English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove the quantitative asymptotic stability of the quasi-linearlystratified densities in the 2D inviscid incompressible porous medium equationon $\bbR^2$. The target density is the sum of the large background linearprofile and the small perturbation that could be both nonlinear andnon-monotone. The admissible degrees of both the quasi-linearity and theinitial density disturbance are estimated with respect to the magnitude ofstratification.  Once we restrict ourselves to the purely linear density stratification, ourmethod robustly applies to the three fundamental domains $\bbR^2,$ $\bbT^2,$and $\bbT\times[-1,1]$. The obtained temporal decay rates to the stratifieddensity on $\bbR^2$ and to the newly found asymptotic density profiles on$\bbT^2$ and $\bbT\times[-1,1]$ are sharp, fully realizing the level of thelinearized system.  We require the initial disturbance to be small in $H^m$ for any integer$m\geq 4$. As an independent topic of interest, we prove a suitable anisotropiccommutator estimate that overcomes the limitation of the standard ones. Thisfurther relaxes the condition on $m$ from an integer $m\geq 4$ to any positivenumber $m>3$ in $\bbR^2$ and $\bbT^2$.

Quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation on the three fundamental domains