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Equivalence Between a Time-Fractional and an Integer-Order Gradient flow: the Memory Effect Reflected in the Energy

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Equivalence Between a Time-Fractional and an Integer-Order Gradient flow: the Memory Effect Reflected in the Energy Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy Marvin Fritza, Ustim Khristenkoa, Barbara Wohlmutha aDepartment of Mathematics, Technical University of Munich, Germany Abstract Time-fractional partial differential equations are nonlocal in time and show an innate memory effect. In this work, we propose an augmented energy functional which includes the history of the solution. Further, we prove the equivalence of a time-fractional gradient flow problem to an integer-order one based on our new energy. This equivalence guarantees the dissipating character of the augmented energy. The state function of the integer- order gradient flow acts on an extended domain similar to the Caffarelli{Silvestre extension for the fractional Laplacian. Additionally, we apply a numerical scheme for solving time-fractional gradient flows, which is based on kernel compressing methods. We illustrate the behavior of the original and augmented energy in the case of the Ginzburg{Landau energy functional. Keywords: energy dissipation, time-fractional gradient flows, well-posedness, history energy, augmented energy, kernel compressing scheme, Cahn{Hilliard equation, Ginzburg{Landau energy 2020 MSC: 35A01, 35A02, 35B38, 35D30, 35K25, 35R11 1. Introduction In this work, we investigate the influence of the history on the energy functional of time-fractional gradient flows, i.e., the standard time derivative is replaced by a derivative of fractional order in the sense of Caputo. By definition, the system becomes nonlocal in time, and the history of the state function plays a significant role in its time evolution. Recently, time-fractional partial differential equations (FPDEs) became of increasing interest. Their innate memory effect appears in many applications, e.g., in the mechanical properties of materials [1], in viscoelasticity [2] and -plasticity [3], in image [4] and signal processing [5], in diffusion [6] and heat progression problems [7], in the modeling of solutes in fractured media [8], in combustion theory [9], bioengineering [10], damping processes [11], and even in the modeling of love [12] and happiness [13]. The theory of gradient flows is well-investigated in the integer case, e.g., see the book [14] and the work [15] regarding the analysis of the porous medium equation as a gradient flow. One of the most important properties of a gradient flow is its energy dissipation, which can be immediately derived from the variational formulation and by the chain rule. This relation is also called the principle of steepest descent. Typical applications are the heat equation with the underlying Dirichlet energy, and the Ginzburg{Landau energy, which results in the arXiv:2106.10985v1 [math.AP] 21 Jun 2021 well-known Cahn{Hilliard [16] and Allen{Cahn equation [17] depending on the choice of the underlying Hilbert space. We also mention the Fokker{Planck [18], and the Keller{Segel equations [19], which can be written and analyzed as gradient flows. Some of their time-fractional counterparts have been investigated in the literature, e.g., the time-fractional gradient flows of type Allen{Cahn [20], Cahn{Hilliard [21], Keller{Segel [22], and Fokker{Planck [23] type. Up to now there is no unified theory for time-fractional gradient flows, and it is not yet known whether the dissipation of energy is fulfilled, see also the discussions in [24{27]. From a straightforward testing of the variational form as in the integer order setting, one can only bound the energy by its initial state but one cannot say whether it is dissipating continuously in time. Several papers investigated the dissipation law of time-fractional phase field equations numerically and proposed weighted schemes in order to fulfill the dissipation of the discrete energy, see [28{35]. ∗Corresponding author Preprint submitted to Elsevier June 22, 2021 Our main contribution is the well-posedness of time-fractional gradient flows and the introduction of a new augmented energy, which is motivated by the memory structure of time-fractional differential equations and therefore, includes an additional term representing the history of the state function. We show that the integer-order gradient flow corresponding to this augmented energy on an extended Hilbert space is equivalent to the original time-fractional model. Consequently, the augmented