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

Abstract

We establish the well-posedness of an initial-boundary value problem for a general class of linear time-fractional, advection-diffusion-reaction equations, allowing space-and time-dependent coefficients as well as initial data that may have low regularity. Our analysis relies on novel energy methods in combination with a fractional Gronwall inequality and properties of fractional integrals.

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

“…This section introduces some notations and states some technical results that will be used in our subsequent regularity analysis. As in our recent paper [1], we define the quadratic operators Q µ 1 and Q µ 2 , for µ ≥ 0 and 0 ≤ t ≤ T , by…”

“…This paper is the sequel to a study [1] of existence and uniqueness of the weak solution to a time-fractional PDE of the form…”

“…Various special cases of this problem occur in descriptions of subdiffusive transport, with the parameter α arising from a continuous-time, random walk model [2, section 3.4] in which the waiting-time distribution is a power law decaying like t −1−α as t → ∞ [3,4,5,6,7,8]. For more details, see our related paper [1]. Our purpose here is to derive estimates for the derivatives of u, motivated by their crucial role in the error analysis of numerical methods [9,10,11,12,13,14,15,16] for applications included in the class (1) of time-fractional problems.…”

Uniqueness of determining the variable fractional order in variable-order time-fractional diffusion equations

“…which are a variable-order analogue of the constant-order fractional integral and differential operators defined in (2.2). To better characterize the temporal singularity of the solution at the initial time, we define the weighted Banach spaces involving time C m γ ((0, T ]; X ) with m ≥ 2, 0 ≤ µ < 1 modified from those in [20]…”

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

“…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.…”