Long time behavior for a dissipative shallow water model

Abstract

We consider the two-dimensional shallow water model derived in [29], describing the motion of an incompressible fluid, confined in a shallow basin, with varying bottom topography. We construct the approximate inertial manifolds for the associated dynamical system and estimate its order. Finally, working in the whole space R^2, under suitable conditions on the time dependent forcing term, we prove the L^2 asymptotic decay of the weak solutions.