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 advectiondiffusion equations 𝑢𝑡 + (𝑏(𝑥, 𝑡)𝑢)𝑥 = 𝑢𝑥𝑥, 𝑥 ∈ R, 𝑡 > 0 with initial data 𝑢(⋅, 0) ∈ 𝐿𝑝 0 (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.