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Variational Formulation of Time-Fractional Parabolic Equations

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Variational Formulation of Time-Fractional Parabolic Equations Variational formulation of time-fractional parabolic equations ∗ Michael Karkulik† Abstract We consider initial/boundary value problems for time-fractional parabolic PDE of order 0 <α< 1 with Caputo fractional derivative (also called fractional diffusion equations in the literature). We prove well-posedness of corresponding variational formulations based entirely on fractional Sobolev-Bochner spaces, and clarify the question of possible choices of the initial value. Key words: Fractional diffusion equation, Initial value/boundary value problem, Well-posedness AMS Subject Classification: 26A33, 35K15, 35R11 1 Introduction Physical phenomena based on standard diffusion, where the mean square displacement of a diffusing particle scales linearly with time x(t)2 t, are typically modeled by partial differential h i∼ equations involving standard (i.e., integer order) differential operators. So-called anomalous diffusion, on the other hand, is characterized by non-linear scaling. For example, a diversive number of systems exhibit anomalous diffusion which follows the power-law x(t)2 tα with h i ∼ 0 <α< 1 (subdiffusion) or 1 <α< 2 (superdiffusion). Systems with such power-laws include ones with constrained pathways such as fractal, disordered, or porous media, polymers, aquifers, and quantum systems, among others. We refer to [18] for an extensive overview on the subject. In the latter work, the authors list various ways how to model anomalous diffusion processes. For arXiv:1704.03257v1 [math.NA] 11 Apr 2017 problems involving external fields or boundary conditions, the most natural way is to consider partial differential equations involving so-called fractional differential operators. In the work at hand, we consider a time-fractional parabolic initial/boundary value problem of the form ∂αu ∆u = f in (0, T ) Ω, t − × u = 0 on (0, T ) ∂Ω, (1) × u = g for 0 Ω, { }× ∗Supported by Conicyt Chile through project FONDECYT 1170672. †Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile, mkarkulik.mat.utfsm.cl, email: [email protected] 1 where (0, T ) is a time interval and Ω Rn a spatial Lipschitz domain. Here, ∆ is the spatial α ⊂ − Laplacian, 1/2 <α< 1, and ∂t is a fractional time derivative of order α. More specifically, we will use the so-called Caputo derivative, which is defined formally by 1 t ∂αu(t) := (t s)−αu′(s) ds. t Γ(1 α) − − Z0 Recently, researchers have started to analyze finite element methods with respect to their ability to approximate solutions of fractional differential equations. While this started with classical Galerkin finite element methods for steady-state fractional diffusion equations as in [9, 12], nu- merical methods for time-dependent fractional partial differential equations include time-stepping methods [8, 13, 14], Discontinuous Galerkin methods [19, 20], as well as methods based on the Laplace transform [17]. It goes without saying that this list is far from being exhaustive. We mention here also the numerical approach from [21] which is based on the extension theory by Caffarelli and Silvestre [4]. The aforementioned numerical methods are usually based on a variational formulation of the equation under consideration. Existing works on variational formu- lations of time-fractional parabolic partial differential equations are scarce; as to our knowledge, the works [27, 24, 1] are of relevance in connection with our model problem (1) (for Semigroup theory for related Volterra integral equations see [23]). These works have in common that (i) their functional analytic setting is not based exclusively on classical Sobolev regularity in time, α but rather involves the operator ∂t , and that (ii) the initial value g is taken from L2(Ω). The goal of the present work is to derive the well-posedness of variational formulations set up in classical Sobolev-Bochner spaces and to clarify the question of regularity needed for the initial data. Now, as our functional analytic setting is based only on Sobolev regularity, a result of this kind is specifically interesting for numerical analysis of the equation (1). Indeed, approximation results for functions with certain Sobolev regularity are well known and ubiquitous in numerical