Sharp asymptotic stability of the incompressible porous medium equation on the whole plane

We study the asymptotic stability of the steady state eₛ(x)=-x₂ of the incompressible porous media equation in R². For every $s>2$, we prove that sufficiently small initial perturbations θ₀∈ Hˢ( R²) of eₛ generate unique global solutions ρ=eₛ+θ, where $θ$ remains uniformly bounded in Hˢ and converges strongly to zero in Hˢ as t→∞. The main innovations of this work are threefold. First, exploiting the structure of the equation, we construct a weighted energy that eliminates the need to control \|∇ u²\|L¹ₜL^∞ₓ. Second, rather than estimating the anisotropic commutators directly, we study the integral terms containing these commutators and establish suitable bounds through refined computations in frequency space. Third, we establish convergence in the highest-order Hˢ norm under uniform energy bounds involving only an L^∞ₜHˢₓ bound on $θ$ and an L²ₜHˢₓ bound on R₁θ.