The Systems of Nonlinear Gradient Flows on Metric Spaces and Its Gamma-Convergence

Abstract

We first establish the explicit structure of nonlinear gradient flow systems on metric spaces and then develop Gamma-convergence of the systems of nonlinear gradient flows, which is a scheme meant to ensure that if a family of energy functionals of several variables depending on a parameter Gamma-converges, then the solutions to the associated systems of gradient flows converge as well. This scheme is a nonlinear system edition of the notion initiated by Sylvia Serfaty in 2011.

1. Introduction

The theory of Gamma-convergence was introduced by De Giorgi in the 1970s. It has become both a standard criterion for the study of variational problems and one of the extremely important topics in the calculus of variations. Gradient flows which are defined on metric spaces were also noticed by De Giorgi, that is, the notion replacing “gradient flows in a differentiable structure” then that of “curve of maximal slope in metric space.” This notion was introduced in [1] then further developed in [2–5]. It turns out to be useful in many applications, in particular for defining gradient flows over the probability measure spaces equipped with the Wasserstein metric.

In 2004, Gamma-convergence of gradient flows on Hilbert spaces was introduced by Sandier and Serfaty in [6]. This abstract method states that if a family of energy functionals {𝐽_𝜀}_𝜀>0  Γ-converges to a limiting functional 𝐽, then, under suitable conditions, solutions of the gradient flow of 𝐽_𝜀 converge to solutions of the gradient flow of 𝐽. This scheme was used successfully for the dynamics of Ginzburg-Landau vortices (cf. [6]), the Cahn-Hilliard equation (cf. [7, 8]), and the Allen-Cahn equation (cf. [9]).

The notion of Gamma-convergence of gradient flows on metric spaces was initiated by Serfaty [9] in 2011. She presented and proved the following.

Proposition 1.1 (cf. [9] Gamma-convergence of gradient flows in the metric spaces setting). Let (𝑋_𝜀,𝑑_𝜀) and (𝑋,𝑑) be complete metric spaces. Let Φ_𝜀 and Φ be functionals defined on metric spaces (𝑋_𝜀,𝑑_𝜀) and (𝑋,𝑑), respectively. Assume that there is a sense of convergence 𝒮 of 𝑢_𝜀∈𝑋_𝜀 to 𝑢∈𝑋 which can be general and with respect to the Γ-liminf convergence of Φ_𝜀 to Φ: 𝑢⇀_𝜀^𝒮𝑢⇒liminf_𝜀→0Φ_𝜀𝑢_𝜀≥Φ(𝑢). (1.1) Let 𝑔_𝜀 and 𝑔 be strong upper gradients of Φ_𝜀 and Φ, respectively. Assume in addition the following relations. (1)Lower bound on the metric derivatives: if 𝑢_𝜀(𝑡)⇀^𝒮𝑢(𝑡), for 𝑡∈[0,𝑇) then for all 𝑠∈[0,𝑇)liminf_𝜀→0^𝑠₀𝑢^′_𝜀^𝑝_𝑑𝜀(𝑡)𝑑𝑡≥^𝑠₀𝑢^′𝑝_𝑑(𝑡)𝑑𝑡. (1.2)(2) Lower bound on the slopes: if 𝑢⇀_𝜀^𝒮𝑢, then liminf_𝜀→0𝑔_𝜀𝑢_𝜀≥𝑔(𝑢). (1.3)Let 𝑢_𝜀(𝑡) be a 𝑝-curve of maximal slope on (0,𝑇) for Φ_𝜀 with respect to 𝑔_𝜀, such that 𝑢_𝜀(𝑡)⇀^𝒮𝑢(𝑡), which is well prepared in the sense that lim_𝜀→0Φ_𝜀𝑢_𝜀(0)=Φ(𝑢(0)). (1.4) Then 𝑢 is a 𝑝-curve of maximal slope with respect to 𝑔 and lim_𝜀→0Φ_𝜀𝑢_𝜀(𝑡)(𝑢(𝑡[),𝑔=Φ))∀𝑡∈0,𝑇_𝜀𝑢_𝜀→𝑔(𝑢)in𝐿^𝑝_loc(),𝑢0,𝑇_{′𝜀𝑑𝜀}→𝑢^′_𝑑in𝐿^𝑝_loc(0,𝑇). (1.5)

Obviously, this scheme (Proposition 1.1.) can be applied to single gradient flow problems only. To the best of our knowledge, nonlinear equations are more difficult than linear equations, the problem of system of differential equations is more complicated and more important than the problem of scalar equations, and the system of nonlinear gradient flows on metric spaces has not appeared elsewhere. This gives us a motivation for studying the systems of nonlinear gradient flows on metric spaces and then establishing its Gamma-convergence structure which can be applied to the problems of nonlinear gradient flow systems on metric spaces. This scheme can be regarded as a “nonlinear system” edition of the notion initiated by Sylvia Serfaty.

This paper is organized as follows. In Section 2, we introduce some necessary knowledge on gradient flows, basic definitions of absolutely continuous curve, metric derivative, strong upper gradient, and curve of maximal slope for functional. In Section 3, we establish the explicit structure of nonlinear gradient flow systems and Gamma-convergence of the systems of nonlinear gradient flows on metric spaces. Finally, we give two examples to illustrate a special case of our main results in Section 4.

2. Basic Definitions and Preliminaries

Let (𝑋,⟨⋅,⋅⟩) be a real Hilbert space with the corresponding norm ‖⋅‖ and let 𝐸∶𝑋→ℝ be a functional defined on 𝑋. We say that 𝐸 is Fréchet differentiable at 𝑥∈𝑋 if there exists 𝑥^∗∈𝑋^∗≡ℒ(𝑋;ℝ) (the space of all bounded linear functionals on 𝑋) such that 𝐸(𝑥+ℎ)−𝐸(𝑥)=𝑥^∗(ℎ)+𝑜(‖ℎ‖), (2.1) where lim_{‖ℎ‖→0}(𝑜(‖ℎ‖)/‖ℎ‖)=0.

