Optimizing exit times
Jean‐Luc Thiffeault, Florence Marcotte, Charles R. Doering, W. R. Young · Bulletin of the American Physical Society · 2015
Consider the advection-diffusion equation for a passive scalar θ(x, t), advected by a steady velocity field u(x), with Dirichlet boundary conditions on some domain Ω: ∂tθ + u · ∇θ = D∆θ, u · n|∂Ω = 0, θ|∂Ω = 0, (1) with ∇ · u = 0. We take θ(x, t0) = θ0(x) ≥ 0, so θ(x, t) ≥ 0. Integrating (1) over Ω, we have ∂t〈θ〉+ 〈u · ∇θ〉 = D〈∆θ〉. (2) The advection term vanishes since the walls are impenetrable, and we have