Existence of quasi-static crack evolution for atomistic systems

We describe the atomistic deformation of a material occupying an open, bounded set $\Omega\subset\mathbb{R}^{d}$ in dimension $d\in\{1,2,3\}$. By $\mathcal{L}\subset\mathbb{R}^{d}$ we denote an arbitrary but fixed lattice, and let $\mathcal{L}(\Omega)=\Omega\cap\mathcal{L}$ be the positions of the atoms (or particles) in the specimen in the reference configuration. The deformation of the particles is described through a map $y\colon\mathcal{L}(\Omega)\to\mathbb{R}^{n}$, where for each $x\in\mathcal{L}(\Omega)$ the vector $y(x)\in\mathbb{R}^{n}$ indicates the position of $x$ in the deformed configuration. (The typical choice is $n=d$.)

We model the interaction of the atoms by classical potentials in the frame of Molecular Mechanics [2, 30]: given a deformation $y\colon\mathcal{L}(\Omega)\to\mathbb{R}^{n}$, we assign a phenomenological energy $\mathcal{E}(y)$ to the deformation by

$\displaystyle\mathcal{E}(y)=\frac{1}{2}\sum_{\underset{x\neq x^{\prime}}{x,x^{\prime}\in\mathcal{L}(\Omega)}}W_{x,x^{\prime}}\big{(}|y(x)-y(x^{\prime})|\big{)},$

(2.1)

where $W_{x,x^{\prime}}\colon[0,\infty)\to\mathbb{R}$ denotes a smooth two-body interaction potential, with $W_{x,x^{\prime}}=W_{x^{\prime},x}$. The factor $\frac{1}{2}$ accounts for the fact that every contribution is counted twice in the sum. It is also possible that the sum ranges only over a subset of all pairs by simply setting $W_{x,x^{\prime}}\equiv 0$ for certain interactions. Whenever $W_{x,x^{\prime}}\neq 0$, the potential is supposed to be repulsive for small distances and attractive for long distances, a classical choice being a potential of Lennard-Jones-type. More specifically, we assume that $W_{x,x^{\prime}}$ satisfies

1. (i)

   $\lim_{r\searrow 0}W_{x,x^{\prime}}(r)=\infty$;

2. (ii)

   $W_{x,x^{\prime}}(r)$ is minimal if and only if $r=|x-x^{\prime}|$ with $W_{x,x^{\prime}}(|x-x^{\prime}|)<0$;

3. (iii)

   $W_{x,x^{\prime}}(r)$ is decreasing for $r<|x-x^{\prime}|$ and increasing for $r>|x-x^{\prime}|$;

4. (iv)

   $\lim_{r\to+\infty}W_{x,x^{\prime}}(r)=\lim_{r\to+\infty}\frac{{\rm d}}{{\rm d}r}W_{x,x^{\prime}}(r)=\lim_{r\to+\infty}\frac{{\rm d}^{2}}{{\rm d}^{2}r}W_{x,x^{\prime}}(r)=0$.

The repulsion effect (i) describes the Pauli repulsion at short distances of the interacting particles due to overlapping electron orbitals, and (ii)-(iv) correspond to attraction at long ranged interactions vanishing at infinity. Some toy examples of lattices and possible configurational energies are given below in Section 3. Let us mention that we restrict our model to pair interactions equilibrated in the reference configuration (see Assumption (ii)) for mere simplicity. The model could be generalized to pre-stressed interactions, and multi-body, non-local, and multiscale energies could be considered, see e.g. [12] for a very general framework. We include, however, non-homogeneous interactions as all potentials $W_{x,x^{\prime}}$ can be different. A possible extension featuring three-body potentials is addressed in Subsection 2.4 below.