Note that if such an 𝑥^∗ exists, then it is unique and we denote 𝐷𝐸(𝑥)≡𝑥^∗. In view of the Riesz representation theorem there exists a unique element 𝑦∈𝑋 such that 𝑦,ℎ=𝑥^∗(ℎ)∀ℎ∈𝑋. (2.2) Moreover ‖𝑥^∗‖_𝑋^∗=‖𝑦‖. 𝐷𝐸(𝑥) is called the differentiable of 𝐸 at 𝑥 (notice that it is a bounded linear functional on 𝑋). We denote ∇_𝑋𝐸(𝑥)≡𝑦 and we call ∇_𝑋𝐸(𝑥) the gradient of 𝐸 at 𝑥. Hence we have 𝐷𝐸(𝑥)(ℎ)=∇_𝑋𝐸(𝑥),ℎ∀ℎ∈𝑋. (2.3) We say that 𝐸 is of class 𝐶¹ on 𝑋 (i.e., 𝐸∈𝐶¹(𝑋;ℝ)) if the map 𝑥→𝐷𝐸(𝑥) is continuous on 𝑋. If 𝐸∈𝐶¹(𝑋;ℝ), then the directional derivative of 𝐸 at 𝑢(∈𝑋) in direction 𝜑 exists and is given by 𝑑𝑑𝑡_𝑡=0𝐸𝑢+𝑡𝜑=∇_𝑋𝐸(𝑢),𝜑. (2.4) Let 𝛾∶ℝ→𝑋 be a differentiable curve in 𝑋 with 𝛾(𝑡₀)=𝑥∈𝑋. Then 𝑑𝑑𝑡_𝑡=𝑡0𝐸𝛾(𝑡)=∇_𝑋𝐸𝛾𝑡₀,𝛾^′𝑡₀. (2.5) The evolution equation 𝑑𝑥𝑑𝑡=−∇_𝑋𝐸(𝑥) (2.6) is called the gradient flow of 𝐸 on Hilbert space (𝑋,‖⋅‖_𝑋). Using the definitions of Hilbert spaces and Gradient flows, it is easy to show the following basic and useful lemma.

Lemma 2.1. Suppose that for each 1≤𝑖≤𝑛, (𝑌_𝑖,⟨⋅,⋅⟩_𝑌𝑖) is an inner product space with induced norm ‖⋅‖_𝑌𝑖. Let one defines 𝑥,𝑦_𝑌≡^𝑛_𝑖=1𝑥_𝑖,𝑦_𝑖𝑌𝑖 (2.7) for each 𝑥=(𝑥₁,…,𝑥_𝑛), 𝑦=(𝑦₁,…,𝑦_𝑛) in 𝑌≡𝑌₁×⋯×𝑌_𝑛. Then one has the following. (i) We can see that (𝑌,⟨⋅,⋅⟩_𝑌) is an inner product space with induced norm ‖𝑥‖²_𝑌=^𝑛_𝑖=1𝑥_𝑖²_𝑌𝑖∀𝑥=𝑥₁,…,𝑥_𝑛∈𝑌. (2.8) Thus, (𝑌,‖⋅‖_𝑌) is a Hilbert space. (ii) Let 𝐹 be a 𝐶¹ functional defined on 𝑌≡𝑌₁×⋯×𝑌_𝑛 and let 𝑥=(𝑥₁,…,𝑥_𝑛)∈𝑌. An element 𝑦=(𝑦₁,…,𝑦_𝑛)∈𝑌 (and denote 𝑦=∇_𝑌𝐹(𝑥)) is called the gradient of 𝐹 at 𝑥 on the Hilbert space 𝑌 if for every differentiable curve 𝛾(𝑡)=(𝛾₁(𝑡),…,𝛾_𝑛(𝑡)) in 𝑌 satisfying 𝛾(𝑡₀)=𝑥, 𝑑𝐹𝛾(𝑡𝑑𝑡)_𝑡=𝑡0=∇_𝑌𝐹𝛾𝑡₀,𝛾^′𝑡_0𝑌=^𝑛_𝑖=1𝑦_𝑖,𝛾^′_𝑖𝑡_0𝑌𝑖. (2.9) Let one denotes 𝑦_𝑖=∇_𝑌𝑖𝐹𝛾𝑡₀=∇_𝑌𝑖𝐹(𝑥)∀1≤𝑖≤𝑛. (2.10) ∇_𝑌𝑖𝐹(𝑥) is called the gradient of 𝐹 at 𝑥 with respect to 𝑌_𝑖. Hence one has 𝑑𝐹𝛾(𝑡𝑑𝑡)_𝑡=𝑡0=^𝑛_𝑖=1∇_𝑌𝑖𝐹𝛾𝑡₀,𝛾^′_𝑖𝑡_0𝑌𝑖. (2.11) The evolution equation (the gradient flow of 𝐹 on (𝑌,‖⋅‖_𝑌)) 𝑑𝑥𝑑𝑡=−∇_𝑌𝐹(𝑥), (2.12) can be expressed as the following system of gradient flows on Hilbert spaces: 𝑑𝑥₁𝑑𝑡=−∇_𝑌1𝐹𝑥₁(𝑡),…,𝑥_𝑛(𝑡)∈𝑌₁,𝑑𝑥₂𝑑𝑡=−∇_𝑌2𝐹𝑥₁(𝑡),…,𝑥_𝑛(𝑡)∈𝑌₂,⋮𝑑𝑥_𝑛𝑑𝑡=−∇_𝑌𝑛𝐹𝑥₁(𝑡),…,𝑥_𝑛(𝑡)∈𝑌_𝑛. (2.13)

Definition 2.2 (𝑝-absolutely continuous curve). Let (𝑋,𝑑) be a complete metric space equipped with the distance 𝑑. A mapping 𝑣∶(𝑎,𝑏)→𝑋 is called a 𝑝-absolutely continuous curve or belongs to 𝐴𝐶^𝑝(𝑎,𝑏;𝑋), 𝑝≥1, if there exists an 𝐿^𝑝(𝑎,𝑏) function 𝑚 such that 𝑑(𝑣(𝑠),𝑣(𝑡))≤^𝑡_𝑠𝑚(𝑟)𝑑𝑟∀𝑎<𝑠≤𝑡<𝑏. (2.14)

Proposition 2.3 (cf. [5]). Let (𝑋,𝑑) be a metric space and let 𝑢∶[𝑎,𝑏]→𝑋. Then (i)𝑢∈𝐴𝐶([𝑎,𝑏];𝑋) if and only if there exists 𝑚∈𝐿¹(𝑎,𝑏), 𝑚≥0, such that 𝑑(𝑢(𝑠),𝑢(𝑡))≤^𝑡_𝑠𝑚(𝜏)𝑑𝜏∀𝑎≤𝑠<𝑡≤𝑏. (2.15)(ii) If 𝑢∈𝐴𝐶([𝑎,𝑏];𝑋), the metric derivative 𝑢^′_𝑑(𝑡)≡lim_ℎ→0𝑑(𝑢(𝑡+ℎ),𝑢(𝑡))|ℎ| (2.16) exists for a.e. 𝑡∈(𝑎,𝑏), |𝑢^′|_𝑑∈𝐿¹(𝑎,𝑏), 𝑑(𝑢(𝑠),𝑢(𝑡))≤^𝑡_𝑠𝑢^′_𝑑𝑑𝜏∀𝑎≤𝑠<𝑡≤𝑏, (2.17) and if 𝑚 satisfies (2.15), then |𝑢^′|_𝑑≤𝑚 a.e. on (𝑎,𝑏).

