Analysis of Staggered Evolutions for Nonlinear Energies in Phase Field Fracture

Abstract

We consider a class of separately convex phase field energies employed in fracture mechanics, featuring non-interpenetration and a general softening behavior. We analyze the time-discrete evolutions generated by a staggered minimization scheme, where fracture irreversibility is modeled by a monotonicity constraint on the phase field variable. After recasting the staggered scheme by means of gradient flows, we characterize the time-continuous limits of the discrete solutions in terms of balanced viscosity evolutions, parametrized by their arc-length with respect to the $L^2$-norm (for the phase field) and the $H^1$-norm (for the displacement field). By a careful study of the energy balance we deduce that time-continuous evolutions may still exhibit an alternate behavior in discontinuity times.

Appendices

Comparing Different Parametrizations

In this appendix we will compare, qualitatively, the evolutions of Theorem 1 with those of [24, Theorem 4.2], or, more precisely, we will compare the evolutions obtained here, employing $H^1$-norm for u and $L^2$-norm for z, with those obtained employing energy norms. As we will see, the evolutions will be qualitatively the same (up to subsequences) even if these norms are not equivalent.

We need to consider the setting of [24], otherwise energy norms would not be defined. Let $\mathcal {J} :[0,T] \times H^1_0 (\varOmega , \mathbb {R}^2) \times H^1 (\varOmega ; [0,1]) \rightarrow [0,+\infty )$ given by

$$\begin{aligned} \mathcal {J} (t,u,z) = \tfrac{1}{2} \int _\varOmega (z^2 + \eta ) \varvec{\sigma }(u + g(t)){:\,} \varvec{\epsilon }(u + g(t)) \, \mathrm {d}x + \tfrac{1}{2} \int _\varOmega |\nabla z |^2 + (z-1)^2 \, \mathrm {d}x , \end{aligned}$$

where we assume that the boundary datum g belongs to $C^{1,1}([0,T]; W^{1,p}(\varOmega ;\mathbb {R}^{2}))$, $p>2$. Note that this energy is separately quadratic, thus it is natural, and technically convenient, to introduce a couple of energy (instrinsic) norms

$$\begin{aligned} \Vert u \Vert ^2_{z} = \int _\varOmega (z^2 + \eta ) \varvec{\sigma }(u){:\,} \varvec{\epsilon }(u)\, \mathrm {d}x , \qquad \Vert z \Vert ^2_{u} = \int _\varOmega | \nabla z |^2 + z^2 ( 1 + \varvec{\sigma }(u){:\,} \varvec{\epsilon }(u)) \, \mathrm {d}x , \end{aligned}$$

which correspond, respectively, to the quadratic part of the energies $\mathcal {J}(t, \cdot , z)$ and $\mathcal {J}(t, u, \cdot )$. Accordingly, we employ the slopes

$$\begin{aligned} |\partial _{u} \mathcal {J}|_z (t,u,z)&= \max \,\{-\partial _{u}\mathcal {J}(t,u,z)[\varphi ]:\,\varphi \in H^1_0(\varOmega ; \mathbb {R}^2),\, \Vert \varphi \Vert _{z}\leqq 1\},\\ |\partial _{z}^{-} \mathcal {J}|_u (t,u,z)&= \max \,\{-\partial _{z} \mathcal {J}(t,u,z)[\xi ]:\,\xi \in H^1(\varOmega ),\,\xi \leqq 0,\,\Vert \xi \Vert _{u}\leqq 1\} . \end{aligned}$$

Let us consider again the alternate scheme (at time $t^k_i$)

$$\begin{aligned}&\displaystyle u^{k}_{i,j+1}:= \mathrm {argmin} \,\{\mathcal {J}(t^{k}_{i},u,z^{k}_{i,j}):\,u\in \mathcal {U}\} , \\&\displaystyle z^{k}_{i,j+1}:= \mathrm {argmin} \,\{\mathcal {J}(t^{k}_{i},u^{k}_{i,j + 1},z):\, z\in \mathcal {Z},\, z\leqq z^{k}_{i,j}\}. \end{aligned}$$