analysis. The property (i) is owed to the fact that there is no rigorous definition of time-fractional derivatives on fractional Sobolev-Bochner spaces available. It sure is true that operators defined between real valued Sobolev spaces L (J) L (J) do extend to vector-valued counterparts 2 → 2 L (J; X) L (J; X) (for X a Hilbert space, this is a classical result of Marcienkiwicz and 2 → 2 Zygmund [16]), but the fact that we are dealing with Sobolev regularity Hα(J; X) in time needs some care and additional analysis. To that end, we will show first that the fractional Caputo derivative is a linear and bounded operator on a time-fractional Sobolev-Bochner space. This way, we can consider a variational formulation of (1) based exclusively on Sobolev regularity, which resembles classical variational formulations for parabolic equations. Regarding the point (ii), the choice of g L (Ω) as initial value is indeed admissible, but one has to bear in mind ∈ 2 the following: While the space L (J; H1(Ω)) L (J; H−1(Ω)), used in variational formulations 2 ∩ 2 of parabolic equations, is continuously embedded in C(J; L2(Ω)), this is no longer true for the equation (1). We will show that the spacee of solutions of our variational formulation of (1) is con- tinuously embedded in C(J; H1−1/α−ε(Ω)) for all ε> 0. Our main result is then well-posedness of the variational formulation, cf. Theorem 2. 2 2 Mathematical setting and main results 2.1 Sobolev and Bochner spaces We denote by Ω Rd a (spatial) Lipschitz domain, and by J = (0, T ) for T > 0 a temporal ⊂ 1 interval. We use Lebesgue and Sobolev spaces L2(Ω) and H (Ω), the tilde denoting vanishing trace on the boundary ∂Ω. The fractional Sobolev spaces Hs(Ω) for s (0, 1) are defined by the s 1 ∈ K-method of interpolation as H (Ω) := [L2(Ω), H (Ω)]s,2, cf.e [26], where e ∞ [B ,B ] := u B u e < , e u 2 := t−2s−1K (t, u)2 dt 0 1 s,2 0 [B0,B1]s,2 [B0,B1] 2 [B0,B1] ∈ | k k ∞ k k s, 0 n o Z with the K-functional 2 2 2 2 K[B0,B1](t, u) := inf u w B0 + t w B1 . w∈B1 k − k k k The topological dual of a Banach space X is denoted by X′, and we define H−s(Ω) := Hs(Ω)′ as the topological duals with respect to the extended L (Ω) scalar product ( , ), and duality 2 · · will be denoted by , . We set H0(Ω) := L (Ω). In time, we additionally use Lebesguee and h· ·i 2 Sobolev spaces L (J) and Hα(J), for α (0, 1]. The L (J) scalar product will also be denoted 2 ∈ 2 by ( , ), but it will always be cleare which scalar product we are using. We use the notation Du · · to denote the distributional derivative of a function u given on J. For α (0, 1) the norm on ∈ the space Hα(J) is given by 2 2 2 2 2 f(s) f(t) f α := f + f α < , where f α := | − | ds dt. k kH (J) k kL2(J) | |H (J) ∞ | |H (J) s t 2α+1 ZJ ZJ | − | α 1 −s s ′ We mention that H (J) = [L2(J),H (J)]α,2 with equivalent norms. We set H (J) := H (J) and H−s(J) := Hs(J) for s (0, 1). For a Banach space X, we use the Bochner space L (J; X) ∈ 2 of functions f : J X which are strongly measurable with respect to the Lebesgue measuree ds → on J eand f 2 := f(s) 2 ds < . k kL2(J;X) k kX ∞ ZJ For w a measurable, positive function on J, we denote by L2(J, w; X) the w-weighted Lebesgue space of strongly measurable functions with norm f 2 := w(s)2 f(s) 2 ds < . k kL2(J,w;X) k kX ∞ ZJ We say that a function f L (J; X) has a weak time-derivative ∂ f L (J; X), if ∈ 2 t ∈ 2 ∂ fϕ = f∂ ϕ for all ϕ C∞(J). (2) t − t ∈ 0 ZJ ZJ 3 Note that this last integral has to be understand in the sense of Bochner. We define the space H1(J; X) as the space of functions with 2 2 2 f 1 := f + ∂ f . k kH (J;X) k kL2(J;X) k t kL2(J;X) For 0 <α< 1, we also use the fractional Sobolev-Bochner space Hα(J; X) of ds-strongly measurable functions f : J X with → 2 2 2 2 2 f(s) f(t) X f α := f + f α < , where f α := k − k ds dt. k kH (J;X) k kL2(J;X) | |H (J;X) ∞ | |H (J;X) s t 2α+1 ZJ ZJ | − | We will also use these Bochner spaces on R instead of J. For a recent introduction to Bochner spaces, we refer to [11]. 2.2 Fractional time derivative on Bochner spaces For 0 <β< 1, we define the left and right-sided Riemann-Liouville fractional integral operators 1 t 1 T D−βu(t) := (t s)β−1u(s) ds and D−βu(t) := (s t)β−1u(s) ds, 0 Γ(β) − T Γ(β) − Z0 Zt where Γ is Euler’s Gamma function. For sufficiently smooth functions u, the left-sided Caputo fractional derivative ∂α for α (0, 1) is defined as ∂αu := Dα−1Du.
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