Sun-dual characterization for unbounded domains

Investigate whether, for unbounded domains, the sun-dual construction associated with the Stokes semigroup reduces to a simple $L^1$-type quotient or instead yields a more exotic topological space.

Background

The paper identifies the sun-dual space of the Stokes operator on Cσ,0(Ω)C_{\sigma,0}(\Omega) for bounded C1,αC^{1,\alpha}-domains with the quotient space L1(Ω,Cd)/∇W1,1(Ω)L^1(\Omega,C^d)/\nabla W^{1,1}(\Omega). The authors explain that extending this characterization to unbounded domains is problematic because compactness of the resolvent is lost, so the sun-reflexivity theorem used in the bounded-domain argument is no longer available. Consequently, it is unresolved whether the sun-dual space in unbounded geometries has an analogous concrete L1L^1-quotient description or a substantially different topological structure.

References

Although this abstract construction is highly robust, its specific $L1$-characterization cannot be seamlessly generalized to unbounded domains. In such geometries, even for $p \in (1,\infty)$, Farwig, Kozono, and Sohr already demonstrated that the Helmholtz decomposition fails and $Lp_{\sigma,n}(\Omega)$ must be replaced by appropriate intersections or sums with $L2$. Furthermore, the loss of compactness of the resolvent on $C_{\sigma,0}(\Omega)$ prevents the use of Phillips' sun-reflexivity theorem . This suggests that in the unbounded setting, the sun-dual construction may not reduce to a simple $L1$-type quotient, but may instead yield a more exotic topological space. We plan to investigate this in the future.

— The $\mathrm{L}^1$-Stokes Semigroup  (2609.08715 - Binz et al., 8 Sep 2026) in Section 1, Introduction