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

@misc{banihashemi2026sharp,
  title = {{Sharp stability and instability of stratified steady states for the incompressible porous media equation}},
  author = {Seyed Abdolhamid Banihashemi and Sepehr Mohammadkhani and Huy Q. Nguyen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37397},
  eprint = {2609.37397},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37397},
}

TY  - UNPB
TI  - Sharp stability and instability of stratified steady states for the incompressible porous media equation
AU  - Seyed Abdolhamid Banihashemi
AU  - Sepehr Mohammadkhani
AU  - Huy Q. Nguyen
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37397
UR  - https://arxiv.org/abs/2609.37397
AB  - We study the stability and instability of stratified steady states ρ_s=ρ_s(y) for the two-dimensional incompressible porous media equation on 𝕋×(−1,1) and 𝕋×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_ℝρ_s' >limsup_|y|→∞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.
ER  -