Weak–strong uniqueness at the energy level for a curved viscoelastic shell

Let $\Omega \subset \mathbb{R}^3$ be a bounded smooth domain whose boundary is parametrised by a smooth regular embedding $\varphi : \omega = \mathbb{T}^2 \to \partial\Omega$, with outer unit normal $n$, and fix a tubular radius $L > 0$. A displacement $\eta(t, y)$ with $\|\eta\|_{L^\infty((0, T) \times \omega)} < L$ moves the boundary to $\varphi_\eta = \varphi + \eta n$, bounding the domain $\Omega_\eta(t)$ given by the Hanzawa transform; the reference surface is curved. All functions on $\omega$ are periodic, with the conventions of Breit, Mensah, Schwarzacher and Su. An incompressible Navier–Stokes fluid in $\Omega_\eta(t)$ interacts with a viscoelastic shell: $$v_t + (v \cdot \nabla) v = \Delta v - \nabla \pi + f, \quad \nabla \cdot v = 0 \ \text{ in } \Omega_\eta(t), \qquad v \circ \varphi_\eta = \eta_t n,$$ $$\eta_{tt} - \partial_t \Delta_y \eta + \Delta_y^2 \eta = g - n^{\mathsf T} \tau \circ \varphi_\eta\, n_\eta\, |\partial_1 \varphi_\eta \times \partial_2 \varphi_\eta|, \qquad \tau = \nabla v + \nabla v^{\mathsf T} - \pi I.$$ With $I = (0, T)$, a weak solution with data $(\eta_0, \eta_*, v_0, f, g)$ has $\eta \in W^{1,\infty}(I; L^2(\omega)) \cap W^{1,2}(I; H^1(\omega)) \cap L^\infty(I; H^2(\omega))$ and $v \in L^\infty(I; L^2(\Omega_\eta)) \cap L^2(I; H^1_{\mathrm{div}}(\Omega_\eta))$, satisfies the kinematic condition, takes the initial values, and satisfies $$\begin{aligned} &-\int_0^T \!\! \int_{\Omega_\eta(t)} v \cdot \phi_t - \int_0^T \!\! \int_\omega \eta_t b_t - \int_0^T \!\! \int_{\Omega_\eta(t)} v \otimes v : \nabla \phi + \int_0^T \!\! \int_{\Omega_\eta(t)} \nabla v : \nabla \phi + \int_0^T \!\! \int_\omega \nabla_y \eta_t \cdot \nabla_y b + \int_0^T \!\! \int_\omega \Delta_y \eta\, \Delta_y b \\ &\quad = \int_{\Omega_{\eta_0}} v_0 \cdot \phi(0) + \int_\omega \eta_* b(0) + \int_0^T \!\! \int_{\Omega_\eta(t)} f \cdot \phi + \int_0^T \!\! \int_\omega g b \end{aligned}$$ for all smooth $b$ and smooth divergence-free $\phi$ on $\mathbb{R}^3$ vanishing at $T$ with $\phi \circ \varphi_\eta = b n$, together with the energy inequality, for almost every $t$, $$E(t) + \int_0^t \!\! \int_{\Omega_\eta(s)} |\nabla v|^2 + \int_0^t \!\! \int_\omega |\nabla_y \eta_t|^2 \le E(0) + \int_0^t \!\! \int_{\Omega_\eta(s)} f \cdot v + \int_0^t \!\! \int_\omega g \eta_t, \qquad E = \tfrac12 \int_{\Omega_\eta} |v|^2 + \tfrac12 \int_\omega \bigl(|\eta_t|^2 + |\Delta_y \eta|^2\bigr).$$ The data are $\eta_0 \in H^3(\omega)$, $\eta_* \in H^1(\omega)$, $v_0 \in H^1_{\mathrm{div}}(\Omega_{\eta_0})$ with $v_0 \circ \varphi_{\eta_0} = \eta_* n$, $g \in L^2(I; L^2(\omega))$ and $f \in L^\infty(I; L^2_{\mathrm{loc}}(\mathbb{R}^3)) \cap L^2(I; H^1_{\mathrm{loc}}(\mathbb{R}^3))$. A strong solution $(\eta_s, v_s, \pi_s)$ is a weak solution with $$\eta_s \in W^{1,\infty}(I; H^1) \cap W^{1,2}(I; H^2) \cap L^\infty(I; H^3) \cap W^{2,2}(I; L^2) \cap L^2(I; H^4),$$ $$v_s \in W^{1,2}(I; L^2(\Omega_{\eta_s})) \cap L^2(I; H^2(\Omega_{\eta_s})) \cap L^\infty(I; H^1(\Omega_{\eta_s})), \qquad \pi_s \in L^2(I; H^1(\Omega_{\eta_s})).$$

Breit, Mensah, Schwarzacher and Su proved that a weak solution $(\eta_w, v_w)$ coincides with a strong solution with the same data if, in addition, $\eta_w \in L^\infty(I; C^{0,1}(\omega))$; the energy class gives only $\eta_w \in L^\infty(I; H^2(\omega))$, which does not embed into $C^{0,1}(\omega)$. For a flat reference geometry, Schwarzacher and Sroczinski's direct approach gives weak–strong uniqueness without the Lipschitz assumption.

Problem. Suppose a strong solution $(\eta_s, v_s, \pi_s)$ exists on $I$. Must every weak solution $(\eta_w, v_w)$ with the same data coincide with it on their common admissible interval, that is, $\eta_w = \eta_s$ and $v_w = v_s$ almost everywhere, without assuming $\eta_w \in L^\infty(I; C^{0,1}(\omega))$?