Establish the missing regularity of weak adjoint velocity solutions

Establish whether every weak solution mathbf{P} of the adjoint velocity difference equation belongs to the higher-regularity space W, thereby justifying testing the variational adjoint equation with mathbf{P} and enabling a direct energy proof of uniqueness among weak adjoint solutions.

Background

The paper formulates the velocity adjoint equation variationally because the natural solution regularity is only P∈L∞(0,Tξ;V1(Ω))\mathbf P\in L^\infty(0,T_\xi;V^1(\Omega)) with ∂t(MαβP)∈L2(0,Tξ;W′)\partial_t(\mathcal M_{\alpha\beta}\mathbf P)\in L^2(0,T_\xi;W'). The variational identity, however, is defined for test functions in the more regular space WW.

The missing inclusion P∈W\mathbf P\in W prevents the standard energy test ϕ=P\boldsymbol\phi=\mathbf P. The paper therefore proves uniqueness indirectly by introducing an auxiliary forward dual problem whose velocity component lies in WW. Establishing the unresolved regularity would provide a direct uniqueness argument and clarify the strong regularity and boundary structure of the adjoint system.

References

A direct energy argument cannot be applied to eq:adjoint:P. Indeed, at the natural level of regularity we only have

\mathbf P\in L\infty(0,T_\xi;V1(\Omega)), \qquad \partial_t(\mathcal M_{\alpha\beta}\mathbf P) \in L2(0,T_\xi;W'),

whereas the variational identity eq:adjoint:P is defined for test functions belonging to W. Since it is not known that \mathbf P\inW, the choice \boldsymbol\phi=\mathbf P is not admissible.

eq:adjoint:P:

$\langle-\partial_t(\mathcal M_{\alpha\beta}\mathbf{P}), \boldsymbol\phi\rangle_{W'\timesW} +\mathcal A_{u}(\mathbf{P},\boldsymbol\phi) =(\theta\nabla\mathbf{S},\boldsymbol\phi), $

— Boundary control for optimal mixing by two-dimensional second-grade fluids  (2609.09946 - Kinra, 9 Sep 2026) in Section 'Adjoint system', proof of Lemma \ref{lem:adjoint}, immediately before the dual-problem uniqueness argument