We remark that, given $\tau _k$, the families $u^{k}_{i,j}$ and $z^k_{i,j}$ are uniquely determined (by strict separate convexity of the energy). Following [24], we interpolate and parametrize the discrete configurations $u^k_{i,j}$ and $z^k_{i,j}$ with respect to the energy norms $\Vert \cdot \Vert ^2_{z}$ (for the displacement field) and $\Vert \cdot \Vert ^2_{u}$ (for the phase field). We remark that in this case it is enough to consider piece-wise affine interpolation, which actually coincides, for both u and z, with the gradient flow in the energy norm. As a result, we get a sequence of arc-length parametrizations $( {\bar{t}}_k , {\bar{u}}_k, {\bar{z}}_k)$, bounded in $W^{1,\infty } ( [ 0, R] ; [0,T] \times H^1_0(\varOmega ; \mathbb {R}^2) \times H^1 (\varOmega ))$ and of uniformly finite length, that is, with R independent of $k \in \mathbb {N}$. Invoking [24, Lemma 4.3], there exists a subsequence (non relabelled) and a limit $( {\bar{t}}, {\bar{u}}, {\bar{z}})$ in $W^{1,\infty } ( [ 0, R] ; [0,T] \times H^1_0(\varOmega ; \mathbb {R}^2) \times H^1 (\varOmega ))$ such that for every sequence $r_{k}$ converging to $r \in [0,R]$ we have

$$\begin{aligned} {\bar{t}}_{k}(r_{k})\rightarrow {\bar{t}} (r) ,\ {\bar{u}}_{k}(r_{k})\rightarrow {\bar{u}}(r) \text { in } H^1_0(\varOmega ; \mathbb {R}^2),\ {\bar{z}}_{k}(r_{k}) \rightharpoonup {\bar{z}}(r) \text { in } H^{1}(\varOmega ).\nonumber \\ \end{aligned}$$

(132)

Moreover, invoking [24, Theorem 4.2] the limit evolution satisfies the following properties:

• $({\bar{a}})$Regularity: $({\bar{t}},{\bar{u}},{\bar{z}})\in W^{1,\infty }([0,R];[0,T]\times H^1_0 ( \varOmega ;\mathbb {R}^{2}) \times H^{1}(\varOmega ; [0,1]))$, and for almost everywhere $r\in [0,R]$

  $$\begin{aligned} \bar{t\,}'(r)+\Vert {\bar{u}}'(r)\Vert _{ {\bar{z}}(r)}+\Vert {\bar{z}}'(r)\Vert _{{\bar{u}} (r)}\leqq 1, \end{aligned}$$

  here the symbol $'$ denotes the derivative with respect to the parametrization variable r;

• $({\bar{b}})$Time parametrization: the function ${\bar{t}}:[0,R]\rightarrow [0,T]$ is non-decreasing and surjective;

• $({\bar{c}})$Irreversibility: the function ${\bar{z}}$ is non-increasing and $0 \leqq {\bar{z}}(r) \leqq 1$ for every $0\leqq r \leqq R$;

• $({\bar{d}})$Equilibrium: for every continuity point $r\in [0,R]$ of $({\bar{t}},{\bar{u}},{\bar{z}})$

  $$\begin{aligned} |\partial _{u} \mathcal {J}|_{{\bar{z}}(r)} ({\bar{t}}(r),{\bar{u}}(r),{\bar{z}}(r))=0\qquad \text {and}\qquad |\partial _{z}^{-} \mathcal {J}|_{{\bar{u}}(r)} ({\bar{t}}(r),{\bar{u}}(r),{\bar{z}}(r))=0 ; \end{aligned}$$

• $({\bar{e}})$Energy-dissipation equality: for every $r \in [0,R]$

  $$\begin{aligned} \mathcal {J}({\bar{t}}(r), \ {\bar{u}}(r), {\bar{z}}(r)) =&\mathcal {J}(0,u_0,z_0)\nonumber \\&-\int _{0}^{r}|\partial _{u} \mathcal {J}|_{{\bar{z}}(\rho )} ({\bar{t}}(\rho ),{\bar{u}}(\rho ),{\bar{z}}(\rho ))\, \Vert {\bar{u}}' (\rho ) \Vert _{{\bar{z}}(\rho )} \mathrm {d}\rho \nonumber \\&-\int _{0}^{r} \!\! |\partial _{z}^{-} \mathcal {J}|_{{\bar{u}}(\rho )} ({\bar{t}}(\rho ),{\bar{u}}(\rho ),{\bar{z}}(\rho ))\, \Vert {\bar{z}}' (\rho ) \Vert _{{\bar{u}}(\rho )} \mathrm {d}\rho \nonumber \\&+\int _{0}^{r} \!\! \mathcal {P}( {\bar{t}}(\rho ),{\bar{u}}(\rho ), {\bar{z}}(\rho )) \,\bar{t\,} '(\rho )\,\mathrm {d}\rho . \end{aligned}$$

  (133)

In [24] the authors showed property $({\bar{d}})$ for every $r\in [0,R]$ with ${\bar{t}}'(r)>0$. However, it is not difficult to see that the same equilibrium condition is verified at continuity points.