In the following, let us give a motivation for defining “gradient flow” on metric spaces. Here we completely follow the nice contents of Section 1.3 in [5]. Note that every solution 𝑢 of the gradient flow 𝑑𝑢𝑑𝑡=−∇_𝑋𝐸(𝑢(𝑡)) (2.18) can be characterized by the following the scalar equations: 𝑑(𝐸(𝑢(𝑡=()𝑑𝑡)))𝐸∘𝑢^′(𝑡)=∇_𝑋𝐸(𝑢(𝑡)),𝑢^′(𝑡∇)=−_𝑋𝐸(𝑢(𝑡)),∇_𝑋𝐸(𝑢(𝑡𝑢))=−^′(𝑡),𝑢^′(𝑡∇)=−_𝑋𝐸(𝑢(𝑡))𝑢^′(𝑡,𝑢_′)(𝑡)=∇_𝑋𝐸(𝑢(𝑡)). (2.19)(2.20) Using Young's inequality, (2.19) and (2.20) are equivalent to (𝐸∘𝑢)^′(𝑡)1=−2𝑢^′(𝑡)²−12∇_𝑋𝐸(𝑢(𝑡))². (2.21) We can impose (2.18), (2.19), (2.20), and (2.21) as a system of differential inequalities in the couple (𝑢,𝑔) by using the following strategies. (i) The function 𝑔 is an upper bound for the modulus of the gradient (𝐸∘𝑣)^′≤𝑔(𝑣)⋅𝑣^′ (2.22) for every regular curve 𝑣∶(0,+∞)↦𝑋. (ii) Impose that the functional 𝐸 decreasing along 𝑢 as much as possible compatibly with (2.22), that is, (𝐸∘𝑢)^′(𝑡)≤−𝑔(𝑢(𝑡))⋅𝑢^′(𝑡)in(0,+∞). (2.23)(iii) Prescribe the dependence of ‖𝑢^′‖ on 𝑔(𝑢), 𝑢^′=𝑔(𝑢)in(0,+∞), (2.24) or even in a single formula (𝐸∘𝑢)^′(𝑡)1≤−2𝑢^′(𝑡)²−12𝑔(𝑢(𝑡))²in(0,+∞). (2.25)

Whereas (2.18), (2.19), and (2.20) make sense only in a Hilbert space framework, the formulas (2.21)~(2.25) are of purely metric nature and can be extended to more general metric space, provided we understand ‖𝑢^′‖ as the metric derivative of 𝑢, |𝑢^′|_𝑑. Of course, the concept of upper gradient provides only an upper estimate for the modulus of ∇_𝑋𝐸 in the regular case, but it is enough to define steepest descent curves, that is, curves which realize the minimal selection of 𝑑/𝑑𝑡(𝐸∘𝑢)(𝑡) compatible with (𝐸∘𝑣)^′(𝑡)≤𝑔(𝑣(𝑡))𝑣^′_𝑑(𝑡) (2.26) for a.e. 𝑡∈(0,+∞).

Suppose that 𝑢∈𝐴𝐶(𝑎,𝑏;𝑋), and 𝑔∘𝑢 is Borel. Using (2.26), we have for any 𝑎<𝑠<𝑡<𝑏, |(𝐸∘𝑢)(𝑡)−(𝐸∘𝑢)(𝑠)|=^𝑡_𝑠(𝐸∘𝑢)^′(𝜏)≤𝑑𝜏_𝑡𝑠(𝐸∘𝑢)^′(𝜏≤)𝑑𝜏_𝑡𝑠𝑔(𝑢(𝜏))𝑢^′_𝑑(𝜏)𝑑𝜏. (2.27) Therefore we say that 𝑔 is a strong upper gradient for 𝐸 if for each 𝑢∈𝐴𝐶(𝑎,𝑏;𝑋), 𝑔∘𝑢 is Borel and (2.27) holds for all 𝑎<𝑠<𝑡<𝑏. Using the ideas (ii) and (iii) and Young's inequality, we say that a locally absolutely continuous function 𝑢∶(𝑎,𝑏)↦𝑋 is a curve of maximal slope for 𝐸 with respect to its strong upper gradient 𝑔 if 𝐸∘𝑢 is a.e. equal to a nonincreasing map 𝜑 and (𝐸∘𝑢)^′(𝑡)=𝜑^′(𝑡)𝑢≤−^′2𝑑(𝑡)2−𝑔(𝑢(𝑡))²2 (2.28) for a.e. 𝑡∈(𝑎,𝑏). Let us present the main definitions and three lemmas as follows.

Definition 2.4 (strong upper gradient). Suppose that (𝑋_𝑖,𝑑_𝑖) is a complete metric space equipped with the distance 𝑑_𝑖 for 𝑖=1,…,𝑛. Let 𝑔_𝑖∶𝑋₁×⋯×𝑋_𝑛→[0,+∞] for each 𝑖 with 1≤𝑖≤𝑛. We say that 𝑔=(𝑔₁,…,𝑔_𝑛) is a strong upper gradient for Φ∶𝑋₁×⋯×𝑋_𝑛→ℝ if for each (𝑢₁,…,𝑢_𝑛)∈𝐴𝐶(𝑎,𝑏;𝑋₁×⋯×𝑋_𝑛), 𝑔_𝑖∘(𝑢₁,…,𝑢_𝑛) is Borel for 𝑖=1,…,𝑛, and Φ𝑢₁,…,𝑢_𝑛(𝑡)−Φ𝑢₁,…,𝑢_𝑛(𝑠)≤^𝑡_𝑠^𝑛_𝑖=1𝑢^′_𝑖𝑑𝑖(𝜏)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏)𝑑𝜏∀𝑎<𝑠≤𝑡<𝑏. (2.29)

Definition 2.5 (a pair of Young's functions). Suppose that 𝐹^∗and 𝐺^∗∶[0,+∞)→[0,+∞) are two differentiable functions. We say that (𝐹^∗,𝐺^∗) is a pair of Young's functions if they satisfy(1)  Young^’s inequality: 𝑠𝑡≤𝐹^∗(𝑡)+𝐺^∗(𝑠)  for all  𝑠,𝑡≥0,(2)  Young^’s equality: 𝑠𝑡=𝐹^∗(𝑡)+𝐺^∗(𝑠)⇔𝑠=𝑓^∗(𝑡)  or  𝑡=𝑔^∗(𝑠),
where 𝑓^∗=(𝐹^∗)^′ and 𝑔^∗=(𝐺^∗)^′.

