Existence of Weak Solutions and Higher-Order Regularity for a Heat-Wave Fluid-Structure Interaction System on a Periodic Strip

In this paper, we study a heat-wave fluid-structure interaction system posed on a periodic strip geometry and investigate whether higher-order $H^s$-estimates for arbitrarily large values of $s$ can be established for the coupled system despite the inherent mismatch of parabolic and hyperbolic regularity. We establish global existence of weak solutions and Sobolev regularity up to the $H^3$-level, which, to the best of our knowledge, is the highest regularity currently established for this system. We further show that, even for arbitrarily regular initial data, neither the semigroup approach through higher-order generator domains $D(A^k)$ with $k\in \mathbb{N}$ nor the PDE-based tangential-normal recovery procedure yields closed estimates beyond $H^3$ due to the complexities of the coupled PDE system.