Moreover, by [24, Proposition 4.1] we have

• $({\bar{f}})$Non-degeneracy: there exists $C>0$ such that for almost every $r\in [0,R]$

  $$\begin{aligned} C < \bar{t\,}'(r) + \Vert {\bar{u}}'(r) \Vert _{{\bar{z}}(r)} + \Vert {\bar{z}}'(r) \Vert _{{\bar{u}}(r)} . \end{aligned}$$

  (134)

Finally, note that, by the separate differentiability of the energy, the equilibrium conditions (d) can be written in an equivalent “norm-free” fashion as

• $({\bar{d}}')$Equilibrium: for every continuity point $r\in [0,R]$ of $({\bar{t}},{\bar{u}},{\bar{z}})$

  $$\begin{aligned} \partial _{u} \mathcal {J} ({\bar{t}}(r), {\bar{u}}(r), {\bar{z}}(r)) [ \varphi ] = 0 , \qquad \text {and} \qquad \partial _{z} \mathcal {J} ({\bar{t}}(r),{\bar{u}}(r),{\bar{z}}(r)) [ \xi ] = 0 , \end{aligned}$$

  for every $\varphi \in H^1_0 ( \varOmega ; \mathbb {R}^2)$ and every $\xi \in H^1(\varOmega )$ with $\xi \leqq 0$.

At this point, consider the subsequence (not relabelled) converging to (t, u, z) and let us re-interpolate the discrete configurations $u^k_{i,j}$ and $z^k_{i,j}$ with respect to the norms $\Vert \cdot \Vert _{H^1}$ (for the displacement field) and $\Vert \cdot \Vert _{L^2}$ (for the phase field) as we did in Section 5.1. In this way we get a new sequence of parametrizations $(t_k , u_k ,z_k)$ bounded in $W^{1,\infty } ( [ 0, S] ; [0,T] \times H^1_0(\varOmega ; \mathbb {R}^2) \times L^2 (\varOmega ))$. Clearly, we can apply Proposition 6 which provides (up to a further subsequence) a limit parametrization $(t , u ,z) \in W^{1,\infty } ( [ 0, S] ; [0,T] \times H^1_0(\varOmega ; \mathbb {R}^2) \times L^2 (\varOmega ) )$ such that

$$\begin{aligned} t_{k}(s_{k})\rightarrow t(s),\quad u_{k}(s_{k})\rightarrow u(s) \text { in } H^1(\varOmega ; \mathbb {R}^2), \quad z_{k}(s_{k}) \rightharpoonup z(s) \text { weakly in } H^{1}(\varOmega ), \end{aligned}$$

for every sequence $s_{k}$ converging to $s\in [0,S]$. The limit (t, u, z) satisfies properties (a)–(e) of Theorem 1.

We recall that $(t_k, u_k, z_k)$ is defined in the points $s^k_{i,j}$ and $s^k_{i,j+\frac{1}{2}}$ (see Section 5.1). In a similar way, the interpolation $({\bar{t}}_k, {\bar{u}}_k, {\bar{z}}_k)$ is defined in points of the form $r^k_{i,j}$ and $r^k_{i,j+\frac{1}{2}}$ (see Section 4.3 in [24]). Moreover, we notice that the interpolation nodes are different since the underlying parametrizations are different. However, the configurations computed by the alternate minimization scheme are the same. Therefore, we have that

$$\begin{aligned} t_{k}(s^{k}_{i,j}) = {\bar{t}}_{k}(r^{k}_{i,j}), \quad u_{k}(s^{k}_{i,j}) = {\bar{u}}_{k}(r^{k}_{i,j}), \quad z_{k}(s^{k}_{i,j}) = {\bar{z}}_{k}(r^{k}_{i,j}). \end{aligned}$$

The same holds for nodes of the form $s^{k}_{i,j+\frac{1}{2}}$ and $r^{k}_{i,j+\frac{1}{2}}$.

