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

Kim-Ngan Le, William McLean, Martin Stynes
2019 Communications on Pure and Applied Analysis  
A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator ut − ∇ · (∂ 1−α t κα∇u − F∂ 1−α t u), where 0 < α < 1. The forcing function F = F(t, x), which is more difficult to analyse than the case F = F(x) investigated previously by other authors. The spatial domain Ω ⊂ R d , where d ≥ 1, has a smooth boundary. Existence, uniqueness and regularity of a mild solution u is proved under the hypothesis that the initial data u 0 lies in L 2 (Ω). For 1/2
more » ... n L 2 (Ω). For 1/2 < α < 1 and u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω), it is shown that u becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived-these are known to be needed in numerical analyses of this problem. t is the standard 2000 Mathematics Subject Classification. 35R11. Riemann-Liouville fractional derivative operator defined by ∂ 1−α t u = (J α u) t , where J β denotes the Riemann-Liouville fractional integral operator of order β, viz., Lemma 3.3. [14, Lemma 3.1 (ii)] If α ∈ (0, 1) and v(x, ·) ∈ L 2 (0, T ) for each x ∈ Ω, then for t ∈ [0, T ], t 0 J α v, v (s) ds ≥ cos(απ/2) t 0 J α/2 v 2 (s) ds. TIME-FRACTIONAL FOKKER-PLANCK EQUATION WITH GENERAL FORCING 2769 Lemma 3.4. [5, Lemma 2.1] Let β ∈ (0, 1). If φ(·, t) ∈ L 2 (Ω) for t ∈ [0, T ], then Proof. As the proof is short, we give it here for completeness. The Cauchy-Schwarz inequality yields
doi:10.3934/cpaa.2019124 fatcat:t6wepwrilbburjil6db2zi3c6i