We analyze the asymptotic stability of the quasi-linearly stratified densities in the 2D inviscid incompressible porous medium equation on R2 with respect to the buoyancy frequency N. Our target density of stratification is the sum of the large background linear profile with its slope N and the small perturbation that could be both non-linear and non-monotone. Quantification in N will be performed not only on how large the initial density disturbance is allowed to be but also on how much the target densities can deviate from the purely linear density stratification without losing their stability. For the purely linear density stratification, our method robustly applies to the three fundamental domains R2, T2, and T×[−1,1], improving both the previous result by Elgindi (2017) [18] on R2 and T2, and the study by Castro et al. (2019) [12] on T×[−1,1]. The obtained temporal decay rates to the stratified density on R2 and to the newly found asymptotic density profiles on T2 and T×[−1,1] are all sharp, fully realizing the level of the linearized system. We require the initial disturbance to be small in Hm for any integer m≥4, which we even relax to any positive number m>3 via a suitable anisotropic commutator estimate.
Junha Kim was supported by a KIAS Individual Grant (MG086501) at Korea Institute for Advanced Study. We sincerely thank the anonymous referees for very careful reading of the manuscript and providing several suggestions, which have been reflected in the paper.