Definition 2.6 (curve of maximal slope). Let (𝐹^∗_𝑖,𝐺^∗_𝑖) be a pair of Young's functions for each 𝑖 with 1≤𝑖≤𝑛. We say that (𝑢₁,…,𝑢_𝑛)∶(𝑎,𝑏)→𝑋₁×⋯×𝑋_𝑛 is a (𝐹^∗₁,…,𝐹^∗_𝑛)-curve of maximal slope for the functional Φ with respect to the strong upper gradient 𝑔=(𝑔₁,…,𝑔_𝑛) if Φ(𝑢₁,…,𝑢_𝑛) is 𝐿¹-a.e. equal to a nonincreasing map 𝜑 and 𝜑^′(𝑡)≤−^𝑛_𝑖=1𝐹^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)+𝐺^∗_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡) (2.30) for 𝐿¹-a.e. 𝑡∈(𝑎,𝑏).

Definition 2.7 (Γ-liminf convergence). Let (𝑋_𝑖,𝜀,𝑑_𝑖,𝜀) and (𝑋_𝑖,𝑑_𝑖) be complete metric spaces for all 1≤𝑖≤𝑛 and 𝜀>0. Suppose that Φ_𝜀∶𝑋_1,𝜀×⋯×𝑋_𝑛,𝜀↦(−∞,+∞] and Φ∶𝑋₁×⋯×𝑋_𝑛↦(−∞,+∞] are functionals defined on 𝑋_1,𝜀×⋯×𝑋_𝑛,𝜀 and 𝑋₁×⋯×𝑋_𝑛, respectively. We say that Φ_𝜀Γ-liminf converges to Φ if 𝑢_1,𝜀,…,𝑢⇀_𝑛,𝜀^𝒮𝑢₁,…,𝑢_𝑛, (2.31) then liminf_𝜀→0Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀≥Φ𝑢₁,…,𝑢_𝑛, (2.32) where (𝑢_1,𝜀,…,𝑢_𝑛,𝜀)∈𝑋_1,𝜀×⋯×𝑋_𝑛,𝜀, (𝑢₁,…,𝑢_𝑛)∈𝑋₁×⋯×𝑋_𝑛, and the sense of convergence 𝒮 can be general.

Lemma 2.8. Suppose that 𝛼≥0 and 𝑎_𝑛≥0 for each 𝑛∈ℕ. If liminf_𝑛→∞𝑎_𝑛≥𝛼 and if 𝑓 is a continuous nondecreasing function on [0,+∞), then liminf_𝑛→∞𝑓𝑎_𝑛≥𝑓(𝛼). (2.33)

Proof. (i) For each 𝑘∈ℕ, 𝑎⁻_𝑘≡inf_𝑛≥𝑘𝑎_𝑛≤𝑎_𝑛, ∀𝑛≥𝑘.
(ii) Since 𝑓 is non-decreasing on [0,∞), and using (i), we have 𝑓𝑎⁻_𝑘≤𝑓𝑎_𝑛,∀𝑛≥𝑘. (2.34) Therefore, 𝑓𝑎⁻_𝑘isalowerboundfor𝑓𝑎_𝑛∣𝑛≥𝑘, (2.35) and so 𝑓𝑎⁻_𝑘≤inf_𝑛≥𝑘𝑓𝑎_𝑛,foreach𝑘∈ℕ. (2.36) (iii) Since 𝑓 is continuous on [0,∞), and using (ii), we have 𝑓(𝛼)≤𝑓liminf_𝑛→∞𝑎_𝑛=𝑓lim_𝑘→∞inf_𝑛≥𝑘𝑎_𝑛=𝑓lim_𝑘→∞𝑎⁻_𝑘=lim_𝑘→∞𝑓𝑎⁻_𝑘≤lim_𝑘→∞inf_𝑛≥𝑘𝑓𝑎_𝑛=liminf_𝑘→∞𝑓𝑎_𝑘. (2.37)

Lemma 2.9 (see [10, Theorem   5.11]). Let 𝑓 be nonnegative and measurable on 𝐸. Then ∫_𝐸𝑓𝑑𝑥=0 if and only if 𝑓=0 a.e. in 𝐸.

Lemma 2.10. Let (𝑋_𝑖,𝑑_𝑖) be a metric space for each 1≤𝑖≤𝑛. Let 𝑋≡𝑋₁×⋯×𝑋_𝑛={(𝑥₁,⋯,𝑥_𝑛)∣𝑥_𝑖∈𝑋_𝑖for1≤𝑖≤𝑛}. Define the function 𝑑∶𝑋×𝑋↦ℝ by 𝑑𝑥,𝑦≡^𝑛_𝑖=1𝑑_𝑖𝑥_𝑖,𝑦_𝑖 (2.38) for each 𝑥=(𝑥₁,…,𝑥_𝑛),𝑦=(𝑦₁,…,𝑦_𝑛)∈𝑋. Then(i)(𝑋,𝑑) is a metric space, (ii) suppose that 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝐴𝐶(𝑎,𝑏;𝑋). The metric derivative |𝑣^′|_𝑑 can be expressed as 𝑣^′_𝑑(𝑡)=^𝑛_𝑖=1𝑣^′_𝑖𝑑𝑖(𝑡) (2.39) for a.e. 𝑡∈(𝑎,𝑏).

3. Main Results

In the following theorem we introduce the systems of explicit nonlinear gradient flows of energy functional with respect to the strong upper gradient on metric spaces and investigate an upper control for some form of velocity of solutions by its dissipation rate of the energy functional. Using this idea, we can see that if motion is driven by energy dissipation and if there are solutions that move without losing much energy, then they must move very slowly for each component solution.

