Global Existence of Quasi-Stratified Solutions for the Confined IPM Equation

Abstract

In this paper, we consider a confined physical scenario to prove global existence of smooth solutions with bounded density and finite energy for the inviscid incompressible porous media (IPM) equation. The result is proved using the stability of stratified solutions, combined with an additional structure of our initial perturbation, which allows us to get rid of the boundary terms in the energy estimates.

Asymptotic stability of the 2D Boussinesq equations without thermal conduction

“…We now turn to the Fourier expansion of functions defined in Ω = R × (0, 1) with vanishing Dirichlet/Neumann boundary value. Following [4,5], we introduce the functional spaces for m…”

On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term

“…Indeed, both difficulties i) and ii) can be bypass if our initial perturbation and velocity have a special structure. We introduce the following spaces, which we used in [6] to characterize our initial data:…”

“…(See for example [9] and [31], in the case of SQG). As in [6], we focus only on our setting and in our specific class of initial data.…”

“…It is enough to assume that N : R + −→ R + satisfies that N ′ (t) N (t) ≥ 1 to obtain: Now, we use the following calculus lemma, which proof can be found in [6]. Then, applying the previous inequality we see that:…”

“…In the context of the 2D Boussinesq system when dissipation is present in at least one of the equations see [12] and [23], where the authors study the global well-posedness and stability/instability of perturbations near a special type of hydrostatic equilibrium. Finally, for other type of problems as the β-plane equation or the IPM equation see [16], [32] and [15], [6] respectively.…”