Since $({\bar{t}}_k, {\bar{u}}_k , {\bar{z}}_k)$ is piecewise affine while $(t_k , u_k ,z_k)$ is not, a direct comparison of the triples $({\bar{t}}, {\bar{u}},{\bar{z}})$ and (t, u, z) is not immediate. Nevertheless, we can show the following “equivalence” of the reparametrizations.

Lemma 15

There exist two positive constants $C_{1}, C_{2}$ such that for every $k\in \mathbb {N}\setminus \{0\}$ and every $i\in \{1,\ldots , k\}$

$$\begin{aligned} C_{1} (s^{k}_{i+1} - s^{k}_{i}) \leqq r^{k}_{i+1} - r^{k}_{i} \leqq C_{2} (s^{k}_{i+1} - s^{k}_{i}) . \end{aligned}$$

(135)

Proof

Using the fact that $\Vert \cdot \Vert _z$ and $\Vert \cdot \Vert _{H^1}$ are equivalent, by Korn’s inequality, while $\Vert \cdot \Vert _{u}$ and $\Vert \cdot \Vert _{H^{1}}$ are equivalent by [24, Lemma 2.3], by (90) we can write

$$\begin{aligned} s^k_{i+1} - s^k_i&= \tau _k + \sum _{j=0}^{\infty } L ( \zeta ^k_{i,j} ) + L ( w^k_{i,j}) \leqq \tau _k + C \sum _{j=0}^{\infty } \Vert z^k_{i,j} - z^k_{i,j+1} \Vert _{H^1} \\&\quad + \Vert u^k_{i,j} - u^k_{i,j+1} \Vert _{H^1} \\&\leqq \tau _k + C' \sum _{j=0}^{\infty } r^k_{i,j+1} - r^i_{i,j} \leqq C' ( r^k_{i+1} - r^k_i ) . \end{aligned}$$

On the other hand, by Proposition 1 and Corollary 1 we have that

$$\begin{aligned} \Vert z^k_{i,j} - z^k_{i,j+1} \Vert _{H^{1}} \leqq \left\{ \begin{array}{ll} C \Vert u^k_{i,j} - u^k_{i,j+1} \Vert _{H^1} &{} \text {if } j\geqq 1,\\ C\tau _{k} &{} \text {if } j=0. \end{array}\right. \end{aligned}$$

Hence, again by equivalence of norms, we get

$$\begin{aligned} r^k_{i+1} - r^k_i&= \tau _k + C \sum _{j=0}^{\infty } \Vert z^k_{i,j} - z^k_{i,j+1} \Vert _{u^{k}_{i,j+1}} + \Vert u^k_{i,j} - u^k_{i,j+1} \Vert _{z^{k}_{i,j}} \\&\leqq C' \Big ( \tau _k + \sum _{j=0}^{\infty } \Vert u^k_{i,j} - u^k_{i,j+1} \Vert _{H^{1}} \Big )\leqq C' ( s^k_{i+1} - s^k_i ) . \end{aligned}$$

This concludes the proof of (135). $\quad \square$

Let us consider an interval of the form $(s^k_{i_k,j_k+\frac{1}{2}} , s^k_{i_k,j_k+1}) \subset (0,S)$ and the corresponding interval $(r^k_{i_k,j_k+\frac{1}{2}} , r^k_{i_k,j_k+1}) \subset (0,R)$. By definition, we have

$$\begin{aligned} t^k_{i_k} = {\bar{t}}_k (r) = t_k (s) \quad \text { and } \quad u^k_{i_k,j_k+1} = {\bar{u}}_k (r) = u_k ( s) , \end{aligned}$$

for every $r \in (r^k_{i_k,j_k+\frac{1}{2}} , r^k_{i_k,j_k+1})$ and every $s \in (s^k_{i_k,j_k+\frac{1}{2}} , s^k_{i_k,j_k+1})$. On the contrary, the phase field interpolations coincide only in the extrema, that is,

$$\begin{aligned} {\bar{z}}_k (r^k_{i_k,j_k+\frac{1}{2}}) = z_k (s^k_{i_k,j_k+\frac{1}{2}}) = z^k_{i,j} \quad {\bar{z}}_k (r^k_{i_k,j_k+1}) = z_k (s^k_{i_k,j_k+1}) = z^k_{i,j+1}.\nonumber \\ \end{aligned}$$