By $\varepsilon:=\min\{|x-x^{\prime}|\colon x,x^{\prime}\in\mathcal{L}(\Omega),\,x\neq x^{\prime}\}$ we denote the minimal interatomic distance. A huge body of literature deals with effective descriptions of atomistic energies of the form (2.1) when the interatomic distance $\varepsilon$ tends to $0$. This passage from atomistic to continuum models is nontrivial even for simple energies and might lead to very different asymptotic behavior, see [5] for taking pointwise limits and [9] for an overview of a variational perspective. For interaction densities of the above form, it has been shown in [14, 24] that effective continuum energies lead to so-called Griffith energy functionals featuring both bulk and surface terms. In the present contribution, the number of atoms and $\varepsilon$ are fixed. The study of an evolutionary model as $\varepsilon\to 0$ is deferred to a forthcoming paper [26].

Following the discussion in [33, 24], we impose Dirichlet boundary condition on a part $\partial_{D}\Omega\subset\partial\Omega$ of the boundary. More precisely, consider a neighborhood $U\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\subset\Omega$ of $\partial_{D}\Omega$ such that $\mathcal{L}_{D}(\Omega):=\mathcal{L}(\Omega)\cap U\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\neq\emptyset$. For a function $g\colon\mathcal{L}_{D}(\Omega)\to\mathbb{R}^{n}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$, we define the admissible deformations by

$$\mathcal{A}(g):=\big{\{}y\colon\mathcal{L}(\Omega)\to\mathbb{R}^{n}\colon\,y(x)=g(x)\ \ \text{ for all }x\in\mathcal{L}_{D}(\Omega);\,\mathcal{E}(y)<\infty\big{\}}.$$

Instead of (or additional to) Dirichlet boundary conditions, one could also consider body or boundary forces. However, as traction tests with body forces are ill-posed for Griffith functionals in a continuum variational formulation, we prefer to focus on boundary value problems also on the atomistic level. For a thorough discussion on the comparison of soft and hard devices in the context of the variational formulation of Griffith’s fracture, we refer the reader to [6, Section 3].

Based on the energy (2.1), we will now introduce an evolutionary model on a process time interval $[0,T]$, where the evolution will be driven by time dependent boundary conditions $g\colon[0,T]\times\mathcal{L}_{D}(\Omega)\to\mathbb{R}^{n}$. The key feature in the modeling is to implement an irreversibility condition along the fracture process for each pair interaction of $(x,x^{\prime})$, once the length $|y(x)-y(x^{\prime})|$ of the interaction exceeds the elastic regime. As long as $|y(x)-y(x^{\prime})|$ does not surpass a threshold $R^{1}_{x,x^{\prime}}=R^{1}_{x^{\prime},x}$ satisfying $R^{1}_{x,x^{\prime}}>|x-x^{\prime}|$, the pair interaction can recover completely after removal of loading. By contrast, if $|y(x)-y(x^{\prime})|$ exceeds $R^{1}_{x,x^{\prime}}$ at some earlier time, the interaction is supposed to be ‘damaged’, and the system should be affected at the present time, i.e., the complete history of the deformation undergone by the atomistic system should be reflected at present time. This corresponds to a model of cohesive type energy á la Barenblatt [13].

To describe this, we fix $t\in[0,T]$, a deformation $y$ at time $t$, and suppose that a finite or infinite family of deformations $(y_{s})_{s<t}$ at previous times is given. The interaction of $(x,x^{\prime})$ at time $t$ is considered to be elastic if $|y_{s}(x)-y_{s}(x^{\prime})|\leq R^{1}_{x,x^{\prime}}$ for all $s<t$. Otherwise, we adopt as memory variable the maximal opening as implemented, e.g. in the continuum models [18, 28], see also [6, Section 5.2]. Writing

$\displaystyle M_{x,x^{\prime}}((y_{s})_{s<t})=\sup_{s<t}|y_{s}(x^{\prime})-y_{s}(x)|\vee|x-x^{\prime}|\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$