Theorem 3.1. Let (𝑋_𝑖,𝑑_𝑖) be a complete metric spaces equipped with distance 𝑑_𝑖 for 𝑖=1,2,…,𝑛. Let Φ be a functional defined on 𝑋₁×𝑋₂×⋯×𝑋_𝑛 and let 𝑔=(𝑔₁,…,𝑔_𝑛) be a strong upper gradient for Φ. Assume that 𝑓_𝑖∶[0,+∞)↦[0,+∞) is a continuous, strictly increasing and surjective function for each 1≤𝑖≤𝑛. Let 𝐹_𝑖 and 𝐺_𝑖 be defined by 𝐹_𝑖(𝑡)=^𝑡₀𝑓_𝑖(𝜏)𝑑𝜏,𝐺_𝑖(𝑡)=^𝑡₀𝑓_𝑖⁻¹(𝜏)𝑑𝜏 (3.1) for each 𝑡≥0 and 1≤𝑖≤𝑛. Suppose that (𝑢₁,,…,𝑢_𝑛)∈𝐴𝐶(𝑎,𝑏;𝑋₁×⋯×𝑋_𝑛) and that is a (𝐹₁,…,𝐹_𝑛)-curve of maximal slope for the functional Φ with respect to the strong upper gradient 𝑔 on (𝑎,𝑏)⊂ℝ. Then one has the following. (i)Φ𝑢₁,…,𝑢_𝑛(𝑠)−Φ𝑢₁,…,𝑢_𝑛(𝑡)=^𝑡_𝑠^𝑛_𝑖=1𝐹_𝑖𝑢^′_𝑖𝑑𝑖(𝜏)+𝐺_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏=)𝑑𝜏_{𝑡𝑠𝑛𝑖=1}𝑢^′_𝑖𝑑𝑖(𝜏)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏)𝑑𝜏 (3.2) for a.e. 𝑎<𝑠≤𝑡<𝑏. (ii)(𝑢₁,…,𝑢_𝑛) satisfies the system of explicit nonlinear gradient flows of Φ with respect to the structure ((𝑓₁,…,𝑓_𝑛),(𝑔₁,…,𝑔_𝑛),𝑋₁×⋯×𝑋_𝑛)𝑓₁𝑢^′_1𝑑1(𝑡)=𝑔₁𝑢₁(𝑡),…,𝑢_𝑛(𝑡),𝑓₂𝑢^′_2𝑑2(𝑡)=𝑔₂𝑢₁(𝑡),…,𝑢_𝑛(𝑡),⋮𝑓_𝑛𝑢^′_𝑛𝑑𝑛(𝑡)=𝑔_𝑛𝑢₁(𝑡),…,𝑢_𝑛(𝑡), (3.3) for a.e. 𝑡∈(𝑎,𝑏). (iii) Assume that function 𝜓_𝑖(𝑡)≡𝑓_𝑖(𝑡)⋅𝑡 is convex and strictly increasing on [0,+∞) for each 1≤𝑖≤𝑛, then one has (a)^𝑛_𝑖=1𝜓_𝑖𝑑_𝑖𝑢_𝑖(𝑡),𝑢_𝑖(𝑠)≤Φ𝑢𝑡−𝑠¹,…,𝑢_𝑛(𝑠)−Φ𝑢₁,…,𝑢_𝑛(𝑡),𝑡−𝑠 (3.4) for all 𝑎<𝑠<𝑡<𝑏, (b)^𝑛_𝑖=1𝜓_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)≤Φ∘𝑢₁,…,𝑢_𝑛^′(𝑡), (3.5) for a.e. 𝑡∈(𝑎,𝑏). (iv) If 𝑓_𝑖(𝑡)=𝑡 for 1≤𝑖≤𝑛, then (ii) can be expressed as 𝑢^′_1𝑑1=𝑔₁𝑢₁,…,𝑢_𝑛,𝑢^′_2𝑑2=𝑔₂𝑢₁,…,𝑢_𝑛,⋮𝑢^′_𝑛𝑑𝑛=𝑔_𝑛𝑢₁,…,𝑢_𝑛, (3.6) which is the system of explicit linear gradient flows of Φ with respect to the structure ((𝑓₁,…,𝑓_𝑛),(𝑔₁,…,𝑔_𝑛),𝑋₁×⋯×𝑋_𝑛).

