The authors study everywhere uniform stability problem for a nonlinear differential equation \(\dot{x}=f\left( x\right) \) on a compact phase space \(X\subset\mathbb{R}^{n}\) using the associated advection equation \[ \frac{\partial\rho}{\partial t}=-\nabla\cdot\left( \rho f\right) . \] Connection between the stability of an ordinary differential equation and a linear partial differential equation provides new computational tools for the analysis of nonlinear dynamical systems. The principal result in the paper states that almost everywhere uniform stability of an attractor set for a continuous dynamical system is equivalent to existence of a positive solution, termed “Lyapunov density,” to an advection type partial differential equation. A finite element based numerical method is used for computing the density in lower dimensions.