Abstract
We prove the nonlinear asymptotic stability of stably stratified solutions to the
Incompressible Porous Media equation (IPM) for initial perturbations in ?H1- (R2) ? ?H s(R2)
with s > 3 and for any 0 < < 1. Such result improves the existing literature, where the
asymptotic stability is proved for initial perturbations belonging at least to H20(R2).
More precisely, the aim of the article is threefold. First, we provide a simplified and improved
proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity
in H1- (R2)? ?Hs(R2) with s > 3 and 0 < < 1. Next, we prove the strong convergence of
the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly,
the asymptotic stability of stratified solutions to (IPM) follows as a byproduct.
A symmetrization of the approximating system and a careful study of the anisotropic properties
of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain
uniform energy estimates. Finally, one of the main new and crucial points is the integrable
time decay of the vertical velocity ku2(t)kL1(R2) for initial data only in ?H 1- (R2) ? ?H s(R2)
with s > 3.
Anno
2024
Autori IAC
Tipo pubblicazione
Altri Autori
Roberta Bianchini, Timhote CrinBarat and Marius Paicu
Editore
Springer.
Rivista
Archive for rational mechanics and analysis (Print)