Proof. According to the definitions of 𝐹_𝑖 and 𝐺_𝑖, (𝐹_𝑖,𝐺_𝑖) is a pair of Young's functions. Recall we assume that (𝑢₁,…,𝑢_𝑛) is a (𝐹₁,…,𝐹_𝑛)-curve of maximal slope for Φ with respect to 𝑔 on (𝑎,𝑏). For a.e. 𝑎<𝑠<𝑡<𝑏, we have −^𝑡_𝑠Φ∘𝑢₁,…,𝑢_𝑛^′(𝜏)𝑑𝜏≥^𝑡_𝑠^𝑛_𝑖=1𝐹_𝑖𝑢^′_𝑖𝑑𝑖(𝜏)+𝐺_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏≥)𝑑𝜏_{𝑡𝑠𝑛𝑖=1}𝑢^′_𝑖𝑑𝑖(𝜏)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏)≥𝑢𝑑𝜏Φ∘₁,…,𝑢_𝑛(𝑡)−Φ∘𝑢₁,…,𝑢_𝑛(𝑠). (3.7) The last inequality holds due to the assumption of strong upper gradient for 𝑔. Thus we easily obtain the following formula: ^𝑡_𝑠^𝑛_𝑖=1𝐹_𝑖𝑢^′_𝑖𝑑𝑖(𝜏)+𝐺_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏=)𝑑𝜏_{𝑡𝑠𝑛𝑖=1}𝑢^′_𝑖𝑑𝑖(𝜏)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏)𝑢𝑑𝜏=Φ∘₁,…,𝑢_𝑛(𝑠)−Φ∘𝑢₁,…,𝑢_𝑛(𝑡), (3.8) for 𝐿¹ a.e. 𝑎<𝑠<𝑡<𝑏. Using the Young's inequality and the Vanishing theorem (Lemma 2.9), we obtain 𝐹_𝑖𝑢^′_𝑖𝑑𝑖(𝜏)+𝐺_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏)=𝑢^′_𝑖𝑑𝑖(𝜏)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏) (3.9) for a.e. 𝑎<𝜏<𝑏 and for 1≤𝑖≤𝑛. By Young's equality, we discover that the system of explicit gradient flows of Φ with respect to the structure ((𝑓₁,…,𝑓_𝑛),𝑔,𝑋₁×⋯×𝑋_𝑛) holds.
We now prove assertion (iii). By using assertions (i), (ii), and the assumption for 𝜓_𝑖, we see that ^𝑛_𝑖=1^𝑡_𝑠𝜓_𝑖𝑢^′_𝑖𝑑𝑖(𝜏)𝑑𝜏=Φ∘𝑢₁,…,𝑢_𝑛(𝑠)−Φ∘𝑢₁,…,𝑢_𝑛(𝑡), (3.10) for a.e. 𝑎<𝑠<𝑡<𝑏. Owing to the metric derivative |𝑢^′_𝑖|_𝑑𝑖 which is the smallest admissible function 𝑚_𝑖 satisfying 𝑑_𝑖𝑢_𝑖(𝑠),𝑢_𝑖(𝑡)≤^𝑡_𝑠𝑚_𝑖(𝜏)𝑑𝜏∀𝑎<𝑠≤𝑡<𝑏 (3.11) and 𝜓_𝑖 is convex and strictly increasing on [0,+∞), by using Jensen's inequality, we find ^𝑛_𝑖=1𝜓_𝑖𝑑_𝑖𝑢_𝑖(𝑡),𝑢_𝑖(𝑠)≤𝑡−𝑠^𝑛_𝑖=1𝜓_𝑖1𝑡−𝑠^𝑡_𝑠𝑢^′_𝑖𝑑𝑖(𝜏)≤1𝑑𝜏𝑡−𝑠^𝑛_𝑖=1^𝑡_𝑠𝜓_𝑖𝑢^′_𝑖𝑑𝑖(𝜏)=𝑢𝑑𝜏Φ∘₁,…,𝑢_𝑛(𝑠)−Φ∘𝑢₁,…,𝑢_𝑛(𝑡).𝑡−𝑠 (3.12) This completes the proof of (a). By passing to the limit 𝑠→𝑡 in (a), we deduce that (b) holds.

In our second main result, we study and establish the abstract structure of “Gamma-convergence of gradient flow systems on metric spaces” which is a nonlinear system edition of the notion (Proposition 1.1) and which can be applied to the problems involving a system of nonlinear gradient flows on metric spaces.

Theorem 3.2 (Gamma-convergence of systems of gradient flows on metric spaces). Let (𝑋_𝑖,𝜀,𝑑_𝑖,𝜀) and (𝑋_𝑖,𝑑_𝑖) be complete metric spaces for all 1≤𝑖≤𝑛 and 𝜀>0. Let Φ_𝜀 and Φ be functionals defined on spaces 𝑋_1,𝜀×⋯×𝑋_𝑛,𝜀 and 𝑋₁×⋯×𝑋_𝑛, respectively. Suppose that the Γ-liminf convergence of Φ_𝜀 to Φ holds. Let 𝑔_𝜀=(𝑔_1,𝜀,…,𝑔_𝑛,𝜀) and 𝑔=(𝑔₁,…,𝑔_𝑛) be strong upper gradients of Φ_𝜀 and Φ, respectively. Let (𝐹^∗_𝑖,𝐺^∗_𝑖) be a pair of Young's functions having continuous, strictly increasing and surjective derivative (𝐹^∗_𝑖)^′(=𝑓^∗_𝑖) for 𝑖=1,…,𝑛. Assume in addition the following relations. (1)Lower bound on the metric derivatives: if (𝑢_1,𝜀,…,𝑢_𝑛,𝜀)(𝑡)⇀^𝒮(𝑢₁,…,𝑢_𝑛)(𝑡), for 𝑡∈[0,𝑇) then for all  𝑠∈[0,𝑇)liminf_𝜀→0𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑠)≥𝑢^′_𝑖𝑑𝑖(𝑠),for𝑖=1,…,𝑛. (3.13)(2)Lower bound on the strong upper gradients: If (𝑢_1,𝜀,…,𝑢_𝑛,𝜀)⇀^𝒮(𝑢₁,…,𝑢_𝑛), then liminf_𝜀→0𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀≥𝑔_𝑖𝑢₁,…,𝑢_𝑛for𝑖=1,…,𝑛. (3.14) Let (𝑢_1,𝜀,…,𝑢_𝑛,𝜀) be a (𝐹^∗₁,…,𝐹^∗_𝑛)-curve of maximal slope on (0,𝑇) for Φ_𝜀 with respect to 𝑔_𝜀=(𝑔_1,𝜀,…,𝑔_𝑛,𝜀) such that (𝑢_1,𝜀,…,𝑢_𝑛,𝜀)(𝑡)⇀^𝒮(𝑢₁,…,𝑢_𝑛)(𝑡), which is well-prepared in the sense that lim_𝜀→0Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(0)=Φ𝑢₁,…,𝑢_𝑛(0). (3.15) Then (i)(𝑢₁,…,𝑢_𝑛) is a (𝐹^∗₁,…,𝐹^∗_𝑛)-curve of maximal slope on (0,𝑇) for Φ with respect to 𝑔=(𝑔₁,…,𝑔_𝑛), (ii)lim_𝜀→0Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)=Φ𝑢₁,…,𝑢_𝑛(𝑡)∀𝑡∈[0,𝑇), (3.16)(iii)lim_𝜀→0^𝑠₀^𝑛_𝑖=1𝐹^∗_𝑖𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑡)+𝐺^∗_𝑖𝑓^∗_𝑖𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑡=)𝑑𝑡_{𝑠0𝑛𝑖=1}𝐹^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)+𝐺^∗_𝑖𝑓^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)𝑑𝑡, (3.17)(iv)lim_𝜀→0^𝑠₀^𝑛_𝑖=1𝐹^∗_𝑖𝑔^∗_𝑖𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)+𝐺^∗_𝑖𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡=)𝑑𝑡_{𝑠0𝑛𝑖=1}𝐹^∗_𝑖𝑔^∗_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)+𝐺^∗_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)𝑑𝑡, (3.18) where 𝑔^∗_𝑖=(𝑓^∗_𝑖)⁻¹.