(2.2)

for the memory variable, the energy contribution with memory reads as

$\displaystyle W_{x,x^{\prime}}\Big{(}M_{x,x^{\prime}}((y_{s})_{s<t})\vee|y(x^{\prime})-y(x)|\Big{)}.$

(2.3)

Other choices, such as that of a cumulative increment, could also be made, but are not discussed in this paper. Note that by implementing the memory variable, the repulsion effect for small distances (Assumption (i)) is lost and (2.3) does not describe correctly the Pauli repulsion at short distances of the interacting particles. Therefore, for the energy contribution of a damaged interaction we rather use

$\displaystyle W_{x,x^{\prime}}(|y(x^{\prime})-y(x)|)\vee W_{x,x^{\prime}}\Big{(}M_{x,x^{\prime}}((y_{s})_{s<t})\vee|y(x^{\prime})-y(x)|\Big{)}.$

Clearly, by Assumption (iii) this is equal to

$\displaystyle W_{x,x^{\prime}}(|y(x^{\prime})-y(x)|)\vee W_{x,x^{\prime}}\Big{(}M_{x,x^{\prime}}((y_{s})_{s<t})\Big{)}.$

(2.4)

This notion is rather simplistic and one drawback lies in the sharp transition between elastic and inelastic behavior. To remedy this, we introduce a second threshold $R^{2}_{x,x^{\prime}}>R^{1}_{x,x^{\prime}}$ and assume that in the transition zone $(R^{1}_{x,x^{\prime}},R^{2}_{x,x^{\prime}})$ the interaction can partially recover, i.e., the energy contribution still partly depends on the current deformation. Let $\varphi_{x,x^{\prime}}\colon[0,\infty)\to[0,1]$ be continuous and increasing with $\varphi_{x,x^{\prime}}(r)=0$ for $r\leq R^{1}_{x,x^{\prime}}$ and $\varphi_{x,x^{\prime}}(r)=1$ for $r\geq R^{2}_{x,x^{\prime}}$. Define the energy contribution as

	$\displaystyle{E}_{x,x^{\prime}}(y;(y_{s})_{s<t}):=$	$\displaystyle\big{(}1-\varphi_{x,x^{\prime}}\big{(}M_{x,x^{\prime}}((y_{s})_{s<t})\big{)}\big{)}W_{x,x^{\prime}}\big{(}|y(x^{\prime})-y(x)|\big{)}$		(2.5)
		$\displaystyle+\varphi_{x,x^{\prime}}\big{(}M_{x,x^{\prime}}((y_{s})_{s<t})\big{)}\Big{(}W_{x,x^{\prime}}(|y(x^{\prime})-y(x)|)\vee W_{x,x^{\prime}}(M_{x,x^{\prime}}((y_{s})_{s<t}))\Big{)}.$

Note that formally this would correspond to (2.4) for the (not admissible) choice $\varphi_{x,x^{\prime}}(r)=0$ for $r\leq R^{1}_{x,x^{\prime}}$ and $\varphi_{x,x^{\prime}}(r)=1$ for $r>R^{1}_{x,x^{\prime}}$. In particular, (2.5) implies

$${E}_{x,x^{\prime}}(y;(y_{s})_{s<t})=W_{x,x^{\prime}}\big{(}|y(x^{\prime})-y(x)|\big{)}\quad\text{ if }M_{x,x^{\prime}}((y_{s})_{s<t})\leq R^{1}_{x,x^{\prime}}$$

$${E}_{x,x^{\prime}}(y;(y_{s})_{s<t})=W_{x,x^{\prime}}(|y(x^{\prime})-y(x)|)\vee W_{x,x^{\prime}}(M_{x,x^{\prime}}((y_{s})_{s<t}))\quad\text{ if }M_{x,x^{\prime}}((y_{s})_{s<t})\geq R^{2}_{x,x^{\prime}}.$$

