  We investigate weak solutions to a fluid-structure interaction (FSI) problem
between the flow of an incompressible, viscous fluid modeled by the
Navier-Stokes equations, and a poroviscoelastic medium modeled by the Biot
equations. These systems are coupled nonlinearly across an interface with mass
and elastic energy, modeled by a reticular plate equation, which is transparent
to fluid flow. We provide a constructive proof of the existence of a weak
solution to a regularized problem. Next, a weak-classical consistency result is
obtained, showing that the weak solution to the regularized problem converges,
as the regularization parameter approaches zero, to a {{classical}} solution to
the original problem, when such a classicalsolution exists. While the
assumptions in the first step only require the Biot medium to be poroelastic,
the second step requires additional regularity, namely, that the Biot medium is
poroviscoelastic. This is the first weak solution existence result for an FSI
problem with nonlinear coupling involving a Biot model for poro(visco)elastic
media.
