Two-Dimensional Grain Boundary Networks: Stochastic Particle Models and Kinetic Limits

Abstract

We study kinetic theories for isotropic, two-dimensional grain boundary networks which evolve by curvature flow. The number densities $f_s(x,t)$ for s-sided grains, $s =1,2,\ldots$, of area x at time t, are modeled by kinetic equations of the form $\partial _t f_s + v_s \partial _x f_s =j_s$. The velocity $v_s$ is given by the Mullins–von Neumann rule and the flux $j_s$ is determined by the topological transitions caused by the vanishing of grains and their edges. The foundations of such kinetic models are examined through simpler particle models for the evolution of grain size, as well as purely topological models for the evolution of trivalent maps. These models are used to characterize the parameter space for the flux $j_s$. Several kinetic models in the literature, as well as a new kinetic model, are simulated and compared with direct numerical simulations of mean curvature flow on a network. The existence and uniqueness of mild solutions to the kinetic equations with continuous initial data is established.

Appendices

Description of the M-Species System as a PDMP

We briefly review the basics of PDMPs, following Davis [5], and then explain how the M-species stochastic particle process of Section 2.1 fits into this framework.

1.1 Background: General Theory of PDMPs

We consider a countable set $\mathcal {S}$ with elements denoted $\mathbf {s}$, a map $\mathbf {d}: \mathcal {S}\rightarrow \mathbb {N}$, and open sets for each $\mathbf {s}$ of the form $M_{\mathbf {s}}\subset \mathbb {R}^{\mathbf {d}(\mathbf {s})}$. The state space is the disjoint union

$$\begin{aligned} E =\coprod _{\mathbf {s}\in \mathcal {S}} M_{\mathbf {s}}= \left\{ (\mathbf {s},\mathbf {x}):\mathbf {s}\in \mathcal {S}, \mathbf {x}\in M_{\mathbf {s}}\right\} . \end{aligned}$$

(79)

The space E has a natural topology. Let $\iota _\mathbf {s}:M_{\mathbf {s}}\rightarrow E$ be the canonical injection defined by $\iota _\mathbf {s}(\mathbf {x}) = (\mathbf {s},\mathbf {x})$. A set $A \subset E$ is open if for every $\mathbf {s}$, $\iota _\mathbf {s}^{-1}(A)$ is open in $M_{\mathbf {s}}$. The collection of all open sets may be used to define the set $\mathcal E$ of Borel subsets of E. This makes $(E,\mathcal E)$ a Borel space.

A PDMP is an E-valued generalized jump process $X(t) = (\mathbf {s}(t),\mathbf {x}(t))$, $t\geqq 0$, that is prescribed by:

• 1. Sufficiently smooth vector fields $\mathbf {v}_\mathbf {s}: M_{\mathbf {s}}\rightarrow \mathbb {R}^{\mathbf {d}(\mathbf {s})}$, $\mathbf {s}\in \mathcal {S}$.

• 2. A measurable function $\lambda :E \rightarrow \mathbb R^+$.

• 3. A transition measure $Q: \mathcal E \times (E \cup \varGamma ^*) \rightarrow [0,1]$. Here $\varGamma ^*$ denotes the exit boundary defined in Eqs. (84)–(83) below.

Points in $M_{\mathbf {s}}$ travel according to flows defined by the vector fields $\mathbf {v}_\mathbf {s}$ until either a Poisson clock with intensity $\lambda (\mathbf {s},\mathbf {x})$ rings or the point $\mathbf {x}(t)$ hits the exit boundary $\varGamma ^*$. When such a critical event occurs the point X(t) jumps to a random new position whose law is given by Q.

