发表机构
University of Maryland, College Park（马里兰大学帕克分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究二维不可压缩多孔介质方程分层稳态的稳定性，证明周期通道上均匀递减稳态在小扰动下非线性稳定，而具有正导数上确界的稳态则非线性不稳定，并给出收敛速率与不稳定性机制。
AI 中文摘要
我们研究二维不可压缩多孔介质方程在$\u0003c{T}\times(-1,1)$和$\u0003c{T}\times\u0003c{R}$上分层稳态$\rho_s=\rho_s(y)$的稳定性与不稳定性。在周期通道上，满足自然边界相容条件的均匀递减稳态在小$H^m$扰动下是非线性稳定的，对每个整数$m>2$成立。解在$L^2$中收敛到初始密度的保测分层，收敛速度为$t^{-m/2}$。相反，每个满足$\u0003c{sup}\rho_s'>0$的稳态在$H^m$中是非线性不稳定的。在无限圆柱上，我们在额外的间隙条件$\u0003c{sup}_{\u0003c{R}}\rho_s' >\u0003c{limsup}_{|y|\to\u0003c{infty}}\rho_s'(y)$下证明非线性不稳定性。在这两种情形中，不稳定性由线性化算子的正特征值产生，这些特征值收敛到$\u0003c{sup}\rho_s'$，该值同时等于其谱界和半群增长界。
英文摘要
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.
