Sharp stability and instability of stratified steady states for the incompressible porous media equation
We study the stability and instability of stratified steady states $ρ_s=ρ_s(y)$ for the two-dimensional incompressible porous media equation on $\mathbb{T}\times(-1,1)$ and $\mathbb{T}\times\mathbb{R}$. On the periodic channel, uniformly decreasing steady states satisfying natural boundary-compatibility conditions are nonlinearly stable under small $H^m$ perturbations, for every integer $m>2$. The solutions converge in $L^2$ to the measure-preserving stratification of the initial density at the rate $t^{-m/2}$. Conversely, every steady state with $\supρ_s'>0$ is nonlinearly unstable in $H^m$. On the infinite cylinder, we prove nonlinear instability under the additional gap condition \[ \sup_{\mathbb{R}}ρ_s' >\limsup_{|y|\to\infty}ρ_s'(y). \] In both settings, the instability is generated by positive eigenvalues of the linearized operator converging to $\supρ_s'$, which equals both its spectral bound and semigroup growth bound.