(136)

Now, up to subsequences (non relabelled), we can assume that, as $k \rightarrow \infty$,

$$\begin{aligned} s^k_{i_k,j_k+\frac{1}{2}} \rightarrow s^- , \qquad s^k_{i_k,j_k+1} \rightarrow s^+ , \qquad r^k_{i_k,j_k+\frac{1}{2}} \rightarrow r^- , \qquad r^k_{i_k,j_k+1} \rightarrow r^+ . \end{aligned}$$

Since parametrizations are different, in general we should distinguish between all the following cases: $s^- = s^+$, $s^- < s^+$, $r^- = r^+$, and $r^- < r^+$; however the situation is much simpler, thanks to the following lemma.

Lemma 16

We have $r^- = r^+$ if and only if $s^- = s^+$.

Proof

Assume that $r^- = r^+$. By compactness, we know that ${\bar{z}}_k ( r^k_{i_k,j_k+\frac{1}{2}} ) = z^k_{i_k,j_k} \rightharpoonup z ( r^-)$ weakly in $H^{1}(\varOmega )$ and that ${\bar{z}}_k ( r^k_{i_k,j_k+1} ) = z^k_{i_k,j_k+1} \rightharpoonup z ( r^+)$ weakly in $H^{1}(\varOmega )$. Since $r^- = r^+$, we have ${\bar{z}}(r^-) = {\bar{z}}(r^+)$ and

$$\begin{aligned} \Vert z^k_{i_k,j_k} - z^k_{i_k,j_k+1} \Vert _{H^1} = \big ( r^k_{i_k,j_k+1} - r^k_{i_k,j_k+\frac{1}{2}} \big ) \rightarrow 0 . \end{aligned}$$

By (88) we know that

$$\begin{aligned} \big ( s^k_{i_k,j_k+1} - s^k_{i_k,j_k+\frac{1}{2}} \big ) = L (\zeta ^k_{i,j}) \leqq C \Vert z^k_{i_k,j_k} - z^k_{i_k,j_k+1} \Vert _{H^1} . \end{aligned}$$

Thus $s^- = s^+$.

Assume that $s^- = s^+$. Hence, arguing as above, by (136) we have $z (s^-) = {\bar{z}} (r^-) = z(s^+) = {\bar{z}} (r^+)$. Moreover, being ${\bar{t}}_k$, $t_k$, ${\bar{u}}_k$ and $u_k$ constant in the corresponding intervals, in the limit we have $t (s^-) = {\bar{t}} (r^-) = t(s^+) = {\bar{t}} (r^+)$ and $u (s^-) = {\bar{u}} (r^-) = u(s^+) = {\bar{u}} (r^+ )$. Then, if $r^- < r^+$ we would contradict the non-degeneracy condition (134). $\quad \square$

As a consequence of Lemma 15, the solutions $({\bar{t}}, {\bar{u}}, {\bar{z}})$ and (t, u, z) coincide in continuity points.

Proposition 9

Let r be a continuity point for $({\bar{t}}, {\bar{u}}, {\bar{z}})$. Then, there exists a continuity point s for (t, u, z) such that $({\bar{t}} (r), {\bar{u}} (r), {\bar{z}}(r) ) = (t (s), u (s), z (s))$.

Viceversa, if s is a continuity point for (t, u, z), then there exists a continuity point r for $({\bar{t}},{\bar{u}},{\bar{z}})$ such that $(t (s), u (s), z (s)) = ({\bar{t}} (r), {\bar{u}} (r), {\bar{z}}(r) )$.