We adopt the choice (2.5) for a continuous function interpolating between $0$ and $1$ also for technical reasons, see the discussion in Remark 4.3. Summarizing, given the history $(y_{s})_{s<t}$, the energy of an admissible configuration $y\in\mathcal{A}(g(t))$ is given by

$\displaystyle\mathcal{E}(y;(y_{s})_{s<t})=\frac{1}{2}\sum_{\underset{x\neq x^{\prime}}{x,x^{\prime}\in\mathcal{L}(\Omega)}}{E}_{x,x^{\prime}}(y;(y_{s})_{s<t}).$

(2.6)

We now present the main result of the paper on a quasi-static evolution of atomistic systems with the irreversibility condition introduced above. We introduce some further notation. Let $N$ be the cardinality of $\mathcal{L}(\Omega)$ and let $\bar{N}$ be the cardinality of $(\mathcal{L}(\Omega)\setminus\mathcal{L}_{D}(\Omega))$, respectively. For convenience, from now on we will denote $y\colon\mathcal{L}(\Omega)\to\mathbb{R}^{n}$ as a single vector $y\in\mathbb{R}^{Nn}$ via

$$y=(y(x_{1}),\ldots y(x_{N}))\in\mathbb{R}^{Nn},$$

where $(x_{1},\ldots,x_{N})$ is a fixed numbering of $\mathcal{L}(\Omega)$ with $\{x_{\bar{N}+1},\ldots,x_{N}\}=\mathcal{L}_{D}(\Omega)$. In this sense, $\mathcal{E}(\cdot;(y_{s})_{s<t})$ in (2.6) is a function defined on $\mathbb{R}^{Nn}$ with variables $y_{1},\ldots,y_{N}$. We also regard boundary conditions $g\colon[0,T]\times\mathcal{L}_{D}(\Omega)\to\mathbb{R}^{n}$ as vectors $g(t)\in\mathbb{R}^{Nn}$ for each $t\in[0,T]$ by setting $g(t,x)=0$ for all $x\in\mathcal{L}(\Omega)\setminus\mathcal{L}_{D}(\Omega)$.

As it is customary in both existence proofs and in numerical approximation schemes, we start with a time-discretization of the problem. Fix a discrete time step size $\tau>0$, which for simplicity satisfies $T/\tau\in\mathbb{N}$. We write $t_{k}^{\tau}=k\tau$ for $k\in\{0,\ldots,T/\tau\}$, and we also define $g^{\tau}_{k}=g(t^{\tau}_{k})$. Suppose that an initial configuration $y_{0}\in\mathcal{A}(g^{\tau}_{0})$ is given. Fix a time $t^{\tau}_{k}$ and suppose that the deformations $(y^{\tau}_{j})_{j<k}$ at the previous times $t^{\tau}_{j}$, $j<k$, have already been found. Then, the deformation at time $t^{\tau}_{k}$, denoted by $y^{\tau}_{k}\in\mathcal{A}(g^{\tau}_{k})$, is given as the minimizer of

$\displaystyle\min_{y\in\mathcal{A}(g^{\tau}_{k})}\Big{(}\mathcal{E}(y;(y^{\tau}_{j})_{j<k})+\frac{\nu}{2\tau}|y-y_{k-1}^{\tau}|^{2}\Big{)}.$

(2.7)

Here, $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^{Nn}$, where $\nu>0$ is a dissipation coefficient. We remark that (2.7) corresponds to the minimizing movement scheme for gradient flows of the energy $\mathcal{E}(\cdot;(y^{\tau}_{j})_{j<k})$ with respect to the $L^{2}$-norm. At the end of the section, see (2.11), we present a variant of (2.7) which corresponds to a model in the Kelvin-Voigt’s rheology. Note that an essential part of our problem is the fact that the energy depends on the complete history, i.e., on the previous deformations $(y^{\tau}_{j})_{j<k}$, which is usually not present in incremental schemes of the form (2.7). Let us mention that this relates to a discrete-time formulation where, in contrast to common variational approaches to fracture in the realm of rate-independent systems, in each step we do not consider absolute minimizers of the energy, but determine local minimizers by penalizing the $L^{2}$-distance between the approximate solutions at two consecutive times. This approach itself is not new and has been adopted, e.g. in [17].

