Uniqueness at finite energy for a viscous fluid coupled to a full von Kármán plate

Let $\mathcal{O} \subset \mathbb{R}^3$ be a bounded domain with smooth boundary $\partial\mathcal{O} = \overline{S} \cup \overline{\Omega}$, where $\Omega \subset \{x_3 = 0\}$ is a bounded smooth planar domain, $S \subset \{x_3 \le 0\}$, and the outward normal on $\Omega$ is $e_3$. Fix a viscosity $\nu > 0$ and a Poisson ratio $0 < \mu < 1/2$. A plate occupying $\Omega$ has in-plane displacement $a : \Omega \times [0, T] \to \mathbb{R}^2$ and transverse displacement $w : \Omega \times [0, T] \to \mathbb{R}$; put $$P(a, w) = \tfrac12(\nabla a + \nabla a^{\mathsf T}) + \tfrac12 \nabla w \otimes \nabla w, \qquad \sigma(a, w) = \mathcal{C} P(a, w), \qquad \mathcal{C}\epsilon = \frac{2}{1 - \mu}\bigl[\mu \operatorname{tr}(\epsilon) I + (1 - \mu)\epsilon\bigr].$$ The fluid velocity $v$ and pressure $p$ and the plate solve the linear Stokes system coupled to the full von Kármán plate with in-plane inertia, without rotational inertia, mechanical damping or external forces: $$v_t - \nu \Delta v + \nabla p = 0, \quad \operatorname{div} v = 0 \ \text{ in } \mathcal{O}, \qquad v = 0 \ \text{ on } S, \quad v = (a_t, w_t) \ \text{ on } \Omega,$$ $$a_{tt} - \operatorname{div} \sigma(a, w) = -\nu (\partial_3 v_\parallel + \nabla_\parallel v_3)|_\Omega, \qquad w_{tt} + \Delta^2 w - \operatorname{div}(\sigma(a, w) \nabla w) = p|_\Omega - 2\nu\, \partial_3 v_3|_\Omega,$$ with $a = 0$ and $w = \partial_{n'} w = 0$ on $\partial\Omega$ and $\int_\Omega w = 0$. Let $X = \{v \in L^2(\mathcal{O})^3 : \operatorname{div} v = 0,\ v \cdot n = 0 \text{ on } S\}$, $V = \{v \in H^1(\mathcal{O})^3 : \operatorname{div} v = 0,\ v = 0 \text{ on } S\}$, let hats denote zero mean on $\Omega$, and let $H^{1/2}_*(\Omega)$ consist of the functions whose extension by zero lies in $H^{1/2}(\mathbb{R}^2)$. The initial data have finite energy: $v_0 \in X$, $a_0 \in H^1_0(\Omega)^2$, $w_0 \in \widehat{H}^2_0(\Omega)$, $a_1 \in L^2(\Omega)^2$, $w_1 \in \widehat{L}^2(\Omega)$ and $v_0 \cdot n = w_1$ on $\Omega$.

A weak solution on $[0, T]$ has $v \in L^\infty(0, T; X) \cap L^2(0, T; V)$, $(a, w) \in L^\infty(0, T; H^1_0(\Omega)^2 \times H^2_0(\Omega))$ with $(a, w)(0) = (a_0, w_0)$, $(a_t, w_t) \in L^\infty(0, T; L^2(\Omega)^2 \times \widehat{L}^2(\Omega)) \cap L^2(0, T; H^{1/2}_*(\Omega)^3)$ and $v|_\Omega = (a_t, w_t)$ for almost every $t$, and satisfies $$\begin{aligned} &-\int_0^T (v, \phi_t)_{\mathcal{O}} + 2\nu \int_0^T (D(v), D(\phi))_{\mathcal{O}} - \int_0^T \bigl[(a_t, b_t)_\Omega + (w_t, d_t)_\Omega\bigr] + \int_0^T (\Delta w, \Delta d)_\Omega \\ &\quad + \int_0^T \!\! \int_\Omega \sigma(a, w) : \bigl[\operatorname{sym} \nabla b + \operatorname{sym}(\nabla w \otimes \nabla d)\bigr] = (v_0, \phi(0))_{\mathcal{O}} + (a_1, b(0))_\Omega + (w_1, d(0))_\Omega \end{aligned}$$ for every $\phi \in L^2(0, T; V)$ with $\phi_t \in L^2(0, T; L^2(\mathcal{O})^3)$, $\phi(T) = 0$ and trace $\phi|_\Omega = (b, d)$, where $b \in L^2(0, T; H^1_0(\Omega)^2)$, $d \in L^2(0, T; \widehat{H}^2_0(\Omega))$ and $(b_t, d_t) \in L^2(0, T; L^2(\Omega)^2 \times \widehat{L}^2(\Omega))$. Here $D(v) = \frac12(\nabla v + \nabla v^{\mathsf T})$, $\operatorname{sym} M = \frac12(M + M^{\mathsf T})$ and $(\cdot, \cdot)$ is the $L^2$ pairing. It also satisfies, for almost every $t$, the energy inequality $$\mathcal{E}(t) + 2\nu \int_0^t \|D(v)\|_{L^2(\mathcal{O})}^2 \le \mathcal{E}(0), \qquad \mathcal{E} = \tfrac12\Bigl(\|v\|_2^2 + \|a_t\|_2^2 + \|w_t\|_2^2 + \|\Delta w\|_2^2 + \int_\Omega \mathcal{C} P(a, w) : P(a, w)\Bigr).$$

With rotational inertia in the plate equation, Chueshov and Ryzhkova proved that finite-energy weak solutions exist and are unique. Without it, they noted that weak solutions satisfying the energy inequality exist, and Ryzhkova-Gerasymova proved well-posedness of strong solutions.

Problem. For every such initial datum and every $T > 0$, is the weak solution $(v, a, w)$ unique?