Proof

Fix $\delta >0$. Since r is a continuity point, by monotonicity of ${\bar{t}}$ we have ${\bar{t}}(r + \delta ) > {\bar{t}} (r)$. Since ${\bar{t}}_k$ converges pointwise to ${\bar{t}}$, we have ${\bar{t}}_k (r + \delta ) - {\bar{t}}_k (r) \geqq \tfrac{1}{2} ( {\bar{t}}(r + \delta ) - {\bar{t}} (r)) > 0$ for every k sufficiently large. As ${\bar{t}}_k$ changes only in parametrization intervals of the form $(r^k_{i,-1} , r^k_{i,0})$ and since $\tau _k \rightarrow 0$, there exist two indexes $i_k< i'_{k}$ such that,

$$\begin{aligned} r^{k}_{i_{k}-1}\leqq r< r^k_{i_k} < r^k_{i_k,0} \leqq \cdots \leqq r^{k}_{i'_{k}} \leqq r + \delta \leqq r^{k}_{i'_{k}+1}. \end{aligned}$$

(137)

Hence for every index $j \in \mathbb {N}$, we have

$$\begin{aligned} r< r^k_{i_k,j+ \frac{1}{2}} \leqq r^k_{i_k,j+1} \leqq r^k_{i_k+1,0} < r + \delta . \end{aligned}$$

Since $\delta$ can be arbitrarily small, we can find a sequence $(i_{k},j_{k})$ such that

$$\begin{aligned} r^k_{i_k,j_k+ \frac{1}{2}} \rightarrow r \quad \text { and } \quad r^k_{i_k,j_k + 1} \rightarrow r . \end{aligned}$$

Then ${\bar{z}}_{k} ( r^k_{i_k,j_k + 1} ) \rightharpoonup {\bar{z}} (r)$ weakly in $H^1(\varOmega )$. Since ${\bar{z}}_{k} ( r^k_{i_k,j_k + 1} ) = z_{k} ( s^k_{i_k,j_k + 1} )$ we have $z _{k}( s^k_{i_k,j_k + 1} ) \rightharpoonup {\bar{z}} (r)$ weakly in $H^1(\varOmega )$. Up to a subsequence (not relabelled) $s^k_{i_k,j_k + 1} \rightarrow s$ and thus $z_{k} ( s^k_{i_k,j_k + 1} ) \rightharpoonup z (s)$. We conclude that $z(s) = {\bar{z}} (r)$. In a similar way ${\bar{t}} (r) = t(s)$ and ${\bar{u}} ( r) = u (s)$.

It remains to show that s is a continuity point for (t, u, z). To this aim, let us set, up to subsequence, $s_{\delta }:=\lim _{k} s^{k}_{i'_{k}}$, where the indexes $i'_{k}$ have been defined in (137). Hence, applying Lemma 15 we deduce that

$$\begin{aligned} |s - s_{\delta }| = \lim _{k\rightarrow \infty }\, |s^{k}_{i_{k}, j_{k}+1} - s^{k}_{i'_{k}}| \leqq \lim _{k\rightarrow \infty }\, |s^{k}_{i_{k}} - s^{k}_{i'_{k}}| \leqq C \lim _{k\rightarrow \infty }\, |r^{k}_{i_{k}} - r^{k}_{i'_{k}}| \leqq C\delta . \end{aligned}$$

By definition of $s_{\delta }$ we have that $t(s_{\delta })= \lim _{k\rightarrow \infty } t_{k}(s^{k}_{i'_{k}}) = \lim _{k\rightarrow \infty } {\bar{t}}_{k}(r^{k}_{i'_{k}})$. By (137) we get that $| {\bar{t}}_{k}(r^{k}_{i'_{k}}) - {\bar{t}}_{k}(r+\delta )|\leqq \tau _{k}$, from which we deduce that $t(s_{\delta }) = {\bar{t}} ( r + \delta ) > {\bar{t}}(r) = t(s)$. This implies that s is of continuity for (t, u, z).

The viceversa can be shown in a similar way. $\quad \square$

On the contrary, in discontinuity points the evolution (t, u, z) and $({\bar{t}}, {\bar{u}}, {\bar{z}})$ interpolate the same configurations but with different paths. To better understand, let us consider an interval of the form $(r^k_{i_{k},j_{k}+\frac{1}{2}}, r^k_{i_{k},j_{k}+1})$ such that $r^k_{i_{k},j_{k}+\frac{1}{2}} \rightarrow r^-$, $r^k_{i_{k},j_{k}+1} \rightarrow r^+$ with $r^- < r^+$. As a consequence both ${\bar{t}}$ and ${\bar{u}}$ are constant in $(r^-, r^+)$ and thus every $r \in (r^-, r^+)$ is not a continuity point. In this case, we have

$$\begin{aligned} {\bar{z}}_{k} ( r^k_{i_{k},j_{k}+\frac{1}{2}} ) \rightharpoonup {\bar{z}} ( r^-) \quad \text { and } \quad {\bar{z}}_{k} ( r^k_{i_{k},j_{k}+1} ) \rightharpoonup {\bar{z}} ( r^+) . \end{aligned}$$