Note that the existence of a minimizer is readily guaranteed since the problem is finite dimensional, the potential is continuous, and the energy is coercive. Moreover, one can also check that for $\tau$ sufficiently small the problem is strictly convex, and therefore the minimizer is unique.

We now introduce piecewise constant and piecewise affine interpolations by $y_{\tau}(t):=y_{k+1}^{\tau}$ for $t\in(t_{k}^{\tau},t_{k+1}^{\tau}]$ and $\hat{y}_{\tau}(t):=y_{k}^{\tau}+(t-k\tau)v_{\tau}(t)$ for $t\in(t_{k}^{\tau},t_{k+1}^{\tau}]$, respectively, where $v_{\tau}(t):=\frac{1}{\tau}(y^{\tau}_{k+1}-y^{\tau}_{k})$. Our goal is to pass to the limit $\tau\to 0$, and to prove that $y_{\tau}$ converges uniformly to a limit evolution $y$, which satisfies a delay differential equation taking the irreversibility of the fracture process into account. We start with the compactness result.

We are now ready to present the main result of the paper.

Here, $\nabla$ is taken with respect to the variables $y_{1},\ldots,y_{\bar{N}}$, i.e., both sides in (2.8) are vectors in $\mathbb{R}^{\bar{N}n}$. More explicitly, (2.8) can be written as

$$\nabla_{y_{i}}\mathcal{E}(y(t);(y(s))_{s<t})=-\nu\partial_{t}y(t,x_{i})\quad\text{for $i=1,\ldots,\bar{N}$ for a.e.\ $t\in[0,T]$}.$$

Although the potentials $W_{x,x^{\prime}}$ are smooth, $\mathcal{E}(\cdot;(y(s))_{s<t})$ is not necessarily differentiable due to the presence of the memory variable. Therefore, strictly speaking, $\nabla$ has to be interpreted as the element of the subdifferential of $\mathcal{E}$ (in the sense of $\lambda$-convex functions) with minimal norm, denoted by $\partial^{\circ}\mathcal{E}$. We refer to (4.16)–(4.17) for details.

Let us highlight that the resulting equation is not an ordinary differential equation of gradient flow type but a delay differential equation as the energy at time $t$ depends on the complete history $(y(s))_{s<t}$.

We close this section by discussing alternative modeling assumptions and possible extensions. Let us start by presenting a variant of the model where the $L^{2}$-dissipation is replaced by a dissipation depending on a (discrete) strain rate. This corresponds to a model in the Kelvin-Voigt’s rheology. For simplicity, we restrict our modeling here to the one-dimensional case ($d=n=1$) as proofs are more involved in higher dimensions. Specifically, we denote the reference configuration by $x_{1}=\varepsilon,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ldots,x_{N}=\varepsilon N$ with $x_{i}-x_{i-1}=\varepsilon$ for $i\in\{2,\ldots,N\}$, and denote deformations $y$ by $y=(y(x_{1}),\ldots,y(x_{N}))\in\mathbb{R}^{N}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$. Let $\mathcal{L}_{D}(\Omega)=\{x_{1},x_{N}\}$, where for convenience we use a different labeling of $\mathcal{L}_{D}(\Omega)$ compared to Subsection 2.3. We introduce the discrete gradient by

$\displaystyle\bar{\nabla}y=\Big{(}\frac{y(x_{i})-y(x_{i-1})}{\varepsilon}\Big{)}_{i=2,\ldots,N}\in\mathbb{R}^{N-1}.$

(2.9)

