General Asymptotic Supnorm Estimates for Solutions of One-Dimensional Advection-Diffusion Equations in Heterogeneous Media

Abstract

We derive general bounds for the large time size of supnorm values ∥u(·,t)∥L∞(ℝ) of solutions to one-dimensional advection-diffusion equations ut+(b(x,t)u)x=uxx,x∈ℝ,t>0 with initial data u(·,0)∈Lp0(ℝ)∩L∞(ℝ) for some 1≤p0<∞ and arbitrary bounded advection speeds b(x,t), introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.