Each vector field $\mathbf {v}_\mathbf {s}$ may be viewed as a first-order differential operator on $M_{\mathbf {s}}$. We assume they define a flow $\varphi _\mathbf {s}(t,\mathbf {x})$ such that

$$\begin{aligned} \frac{\partial }{\partial t} h\left( \mathbf {s},\varphi _\mathbf {s}(t,\mathbf {x}) \right) = \mathbf {v}_\mathbf {s}\left( h\left( \varphi _\mathbf {s}(t,\mathbf {x})\right) \right) , \quad \varphi _\mathbf {s}(0,\mathbf {x}) = \mathbf {x}, \end{aligned}$$

(80)

for all sufficiently smooth test functions $h$ and for t in a maximal interval of existence. The flow terminates only when $\mathbf {x}(t)$ hits

$$\begin{aligned} \partial ^*M_{\mathbf {s}}=\left\{ \mathbf {y}\in \partial M_{\mathbf {s}}: \varphi _\mathbf {s}(t^-, \mathbf {x}) = \mathbf {y}\quad \hbox { for some } (t, \mathbf {x}) \in \mathbb {R}_+ \times M_{\mathbf {s}}\right\} . \end{aligned}$$

(81)

The exit boundary is the disjoint collection

$$\begin{aligned} \varGamma ^* = \coprod _{\mathbf{s} \in \mathcal {S}} \partial ^{*}M_{\mathbf {s}}= \left\{ (\mathbf {s},\mathbf {x}):\mathbf{s} \in \mathcal {S}, \mathbf {x}\in \partial ^*M_{\mathbf {s}}\right\} , \end{aligned}$$

(82)

At a given state $(\mathbf {s},\mathbf {x}) \in E$ we define the first exit time

$$\begin{aligned} t_\mathbf {s}^*(\mathbf{x}) = \sup \{t >0: \varphi _\mathbf {s}(t, \mathbf {x}) \in M_{\mathbf {s}}\}, \end{aligned}$$

(83)

and the survivor function

$$\begin{aligned} \mathcal {F}_{\left( \mathbf {s},\mathbf {x}\right) }(t) = {\left\{ \begin{array}{ll} \exp \left( -\int _0^t \lambda \left( \mathbf {s},\varphi _\mathbf {s}\left( \tau ,\mathbf {x}\right) \right) \,\mathrm{d}\tau \right) , &{}\quad t<t_\mathbf {s}^*(\mathbf {x}), \\ 0, &{}\quad t \geqq t^*(\mathbf {x}). \end{array}\right. } \end{aligned}$$

(84)

The stochastic process $(X(t))_{t \geqq 0}$ with initial condition $X(0) = (\mathbf {s}_0,\mathbf {x}_0)$ is defined as follows. Choose a random time $T_0$ such that $\mathbb {P}[T_0>t] = \mathcal {F}_{\left( \mathbf {s}_0,\mathbf {x}_0\right) }(t)$ and an E-valued random variable $(\mathbf {s}_1,\mathbf {x}_1)$ with law $Q(\cdot \,; \varphi _{\mathbf {s}_0}(T_0,\mathbf {x}_0))$ that is independent of $T_0$. The trajectory of X(t) for $t \leqq T_0$ is then

$$\begin{aligned} X(t) = {\left\{ \begin{array}{ll} \left( \mathbf {s}_0,\varphi _{\mathbf {s}_0}(t,\mathbf {x}_0)\right) , &{}\quad t<T_0, \\ (\mathbf {s}_1,\mathbf {x}_1), &{}\quad t = T_0. \end{array}\right. } \end{aligned}$$

(85)

At $t=T_0$, we repeat this process, replacing the jump time $T_0$ in the algorithm above with $T_1-T_0$ and the state $(\mathbf {s}_0,\mathbf {x}_0)$ with $(\mathbf {s}_1,\mathbf {x}_1)$. Iterating this process, jump by jump, yields a cadlag process X(t), $t \in [0, \infty )$.

Under modest assumptions, it can be shown that $X(t)_{t \geqq 0}$ is a strong Markov process [5, §3]. We only require that $Q\left( A;(\mathbf {s},\mathbf {x})\right)$ is a measurable function of $(\mathbf {s},\mathbf {x})$ for each Borel set $A \in \mathcal E$ and a probability measure on $(E, \mathcal E)$ for each $(\mathbf {s},\mathbf {x}) \in E \cup \varGamma ^*$. The rate function $\lambda :E \rightarrow \mathbb R^+$ must be measurable with a little integrability: specifically, for each state $(\mathbf {s}, \mathbf {x}) \in E$ we require the existence of $\varepsilon > 0$ such that the function $\tau \rightarrow \lambda (\mathbf {s}, \varphi _\mathbf {s}(\tau , \mathbf {x}))$ is summable for $\tau \in [0,\varepsilon )$. These conditions are easily verified in our model.