We also introduce the vector $\bar{\rm D}y\in\mathbb{R}^{N-2}$ defined by

$$(\bar{\rm D}y)_{i}=\frac{2y(x_{i})-y(x_{i-1})-y(x_{i+1})}{\varepsilon},i\in\{2,\ldots,N-1\}.$$

(2.10)

We replace the time-incremental scheme described in (2.7) by

$\displaystyle\min_{y\in\mathcal{A}(g^{\tau}_{k})}\Big{(}\mathcal{E}(y;(y^{\tau}_{j})_{j<k})+\frac{\nu}{2\tau}|\bar{\nabla}y-\bar{\nabla}y_{k-1}^{\tau}|^{2}\Big{)}.$

(2.11)

In the limit of vanishing time steps $\tau\to 0$, this leads to the delay differential equation

$\displaystyle\partial^{\circ}\mathcal{E}(y(t);(y(s))_{s<t})=-\nu\bar{\rm D}\partial_{t}y(t)\quad\text{ on $\{x_{2},\ldots,x_{N-1}\}$ \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}for a.e.\ $t\in[0,T]$},$

(2.12)

where $\partial^{\circ}\mathcal{E}$ is explained in (4.16)–(4.17) and is taken with respect to $(y_{2},\ldots,y_{N-1})$. The proof is similar to the one of Theorem 2.2 and the necessary adaptions are given in Remark 4.4. Let us mention that a more realistic modeling assumption would be that $\nu$ is different for each contribution and depends on the memory variable $M_{x,x^{\prime}}$, with $\nu(M_{x,x^{\prime}})$ decreasing in $M_{x,x^{\prime}}$.

The model developed so far was restricted to pair interactions for the sake of simplicity. However, for a realistic description of brittle materials also multi-body interactions should be taken into account. Here, we present one possible simple extension incorporating three-body interactions. Let $x,x^{\prime},x^{\prime\prime}\in\mathcal{L}(\Omega)$ be three distinct neighboring atoms in the reference configuration with $|x^{\prime}-x|=\varepsilon$ and $|x^{\prime}-x^{\prime\prime}|=\varepsilon$. Consider the angle between the two edges $(x^{\prime},x)$ and $(x^{\prime},x^{\prime\prime})$ under the deformation $y$, given by

$$\theta_{x,x^{\prime},x^{\prime\prime}}(y):=\arccos\big{\langle}y(x)-y(x^{\prime}),y(x^{\prime\prime})-y(x^{\prime})\big{\rangle}\in[0,\pi]\,.$$

Let $W_{x,x^{\prime},x^{\prime\prime}}\colon[0,\pi]\to\mathbb{R}$ be a continuous three-body interaction potential, see [11, 35, 36] for possible choices. We then assume that the three-body energy contribution of $x,x^{\prime},x^{\prime\prime}\in\mathcal{L}(\Omega)$ has the form

$$E_{x,x^{\prime},x^{\prime\prime}}(y,(y_{s})_{s<t}):=W_{x,x^{\prime},x^{\prime\prime}}\big{(}\theta_{x,x^{\prime},x^{\prime\prime}}(y)\big{)}\Big{[}1-\big{(}\varphi\big{(}M_{x,x^{\prime}}((y_{s})_{s<t})\big{)}\vee\varphi\big{(}M_{x^{\prime},x^{\prime\prime}}((y_{s})_{s<t})\big{)}\big{)}\Big{]}\,.$$

This reflects the assumption that the three-body-interaction is only active as long as the two interactions $(x,x^{\prime})$ and $(x^{\prime},x^{\prime\prime})$ are not ’damaged’. The total energy in (2.6) then changes to

$\displaystyle\mathcal{E}(y;(y_{s})_{s<t})=\frac{1}{2}\sum_{\underset{x\neq x^{\prime}}{x,x^{\prime}\in\mathcal{L}(\Omega)}}{E}_{x,x^{\prime}}(y;(y_{s})_{s<t})+\sum_{\underset{|x-x^{\prime}|=|x^{\prime\prime}-x^{\prime}|=\varepsilon}{x,x^{\prime},x^{\prime\prime}\in\mathcal{L}(\Omega),x\neq x^{\prime\prime}}}E_{x,x^{\prime},x^{\prime\prime}}(y,(y_{s})_{s<t})\,.$

(2.13)

Note that Theorem 2.2 stated above still holds for this extended model, as we briefly indicate after the proof of Theorem 2.2, see Remark 4.5.

