Long time behavior for a dissipative shallow water model

Abstract

Abstract. We consider the two-dimensional shallow water model derived in [20], 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, considering the whole domain R 2 and under suitable conditions on the time dependent forcing term, we prove the L2 asymptotic decay of the weak solutions.

Global well‐posedness of inviscid lake equations in the Besov spaces

“…For the degenerate case, namely, the case when $$$ b $$$ is strictly positive in $$$ \Omega $$$ and vanishes on the shore $$$ \mathrm{\partial \Omega } $$$, Bresch and Métivier 7 proved the global existence and uniqueness of solutions. In addition to the issue of well‐posedness, the problem of vanishing viscosity limit of the corresponding viscous models was also addressed in Jiu and Niu, 8 while the case of degenerate topography was investigated in Jiu et al 9 Recently, for the viscous lake equations, Sciacca et al 10 constructed the approximate inertial manifolds for the associated dynamical system and proved the $$$ {L}_2 $$$ asymptotic decay of the weak solutions. Except for the lake equations, there are other viscous shallow water equations with a more general diffusion, which have been widely studied by many mathematicians.…”

“…Except for the lake equations, there are other viscous shallow water equations with a more general diffusion, which have been widely studied by many mathematicians. One can refer to Sciacca et al, 10 Bresch et al, 11 Bresch and Noble, 12 and Chen et al 13 for more studies.…”