energy is monotonically decreasing in time. We note that the state function of the augmented gradient flow acts on an extended domain similar to the Caffarelli–Silvestre approach [36] of the fractional Laplacian using harmonic extensions. This technique of dimension extension has also be used in the analysis of random walks [37] and embeds a long jump random walk to a space with one added dimension. In Section 2, we state some preliminary results on fractional derivatives and Bochner spaces. Moreover, we state and prove a theorem of well-posedness of fractional gradient flows. We state the main theorem of the equivalence of the fractional and the extended gradient flows in Section 3 and give a complete proof. Afterwards, we give two corollaries, one stating the consequence of energy dissipation and the other concerning the limit case α = 1 in the fractional order. In Section 4, we present an algorithm to solve the time-fractional system based on rational approximations. Lastly, we illustrate in Section 5 some simulations in order to show the influence of the history energy. 2. Analytical Preliminaries and Well-Posedness of Time-Fractional Gradient Flows In the following, let H be a separable Hilbert space and X a Banach space such that it holds X ,, H, X0; ! ! where the embedding , is continuous and dense, and ,, is additionally compact. We apply the Riesz repre- ! ! sentation theorem to identify H with its dual. In this regard, (X; H; X0) forms a Gelfand triple, e.g., k k j k H (Ω);H − (Ω);H− (Ω) with k j > 0: ≥ The duality pairing in X is regarded as a continuous extension of the scalar product of the Hilbert space H in the sense u; v X X = (u; v)H u H; v X; h i 0× 8 2 2 We call a function Bochner measurable if it can be approximated by a sequence of Banach-valued simple functions and consequently, we define the Bochner spaces Lp(0;T ; X) as the equivalence class of Bochner measurable functions u : (0;T ) X such that t u(t) p is Lebesgue integrable. Note that L2(0;T ; X) is ! 7! k kX a Hilbert space if X is a Hilbert space, e.g., see [38]. The Sobolev{Bochner space W 1;p(0;T ; X) consists of functions in Lp(0;T ; X) such that their distributional time derivatives are induced by functions in Lp(0;T ; X). 2.1. Fractional derivative Let us introduce the linear continuous Riemann{Liouville integral operator L (L1(0;T ; X)) of order Iα 2 α (0; 1) of a function u L1(0;T ; X), defined by 2 2 u := g u; (2.1) Iα α ∗ where the singular kernel g L1(0;T ) is given by g (t) = tα 1=Γ(α), and the operator denotes the convolu- α 2 α − ∗ tion on the positive half-line with respect to the time variable. Note that the operator has a complementary Iα element in the sense α 1 αu = 1u = 1 u; (2.2) I I − I ∗ see [39]. Then, the fractional derivative of order α (0; 1) in the sense of Caputo is defined by 2 α @t u := g1 α @tu = 1 α@tu; (2.3) − ∗ I − see, e.g., [39, 40]. In the limit cases α = 0 and α = 1, we define @0u = u u and @1u = @ u, respectively. One t − 0 t t can write (2.3) as 1 t @ u(s) @αu(t) = s ds; t Γ(1 α) (t s)α − Z0 − 2 in X for a.e. t (0;T ). The infinite-dimensional valued integral is understood in the Bochner sense. We 2 note that the Caputo derivative requires a function which is absolutely continuous. But this definition can be generalized to a larger class of functions which coincide with the classical definition in case of absolutely continuous functions, see [21, 41]. Similar to before, we define the fractional Sobolev{Bochner space W α,p(0;T ; X) as the functions in Lp(0;T ; X) such that their α-th fractional time derivative is in Lp(0;T ; X). Let us remark that by (2.2) it follows α ( α@t u)(t) = α 1 α@tu = 1@tu(t) = u(t) u0: (2.4) I I I − I − As in the integer-order setting, there are continuous and compact embedding results [41{44]. In particular, provided that X is compactly embedded in H, it holds α,p p 1 1 W 0 (0;T ; X0) L (0;T ; X) , C([0;T ]; H); + = 1; α > 0; \ ! p p0 (2.5) α,1 p r W (0;T ; X) L (0;T ; X0) ,, L (0;T ; H); 1 r < p; α > 0: \ ! ≤ Moreover, it holds the following version of the Gr¨onwall{Bellman inequality in the fractional setting. 1 Lemma 1 (cf. [21, Corollary 1]). Let w; v L (0;T ; R 0), and a; b 0. If w and v satisfy the inequality 2 ≥ ≥ w(t) + g v(t) a + b (g w)(t) a.e. t (0;T ); α ∗ ≤ α ∗ 2 then it holds w(t) + v(t) a C(α; b; T ) for almost every t (0;T ). ≤ · 2 d d We mention the following lemma which provides an alternative to the classical chain rule dt f(u) = f 0(u) dt u to the fractional setting for λ-convex (or semiconvex) functionals f : X R with respect to H, i.e., x ! 7! f(x) λ x 2 is convex for some λ R.
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