In this subsection, we analyze the scheme (2.7) for a given family of deformations $(y^{\tau}_{j})_{j<k}=(y^{\tau}_{j})_{j\leq k-1}$. For convenience, we denote the memory variable $M_{x,x^{\prime}}((y^{\tau}_{j})_{j<k})$ simply by $M^{x,x^{\prime}}_{k-1}$. In a similar fashion, we express the energy for fixed $x,x^{\prime}\in\mathcal{L}(\Omega),x\neq x^{\prime}$, by

$\displaystyle E^{x,x^{\prime}}_{k-1}(\cdot):=E_{x,x^{\prime}}(\;\cdot\;;(y^{\tau}_{j})_{j<k}),$

(4.1)

and for the sum over all contributions we write

$\displaystyle\mathcal{E}_{k-1}(\cdot):=\mathcal{E}(\;\cdot\;;(y^{\tau}_{j})_{j<k})\,.$

(4.2)

Furthermore, in this subsection we drop the superscript $\tau$, i.e., we write $y_{k}$ and $t_{k}$ in place of $y_{k}^{\tau}$ and $t_{k}^{\tau}$, respectively. By convention, in proofs we often drop the index $x,x^{\prime}$ for $W_{x,x^{\prime}}$, $\varphi_{x,x^{\prime}}$, $E^{x,x^{\prime}}_{k-1}$, $M^{x,x^{\prime}}_{k-1}$, $R^{1}_{x,x^{\prime}}$, $R^{2}_{x,x^{\prime}}$ if no confusion arises.

We begin by proving a lemma that follows from the very definition of the potential in (2.5) and the incremental minimization scheme described in (2.7).

As a next step, we prove an estimate that will enable us to control the time derivative of the deformation. As a preparation, consider $g_{i}=g_{i}^{\tau}=g(t_{i}^{\tau})$ for every $i\in\{0,\dots,T/\tau\}$. Then, using $g\in H^{1}([0,T];\mathbb{R}^{Nn})$, the fundamental theorem of calculus, and Jensen’s inequality it follows that

$$\sum_{i=1}^{T/\tau}\frac{|g_{i}-g_{i-1}|^{2}}{\tau}=\sum_{i=1}^{T/\tau}\tau\left|\frac{1}{\tau}\int_{(i-1)\tau}^{i\tau}\partial_{t}g(s)\,{\rm d}s\right|^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\leq\int_{0}^{T}|\partial_{t}g\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(s)|^{2}\,{\rm d}s=\|\partial_{t}g\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\|^{2}_{L^{2}([0,T];\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{R}^{Nn})}\,,$$

(4.4)

and with Hölder’s inequality

$$\sum_{i=1}^{T/\tau}|g_{i}-g_{i-1}|\leq\sqrt{\tau}\left(\sum_{i=1}^{T/\tau}\frac{|g_{i}-g_{i-1}|^{2}}{\tau}\right)^{\frac{1}{2}}\sqrt{T/\tau}\leq\sqrt{T}\|\partial_{t}g\|_{L^{2}([0,T];\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{R}^{Nn})}.$$

(4.5)

We thus have the estimates

$\displaystyle\sum_{i=1}^{T/\tau}\frac{|g_{i}-g_{i-1}|^{2}}{\tau}\leq C,\qquad\sum_{i=1}^{T/\tau}|g_{i}-g_{i-1}|\leq C\,,$

(4.6)

for a constant $C$ independent of $\tau$.