By the non-degeneracy property of $({\bar{t}}, {\bar{u}}, {\bar{z}})$, we deduce that ${\bar{z}} ( r^-) \ne {\bar{z}} ( r^+)$. Moreover, up to subsequence we have

$$\begin{aligned} z_{k} ( s^k_{i_{k},j_{k}+\frac{1}{2}} ) \rightharpoonup z ( s^-) \quad \text { and } \quad z_{k} ( s^k_{i_{k},j_{k}+1} ) \rightharpoonup z ( s^+) . \end{aligned}$$

Since $z\in W^{1,\infty }([0,S]; L^{2}(\varOmega ))$, we get that $s^- \ne s^+$. Recalling that t and u are constant in $(s^-,s^+)$, the energy balance of Theorem 1 reads as

$$\begin{aligned} \mathcal {J} ( t(s^-) , u(s^-) , z(s^+) ) =&\ \mathcal {J} ( t(s^-) , u(s^-) , z(s^-) )\\&- \int _{s^-}^{s^+} \Vert z' (\sigma ) \Vert _{L^2} \, | \partial ^-_v \mathcal {J}| ( t(s^-) , u(s^-) , z(\sigma ) ) \, \mathrm {d}\sigma . \end{aligned}$$

Thus z is a (normalized) unilateral gradient flow in $L^2$, with initial datum $z(s^-)$. On the contrary, ${\bar{z}}$ is the affine interpolation of ${\bar{z}} ( r^-)$ and ${\bar{z}} ( r^+)$. Thus, in general, ${\bar{z}}$ and z do not coincide in the corresponding intervals $(r^-,r^+)$ and $(s^-,s^+)$ even if they coincide in the extrema.

On the Alternate Behavior in Discontinuity Points

Let us consider the set U defined in (119) and denote by ${\mathbf {1}}_{U}$ its characteristic function. From (127) we know that

$$\begin{aligned} \mathcal {F}( t(s), u (s), z (s)) \leqq&\ \mathcal {F}( 0, u_{0}, z_{0}) - \int _{0}^{s} \!\! |\partial _{u} \mathcal {F}| ( t (\sigma ), u ( \sigma ), z ( \sigma ))\, \Vert u' (\sigma ) \Vert _{H^1} \,\mathrm {d}\sigma \\&- \int _{0}^{s} \!\! | \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) \, {\mathbf {1}}_{U}(\sigma ) \, \mathrm {d}\sigma \\&- \int _{0}^{s} \!\! \liminf _{k\rightarrow \infty } | \partial _{z}^{-} \mathcal {F}| ( {\underline{t}}_{k}(\sigma ), {\underline{u}}_{k}( \sigma ), z_{k}( \sigma )) \, {\mathbf {1}}_{U^{c}} (\sigma ) \, \mathrm {d}\sigma . \end{aligned}$$

Note that $z' (\sigma ) = 0$ for every $\sigma \in U^c$ (see Lemma 9), thus, being $\Vert z' \Vert _{L^2} \leqq 1$ almost everywhere in [0, S], we have $\Vert z' \Vert _{L^2} \leqq {\mathbf {1}}_{U}$ almost everywhere in [0, S]; as a consequence, the above estimate together with (22) yields

$$\begin{aligned}&\mathcal {F}( t(s), u (s), \ z (s))\\&\quad \leqq \mathcal {F}( 0, u_{0}, z_{0}) - \int _{0}^{s} \!\! |\partial _{u} \mathcal {F}| ( t (\sigma ), u ( \sigma ), z ( \sigma ))\, \Vert u' (\sigma ) \Vert _{H^1} \,\mathrm {d}\sigma \\&\qquad - \int _{0}^{s} \!\! | \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) \, {\mathbf {1}}_{U}(\sigma ) \, \mathrm {d}\sigma \\&\qquad +\int _{0}^{s} \mathcal {P}(t(\sigma ), u(\sigma ), z(\sigma )) \, t'(\sigma ) \, \mathrm {d}\sigma . \\&\quad \leqq \ \mathcal {F}( 0, u_{0}, z_{0}) - \int _{0}^{s} \!\! |\partial _{u} \mathcal {F}| ( t (\sigma ), u ( \sigma ), z ( \sigma ))\, \Vert u' (\sigma ) \Vert _{H^1} \,\mathrm {d}\sigma \\&\qquad - \int _{0}^{s} \!\! | \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) \, \Vert z' (\sigma ) \Vert _{L^2} \, \mathrm {d}\sigma \\&\qquad +\int _{0}^{s} \mathcal {P}(t(\sigma ), u(\sigma ), z(\sigma )) \, t'(\sigma ) \, \mathrm {d}\sigma \\&\quad = \ \mathcal {F}( t(s), u (s), z (s)) . \end{aligned}$$

