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.
Highlights
In this work, we obtain very general large time estimates for supnorm values of solutions u(⋅, t) to parabolic initial value problems of the form ut + (b (x, t) u)x = uxx, x ∈ R, t > 0, (1a)u (⋅, 0) = u0 ∈ Lp0 (R) ∩ L∞ (R), 1 ≤ p0 < ∞, (1b) for arbitrary continuously differentiable advection fields b ∈ L∞(R × [0, ∞[)
[0, T∗[ → R which is bounded in each strip ST = R × [0, T], 0< and sTat
Let us illustrate with the important case p0 = 1, where one has
Summary
We obtain very general large time estimates for supnorm values of solutions u(⋅, t) to parabolic initial value problems of the form ut + (b (x, t) u)x = uxx, x ∈ R, t > 0,. We observe on the right hand side of (4) that |u(x, t)| is pushed to grow at points (x, t) where bx(x, t) < 0 If this condition persists long enough, large values of |u(x, t)| might be generated, at sites where −bx(x, t) ≫ 1. That u(⋅, t) stays uniformly bounded for all time in this case.[2] Estimates similar to (6) can be shown to hold for the n-dimensional problem ut + div (b (x, t) u) = Δu, u (⋅, 0) ∈ Lp (Rn) ∩ L∞ (Rn) , (8). More involving applications, such as problems with superlinear advection or degenerate diffusion, which require considerable extra work, will be studied in the future
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