg equations with generalized viscosity coefficients of the form μ₁(ρ) = ρ^β for β > 0, including a suitable auxiliary inequality, an approximation scheme incorporating the corresponding Korteweg term and additional regularizations, and a justified renormalized-solution limit passage.

Background

The paper proves global weak-solution existence for three-dimensional compressible Navier–Stokes–Korteweg equations with viscosity coefficient μ₁(ρ) = ρ and capillarity function k(ρ) = κρ^α, for −1 ≤ α ≤ 0. Remark 10 considers the substantially broader class μ₁(ρ) = ρ^β with β > 0. Extending the analysis would first require an appropriate multidimensional auxiliary inequality involving the exponents α and β.

The proposed extension also requires an approximation scheme compatible with the generalized viscosity and Korteweg terms, together with a modified renormalized weak-solution framework incorporating density truncation. The authors emphasize that the compactness and limit arguments depend sensitively on this structure, so the extension cannot be obtained by a direct application of the present proof.