Fractal exponential attractor conjecture for the flow-plate evolution

Let $(T_t,\mathbf H)$ be the evolution associated with the flow-plate system, let $k_*$ be the damping threshold from the main theorem, and let ${\mathbf A}_{\text{exp}}$ denote its exponential attractor. Assume the hypotheses of the main theorem and that $k>k_*$, where $k_*$ depends on the intrinsic parameters of the flow-plate system.

Fractal exponential attractor conjecture. The evolution $(T_t,\mathbf H)$ has a fractal exponential attractor, and

$${\mathbf A}_{\text{exp}}\subset (H^4\cap H_0^2)(\Omega)\times H_0^2(\Omega)\times L^2(-t^*,0;(H^4\cap H_0^2(\Omega)))$$

with ${\mathbf A}_{\text{exp}}$ bounded in this topology.

This conjecture proposes further regularity and finite-dimensional long-time behavior for the exponential attractor of the delay formulation under sufficiently large damping. The source presents it as an open direction motivated by the corresponding decomposition for the delay system.