Proof. Owing to the fact that (𝑢_1,𝜀,…,𝑢_𝑛,𝜀) is a (𝐹^∗₁,…,𝐹^∗_𝑛)-curve of maximal slope for Φ_𝜀 with respect to 𝑔_𝜀 on [0,𝑇), we recall that Theorem 3.1-(i) and (ii) yields Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(0)−Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑠)=^𝑠₀^𝑛_𝑖=1𝐹^∗_𝑖𝑢^′_{𝑖,𝜀𝑑i,𝜀}(𝑡)+𝐺^∗_𝑖𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)=𝑑𝑡_{𝑠0𝑛𝑖=1}𝑢^′_{𝑖,𝜀𝑑i,𝜀}(𝑡)𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)𝑑𝑡, (3.19) for a.e. 0<𝑠<𝑇; 𝑓^∗_𝑖𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑡)=𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡), (3.20) for a.e. 𝑡∈[0,𝑇) and for 1≤𝑖≤𝑛.
Passing to the liminf_𝜀→0 to (3.19) and applying Fatou's lemma, we deduce that liminf_𝜀→0Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(0)−Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑠≥_𝑛𝑖=1)liminf_𝜀→0^𝑠₀𝐹^∗_𝑖𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑡)𝑑𝑡+liminf_𝜀→0^𝑠₀𝐺^∗_𝑖𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)≥𝑑𝑡_{𝑛𝑖=1𝑠0}liminf_𝜀→0𝐹^∗_𝑖𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑡)𝑑𝑡+^𝑠₀liminf_𝜀→0𝐺^∗_𝑖𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡≥)𝑑𝑡_{𝑛𝑖=1𝑠0}𝐹^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)𝑑𝑡+^𝑠₀𝐺^∗_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)𝑑𝑡. (3.21) The last inequality is achieved by the assumptions of (1) and (2) as well as Lemma 2.8.
Using the fact that each (𝐹^∗_𝑖,𝐺^∗_𝑖) is a pair of Young's functions (hence Young's inequality holds), (3.21), and the strong upper gradient assumption of 𝑔 for Φ, we can check that liminf_𝜀→0Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(0)−Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑠≥_{𝑛𝑖=1𝑠0})𝐹^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)𝑑𝑡+^𝑠₀𝐺^∗_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)≥𝑑𝑡_{𝑠0𝑛𝑖=1}𝑢^′_𝑖𝑑𝑖(𝑡)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)𝑢𝑑𝑡≥Φ₁,…,𝑢_𝑛(0)−Φ𝑢₁,…,𝑢_𝑛(𝑠). (3.22) We recall that (𝑢_1,𝜀,…,𝑢_𝑛,𝜀) is well prepared, and using (3.22), we can deduce that limsup_𝜀→0Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑠)≤Φ𝑢₁,…,𝑢_𝑛(𝑠). (3.23)
Using the Γ-liminf convergence of Φ_𝜀 to Φ and (3.23), we obtain lim_𝜀→0Φ_𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑠)=Φ𝑢₁,…,𝑢_𝑛(𝑠),∀𝑠∈[0,𝑇). (3.24)
Combining (3.24) with (3.22), we can now conclude that, for each 𝑠∈[0,𝑇), ^𝑠₀^𝑛_𝑖=1𝐹^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)+𝐺^∗_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡=)𝑑𝑡_{𝑠0𝑛𝑖=1}𝑢^′_𝑖𝑑𝑖(𝑡)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)𝑢𝑑𝑡=Φ₁,…,𝑢_𝑛(0)−Φ𝑢₁,…,𝑢_𝑛(𝑠). (3.25) Using the Young's inequality and the Vanishing theorem (Lemma 2.9) again, we conclude that, for a.e. 𝑡∈[0,𝑇), 𝐹^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)+𝐺^∗_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)=𝑢^′_𝑖𝑑𝑖(𝑡)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡), (3.26) for all 1≤𝑖≤𝑛. Moreover, by Young's equality, we have, for a.e. 𝑡∈[0,𝑇), 𝑓^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)=𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡), (3.27) for each 1≤𝑖≤𝑛.
Next, differentiating formulas (3.25) with respect to variable 𝑠, we see that (𝑢₁,…,𝑢_𝑛) is a (𝐹^∗₁,…,𝐹^∗_𝑛)-curve of maximal slope for Φ with respect to 𝑔 on [0,𝑇). Using formula (3.19), (3.24), and (3.25) we can check that lim_𝜀→0^𝑠₀^𝑛_𝑖=1𝐹^∗_𝑖𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑡)+𝐺^∗_𝑖𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡=)𝑑𝑡_{𝑠0𝑛𝑖=1}𝐹^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)+𝐺^∗_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)𝑑𝑡, (3.28) for all 𝑠∈[0,𝑇). Finally, we recall formulas (3.20) and (3.27), and using (3.28), we obtain lim_𝜀→0^𝑠₀^𝑛_𝑖=1𝐹^∗_𝑖𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑡)+𝐺^∗_𝑖𝑓^∗_𝑖𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑡=)𝑑𝑡_{𝑠0𝑛𝑖=1}𝐹^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)+𝐺^∗_𝑖𝑓^∗_𝑖𝑢^′_𝑖𝑑𝑖(𝑡)𝑑𝑡,lim_𝜀→0^𝑠₀^𝑛_𝑖=1𝐹^∗_𝑖𝑔^∗_𝑖𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)+𝐺^∗_𝑖𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)=𝑑𝑡_{𝑠0𝑛𝑖=1}𝐹^∗_𝑖𝑔^∗_𝑖𝑔_𝑖u₁,…,𝑢_𝑛(𝑡)+𝐺^∗_𝑖𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)𝑑𝑡 (3.29) and complete the proof of Theorem 3.2.

4. Examples

In this section, we present two examples to illustrate a special case of our main results.

