R. Bianchini, Crin-Barat T., M. Paicu. **Relaxation approximation and asymptotic stability of stratified solutions to the IPM equation** (2022)

**Abstract.** We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in \dot{H}^{1-\tau}(\mathbb{R}^2) \cap \dot{H}^s(\mathbb{R}^2) with s > 3 and for any 0 \lt \tau \lt 1. Such result improves the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to H^{20} (\mathbb{R}^2) . 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 H^{1 - \tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2) with s > 3 and 0 \lt \tau \lt 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 \|u_2(t)\|_{L^\infty (\mathbb{R}^2)} for initial data only in \dot H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2) with s >3.

arxiv: 2210.02118