Density of the compact-operator domain in the Sobolev–Lebesgue intersection

Establish whether the domain D(L) of the compact operator L constructed by Brzeźniak and collaborators is dense in the intersection V∩L^p_σ for the relevant unbounded-domain function spaces, so that it can yield the simultaneous approximation required for the analysis.

Background

The paper discusses an alternative compact operator L from the cited work of Brzeźniak et al. Its domain is separately dense in the Sobolev space V and the divergence-free Lebesgue space Lp_σ, and the associated embedding has useful compactness properties. However, simultaneous approximation requires density in the intersection V∩Lp_σ, which the authors explicitly state they have not established. Consequently, the compactness result alone cannot replace the resolvent-based approximation argument developed in the paper.

References

Although we are able to establish separately that D(L) is dense in V and in Lp_{\sigma}, we have not been able to show that D(L) is dense in the intersection V\cap Lp_{\sigma}. Consequently, the compactness result for L obtained in does not, by itself, immediately yield the simultaneous approximation required here.

— Divergence-Free Approximation in Sobolev and Lebesgue Spaces on General Unbounded Domains with Applications to Energy Equality in Fluid Dynamics  (2609.02769 - Khan et al., 2 Sep 2026) in Section 4, subsection “General unbounded domains,” Remark following Theorem 2 (Resolvent approximation)