Example 4.1. Let 𝑝>1. If 𝑓^∗_𝑖(𝑡)=𝑡^𝑝−1 for each 1≤𝑖≤𝑛, then (𝑓^∗_𝑖)⁻¹(𝑡)=𝑡^𝑞−1 for 𝑖=1,2,…,𝑛, where (1/𝑝)+(1/𝑞)=1. Hence 𝐹^∗_𝑖(𝑡)=𝑡^𝑝𝑝,𝐺^∗_𝑖(𝑡)=𝑡^𝑞𝑞foreach1≤𝑖≤𝑛. (4.1) In this case, Theorem 3.1 can be expressed as following. (i) One has Φ𝑢₁,…,𝑢_𝑛(𝑠)−Φ𝑢₁,…,𝑢_𝑛(𝑡)=^𝑡_𝑠^𝑛_𝑖=1𝑢^′_𝑖^𝑝_𝑑𝑖(𝜏)𝑝+𝑔^𝑞_𝑖𝑢₁,…,𝑢_𝑛(𝜏)𝑞=𝑑𝜏_{𝑡𝑠𝑛𝑖=1}𝑢^′_𝑖𝑑𝑖(𝜏)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏)𝑑𝜏. (4.2)(ii) The system of explicit nonlinear gradient flows of Φ with respect to the structure ((𝑓^∗₁,…,𝑓^∗_𝑛),(𝑔₁,…,𝑔_𝑛),𝑋₁×⋯×𝑋_𝑛) is 𝑢^′_𝑖𝑑^𝑝−1_𝑖(𝑡)=𝑔_𝑖𝑢₁(𝑡),…,𝑢_𝑛(𝑡),for𝑖=1,2,…,𝑛. (4.3)(iii)𝜓_𝑖(𝑡)=𝑡^𝑝 for 𝑖=1,…,𝑛 (𝜓_𝑖 is convex and strictly increasing on [0,+∞) for each 1≤𝑖≤𝑛). Then(a)^𝑛_𝑖=1𝑑_𝑖𝑢_𝑖(𝑡),𝑢_𝑖(𝑠)𝑡−𝑠^𝑝≤Φ𝑢₁,…,𝑢_𝑛(𝑠)−Φ𝑢₁,…,𝑢_𝑛(𝑡)𝑡−𝑠 (4.4) for all 𝑎<𝑠<𝑡<𝑏,(b)^𝑛_𝑖=1𝑢^′_𝑖𝑑𝑖(𝑡)^𝑝≤Φ∘𝑢₁,…,𝑢_𝑛^′(𝑡)fora.e.𝑡∈(𝑎,𝑏). (4.5) Under the hypotheses of Theorem 3.2, we have (i)lim_𝜀→0^𝑠₀^𝑛_𝑖=1𝑢^′_{𝑖,𝜀𝑑𝑖,𝜀}(𝑡)^𝑝𝑝+𝑢^′_{𝑖,𝜀𝑑}^𝑝−1_𝑖,𝜀(𝑡)^𝑞𝑞=𝑑𝑡_{𝑠0𝑛𝑖=1}𝑢^′_𝑖𝑑𝑖(𝑡)^𝑝𝑝+𝑢^′_𝑖𝑑^𝑝−1_𝑖(𝑡)^𝑞𝑞𝑑𝑡, (4.6) that is, lim_𝜀→0∫^𝑠₀∑^𝑛_𝑖=1|𝑢^′_𝑖,𝜀|^𝑝_{𝑑𝑖,𝜀}(𝑡)𝑑𝑡=∫^𝑠₀∑^𝑛_𝑖=1|𝑢^′_𝑖|^𝑝_𝑑𝑖(𝑡)𝑑𝑡.(ii)lim_𝜀→0^𝑠₀^𝑛_𝑖=1𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)^𝑞−1𝑝𝑝+𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)^𝑞𝑞=𝑑𝑡_{𝑠0𝑛𝑖=1}𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)^𝑞−1𝑝𝑝+𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)^𝑞𝑞𝑑𝑡, (4.7) that is, lim_𝜀→0∫^𝑠₀∑^𝑛_𝑖=1(𝑔_𝑖,𝜀(𝑢_1,𝜀,…,𝑢_𝑛,𝜀)(𝑡))^𝑞𝑑𝑡=∫^𝑠₀∑^𝑛_𝑖=1(𝑔_𝑖(𝑢₁,…,𝑢_𝑛)(𝑡))^𝑞𝑑𝑡.

Example 4.2. Considering the case 𝑝=2 in Example 4.1, we have(i)Φ𝑢₁,…,𝑢_𝑛(𝑠)−Φ𝑢₁,…,𝑢_𝑛(𝑡)=^𝑡_𝑠^𝑛_𝑖=1𝑢^′_𝑖²_𝑑𝑖(𝜏)2+𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏)²2=𝑑𝜏_{𝑡𝑠𝑛𝑖=1}𝑢^′_𝑖𝑑𝑖(𝜏)𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝜏)𝑑𝜏. (4.8)(ii)The system of explicit linear gradient flows of Φ with respect to the structure is 𝑢^′_1𝑑1(𝑡)=𝑔₁𝑢₁(𝑡),…,𝑢_𝑛(𝑡)𝑢^′_2𝑑2(𝑡)=𝑔₂𝑢₁(𝑡),…,𝑢_𝑛(𝑡)⋮𝑢^′_𝑛𝑑𝑛(𝑡)=𝑔_𝑛𝑢₁(𝑡),…,𝑢_𝑛(𝑡). (4.9)(iii)^𝑛_𝑖=1𝑑_𝑖𝑢_𝑖(𝑡),𝑢_𝑖(𝑠)𝑡−𝑠²≤Φ𝑢₁,…,𝑢_𝑛(𝑠)−Φ𝑢₁,…,𝑢_𝑛(𝑡)𝑡−𝑠 (4.10) for all 𝑎<𝑠<𝑡<𝑏. Moreover, we have ^𝑛_𝑖=1𝑢^′_𝑖²_𝑑𝑖(𝑡)≤Φ∘𝑢₁,…,𝑢_𝑛^′(𝑡)fora.e.𝑡∈(𝑎,𝑏). (4.11)(iv)lim_𝜀→0^𝑠₀^𝑛_𝑖=1𝑢^′_𝑖,𝜀²_{𝑑𝑖,𝜀}(𝑡)𝑑𝑡=^𝑠₀^𝑛_𝑖=1𝑢^′_𝑖²_𝑑𝑖(𝑡)𝑑𝑡. (4.12)(v)lim_𝜀→0^𝑠₀^𝑛_𝑖=1𝑔_𝑖,𝜀𝑢_1,𝜀,…,𝑢_𝑛,𝜀(𝑡)²𝑑𝑡=^𝑠₀^𝑛_𝑖=1𝑔_𝑖𝑢₁,…,𝑢_𝑛(𝑡)²𝑑𝑡. (4.13)

Acknowledgment

Research by authors was supported by National Science Council of Taiwan under Grant no. NSC 100-2115-M-030-001.