Sharp stability and instability of stratified steady states for the incompressible porous media equation

We study the stability and instability of stratified steady states ${\rho }_s={\rho }_s(y)$ 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 $t^{-m\mathrm{/}2}$. Conversely, every steady state with $\sup {\rho }_s^{\mathrm{\prime }}>0$ is nonlinearly unstable in $H^m$. On the infinite cylinder, we prove nonlinear instability under the additional gap condition $$\underset{\mathbb{R}}{\sup }{\rho }_s^{\mathrm{\prime }}>\underset{\mathrm{\mid }y\mathrm{\mid }\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{\rho }_s^{\mathrm{\prime }}(y)\mathrm{.}$$ In both settings, the instability is generated by positive eigenvalues of the linearized operator converging to $\sup {\rho }_s^{\mathrm{\prime }}$, which equals both its spectral bound and semigroup growth bound.