Regularity theory for time-fractional advection–diffusion–reaction equations

Abstract

We investigate the behavior of the time derivatives of the solution to a linear timefractional, advection-diffusion-reaction equation, allowing space-and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals.

Well-Posedness of Time-Fractional Advection-Diffusion-Reaction Equations

“…Our existence theorem is stated as follows. Note the weak continuity at t = 0 asserted in part 5; we show in the companion paper [26] that the solution u is continuous on the closed interval [0, T ] provided u 0 ∈Ḣ µ (Ω) for some µ > 0.…”

Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing

“…the comment following Assumption 6.1). By contrast, the results in [11,12] hold for the full range of values 0 < α < 1, but with constants that blow up as α → 1. Also, the analysis is significantly longer than the one presented here.…”

“…The alternative and longer analysis in [12,Theorems 12 and 13] shows that these bounds can be improved to t q ∆u (q) (t) ≤ C u 0 H 2 (Ω) +M t η−α and t q u (q) (t) ≤ C t α u 0 H 2 (Ω) +M t η ,…”

“…To the best of our knowledge, the well-posedness and regularity properties of solutions to (1) are open questions at present, apart from a pair of recent preprints [11,12] which treat a wider class of problems that includes (1) as a special case. The analysis in [11,12] proceeds along broadly similar lines to here-relying on Galerkin approximation, a fractional Gronwall inequality and compactness arguments-but employs a different sequence of a priori estimates and uses neither the weighted L 2 -norm of Definition 2.2 nor the Aubin-Lions-Simon lemma (Lemma 3.8). An interesting consequence of the approach taken here is that the constants in our estimates remain bounded as α → 1, which one expects since in this limit (1) becomes the classical Fokker-Planck equation.…”

Sharp pointwise-in-time error estimate of L1 scheme for nonlinear subdiffusion equations

“…Maskari and Karaa [5] obtain that if u 0 ∈ Ḣν (Ω) with ν ∈ (0, 2], the solution of problem (1.1) satisfies ∂ t u(t) L 2 (Ω) ≤ Ct να/2−1 , which implies that there exists a parameter σ ∈ (0, α] and u t → ∞ as t → 0 + . One can refer to more works [6,7,8,9,10,11,12] on the discussion of the regularity of solutions.…”