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We consider the interaction of a viscous incompressible fluid with a flexibleshell in three space dimensions. The fluid is described by thethree-dimensional incompressible Navier--Stokes equations in a domain that ischanging in accordance with the motion of the structure. The displacement ofthe latter evolves along a visco-elastic shell equation. Both are coupledthrough kinematic boundary conditions and the balance of forces.  We prove a counterpart of the classical Ladyzhenskaya-Prodi-Serrin conditionyielding conditional regularity and uniqueness of a solution.  Our result is a consequence of the following three ingredients which might beof independent interest: {\bf (i)} the existence of local strong solutions,{\bf (ii)} an acceleration estimate (under the Serrin assumption) ultimatelycontrolling the second-order energy norm, and {\bf (iii)} a weak-stronguniqueness theorem. The first point, and to some extent, the last point werepreviously known for the case of elastic plates, which means that the relaxedstate is flat. We extend these results to the case of visco-elastic shells,which means that more general reference geometries are considered such ascylinders or spheres. The second point, i.e. the acceleration estimate forthree-dimensional fluids is new even in the case of plates.

Ladyzhenskaya-Prodi-Serrin condition for fluid-structure interaction systems