Therefore, all inequalities turn into equalitites and thus

$$\begin{aligned} 0 \leqq \int _{0}^{s} \!\! | \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) \, ( {\mathbf {1}}_{U}(\sigma ) - \Vert z' (\sigma ) \Vert _{L^2} ) \, \mathrm {d}\sigma = 0 ; \end{aligned}$$

hence $| \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) \, ( {\mathbf {1}}_{U}(\sigma ) - \Vert z' (\sigma ) \Vert _{L^2} ) =0$ almost everywhere in [0, S].

Therefore, if $| \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) \ne 0$ for every $\sigma \in (\sigma _1, \sigma _2) \subset U$ then $\Vert z' (\sigma ) \Vert _{L^2} = 1$ almost everywhere in $(\sigma _1, \sigma _2)$ and then both $t'(\sigma )=0$ and $u'(\sigma )=0$ almost everywhere in $(\sigma _1, \sigma _2)$, by (a) in Theorem 1. This means that in the discontinuity interval $(\sigma _1, \sigma _2)$ only z changes, following a normalized, unilateral $L^2$ gradient flow. In view of this observation, excluding the cases in which $| \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) = 0$, we expect that in the presence of time-discontinuties the limit evolution is still alternate, and thus it is not simultaneous in u and z. More precisely, consider a discontinuity at time t, with a transition from $(u^-,z^-)$ to $(u^+,z^+)$, parametrized in the interval $(\sigma ^-, \sigma ^+)$. With t being the limit of continuity points, the left limit $(u^-,z^-)$ is an equilibrium configuration at time t. We expect that the parametrizations of u and z, in a right neighborhood of $\sigma ^-$, provide an alternate interpolation of sequences $z^-_m \searrow z^-$, with $z^-_m \ne z^-$, and $u^-_m \rightarrow u^-$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} u^-_{m-1} \in \text{ argmin } \, \big \{ \mathcal {F}( t , u , z^-_{m} ) \big \}, \\ z^-_{m-1} \in \text{ argmin } \, \big \{ \mathcal {F}( t, u^-_{m-1}, z \, ) : z \leqq z^-_{m} \big \}. \end{array}\right. } \end{aligned}$$

The non-degeneracy condition $z^-_m \ne z^-$ is due to the fact that $(u^-,z^-)$ is an equilibrium configuration, and thus a separate minimizer of the energy $\mathcal {F}( t , \cdot , \cdot )$. Indeed, if $z^-_m = z^-$, for some index $m \in \mathbb {N}$, then, by uniqueness of the minimizer, $u^-_{m-1} = u^-$ and then $z^-_{m-1} = z^-$. By induction and by monotonicity, $z^-_m=z^-$ for every index $m \in \mathbb {N}$ and then $u^-_m = u^-$ for every $m \in \mathbb {N}$; thus, there would be no transition between $(u^-,z^-)$ to $(u^+,z^+)$. In a similar way, we expect sequences $z^+_m \nearrow z^+$ and $u^+_m \rightarrow u^+$ in a left neighborhood $\sigma ^+$ such that

$$\begin{aligned} \quad {\left\{ \begin{array}{ll} u^+_{m+1} \in \text{ argmin } \, \big \{ \mathcal {F}( t , u , z^+_{m} ) \big \}, \\ z^+_{m+1} \in \text{ argmin } \, \big \{ \mathcal {F}( t, u^+_{m+1}, z) : z \leqq z^+_{m} \big \}. \end{array}\right. } \end{aligned}$$

However, in this case we cannot exclude that $z^+_m = z^+$ for some index $m \in \mathbb {N}$. We remark that this qualitative behavior is confirmed by numerical computations.