Uniqueness of arbitrary energy-level weak solutions

Establish uniqueness for arbitrary weak solutions of the heat-wave fluid-structure interaction system in the energy space \(\mathbb H=L^2(\Omega_f)\times V_s\times L^2(\Omega_s)\), beyond the class of weak solutions obtained through the Galerkin approximation.

Background

The paper proves existence of global weak solutions and establishes uniqueness only within the class obtained as limits of the Galerkin approximations. For two arbitrary weak solutions in the energy space, the available regularity gives only $w_t\in L^\infty(0,T;L^2(\Omega_s))$, whereas the natural energy argument would require $w_t\in L^2(0,T;V_s)$ in order to test the difference equation with $(\widehat u,\widehat w_t)$.

The authors note that stronger initial regularity, specifically initial data in the generator domain $D(A)$, supplies the additional regularity needed for the energy argument and yields uniqueness for such more regular solutions. Whether uniqueness holds for all weak solutions at the natural energy level is therefore left unresolved.