Global smooth attractors for dynamics of thermal shallow shells without vertical dissipation


Nonlinear shallow shell models with thermal effects are considered. Such models provide basic prototypes for elastic bodies appearing in the flow/fluid structure interactions. It is assumed that shells are thin and do not account for the regularizing effects of rotary inertia. The nonlinear effects in the model become supercritical, and this raises a first fundamental question of Hadamard well-posedness in the class of weak solutions. The first main result of the present paper addresses the issue of generation of a nonlinear semigroup for such a model. The second result describes longtime behavior of the resulting dynamical system. It is shown that longtime dynamics admits finite-dimensional and smooth global attractors. This result is obtained with-out imposing any mechanical dissipation affecting the vertical displacements of the shell where the latter satisfy free boundary conditions. This particular feature, along with supercritical nonlinearity, leads to substantial challenges in the analysis. The resolution of the encountered difficulties rests on recently developed mathematical tools such as (1) maximal regularity for thermal shells with free boundary conditions, (2) "hidden" trace regularity propagated by thermal effects, (3) compensated compactness and related theory of quasi-stable systems derived from books by Chueshov and by Chueshov and Lasiecka.

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Transactions of the American Mathematical Society