1.2 The M-Species Model as a PDMP

We now show the M-species model defined in Section 2.1 is a PDMP. Define the countable set of species indices

$$\begin{aligned} \mathcal {S}= \bigcup _{m \in \mathbb N}\{1, \ldots , M\}^{m}. \end{aligned}$$

(86)

It is convenient to introduce notation that makes explicit the distinction between the number of particles in a state $(\mathbf {s},\mathbf {x})$ and the fixed parameter N that is the normalizing factor in the empirical measure (15). We denote the number of particles in the state $(\mathbf {s},\mathbf {x})$ by $|\mathbf {s}|$ and write

$$\begin{aligned} {(}\mathbf {s},\mathbf {x}) = \left( s_1, \ldots , s_{|\mathbf {s}|}; x_1, \ldots , x_{|\mathbf {s}|})\right) , \end{aligned}$$

(87)

and the associated empirical measures is

$$\begin{aligned} \mu ^N_\sigma (\mathbf {s},\mathbf {x}) = \frac{1}{N}\sum _{i=1}^{|\mathbf {s}|} \delta _{x_i} \mathbf {1}_{s_i=\sigma }, \quad \sigma =1, \ldots , M. \end{aligned}$$

(88)

For the M-species process, $N(t) = |\mathbf {s}|(t)$, and Eqs. (2),(15) and Eqs. (89)–(90)are consistent.

Similarly, each open set $M_{\mathbf {s}}= \mathbb {R}_+^{|\mathbf {s}|}$ and

$$\begin{aligned} E =\coprod _{\mathbf {s}\in \mathcal {S}}\mathbb {R}_+^{|\mathbf {s}|} = \left\{ (\mathbf {s},\mathbf {x}): \mathbf {s}\in \mathcal S, \, \mathbf {x}\in \mathbb {R}_+^{|\mathbf {s}|}\right\} . \end{aligned}$$

(89)

The velocity fields $\mathbf {v}_\mathbf {s}$ on E are obtained from the velocity fields $v_s$, $s=1,\ldots ,M$ of the M-species model,

$$\begin{aligned} \mathbf {v}_\mathbf {s}= \sum _{i = 1}^{|\mathbf {s}|} v_{s_i}(x_i)\frac{\partial }{\partial x_i}, \end{aligned}$$

(90)

and the exit boundary is

$$\begin{aligned} \varGamma ^* = \{(\mathbf {s},\mathbf {x})\in E| \;\hbox {there exists } \; (s_i,x_i)\; \hbox {such that} \;x_i = 0,\, s_i \in S_-\}. \end{aligned}$$

(91)

In order to define the transition kernel Q, we first describe the finite set of ‘neighbors’ $E^\partial _{\mathbf {s},\mathbf {x}}$ for each state $(\mathbf {s},\mathbf {x}) \in \varGamma ^*$. Each point $(\mathbf {s},\mathbf {x})$ has a finite number, p, of particles with size zero. Let us label these particles with indices $i=k_1$,$k_2$,$\ldots$, $k_p$, ordered such that the species $s_{k_1} \leqq s_{k_2}\leqq \ldots s_{k_p}$. Let us begin by discussing the case when $p=1$ (this is the most important case, since boundary events happen at distinct times with probability 1). When $p=1$, the set $E^\partial _{\mathbf {s},\mathbf {x}}$ may be decomposed into $M_-$ subsets, corresponding to boundary events at $M_-$ species. More precisely, a boundary event occurs at species l, if the size $x_{j_1}=0$ and the associated species $s_{j_1}=l$. According to the rules of Section 2.1, at such a boundary event, $K^{(l)}$ random variables $(S_j,X_j)$ are chosen, and mutated as in Eq. (8). Each such mutation gives rise to a neighbor $(\mathbf {r},\mathbf {y})$ of $(\mathbf {s},\mathbf {x})$. Since the $X_j$ are a random collection of $K^{(l)}$ points of $\mathbf {x}$, we may write $X_j = x_{i_j}$, for indices $i_1,\ldots , i_{K^{(l)}}$. Then $(\mathbf {r},\mathbf {y})$ is obtained from $(\mathbf {s},\mathbf {x})$ in two ‘sub-steps’:

• (i) Pure mutation: $\mathbf {x}$ is unchanged. The coordinates of $\mathbf {s}$ are changed as follows: $s_{i_j} \mapsto R^{(l)}_j$, $j=1, \ldots , K^{(l)}$. Call this intermediate state $\hat{\mathbf {s}}$.

• (ii) Removal of zero size: $\mathbf {x}$ is changed to $\mathbf {y}$ by deleting the particle $x_{j_1}$ with size zero.

The probability $p^\partial (\cdot ; \mathbf {s},\mathbf {x})$ of each transition $(\mathbf {s},\mathbf {x}) \mapsto (\mathbf {r},\mathbf {y}) \in E^\partial _{\mathbf {s},\mathbf {x}}$ is given by the rules of Section 2.1. Finally, observe that these rules extend naturally to degenerate boundary points, where $0=x_{k_1} = x_{k_2} = \ldots x_{k_p}$. In this case, according to the rules of Section 2.1, we order the points $x_{j_1},\ldots , x_{j_p}$ so that the species $s_{j_1}< s_{j_2} < s_{j_p}$, and mutate and remove particles p times in sequence as above.

Similarly, given an interior point $(\mathbf {s},\mathbf {x}) \in E$ we can use the mutation matrix $R^{(0)}$ and the weights $w^{(0)}$ to define a set of interior points $E^{(0)}_{\mathbf {s},\mathbf {x}}$ that $(\mathbf {s},\mathbf {x})$ jumps to along with the corresponding probabilities $p^{(0)}(\cdot ;\mathbf {s},\mathbf {x})$. In this case, the transition involves only a mutation and no removal of zero sizes.

In summary, the transition kernel is given by

$$\begin{aligned} Q(A; \mathbf {s},\mathbf {x}) = {\left\{ \begin{array}{ll} \int _A p^\partial (\mathbf {r},\mathbf {y}; \mathbf {s},\mathbf {x}) \mathbf {1}_{E^\partial _{\mathbf {s},\mathbf {x}}}(\mathbf {r},\mathbf {y}) \,\mathrm{d}(\mathbf {r},\mathbf {y}), &{}\quad (\mathbf {s},\mathbf {x}) \in \varGamma ^*, \\ \int _Ap^{(0)}(\mathbf {r},\mathbf {y}; \mathbf {s},\mathbf {x}) \mathbf {1}_{E^{(0)}_{\mathbf {s},\mathbf {x}}}(\mathbf {r},\mathbf {y}) \, \mathrm{d}(\mathbf {r},\mathbf {y}) , &{}\quad (\mathbf {s},\mathbf {x}) \in E. \end{array}\right. } \end{aligned}$$

(92)

Since each particle carries an independent Poisson-$\beta$ clock $\beta$, the first time T that a clock rings follows the distribution $T \sim \min _{1\leqq i\leqq |\mathbf {s}|} \mathrm {Poisson}(\beta ) = \mathrm {Poisson}(|\mathbf {s}|\beta )$. Thus

$$\begin{aligned} \lambda (\mathbf {s},\mathbf {x}) = \beta |\mathbf {s}|. \end{aligned}$$

(93)

This completes the description of the M-species model as a PDMP.

1.3 Conservation of Total Area and Zero Polyhedral Defect

A benefit of using a finite particle system for grain boundary coarsening is the conservation of area and zero polyhedral defect. Using the notation presented above, we may write area of polyhedral defect in an N particle system as function $A, P : E \rightarrow [0,\infty )$ given by

$$\begin{aligned} A^N[(\mathbf {s},\mathbf {x})] = \sum _{i = 1}^{|\mathbf{s}|} x_i \quad P^N[(\mathbf {s},\mathbf {x})] = \sum _{i = 1}^{|\mathbf{s}|} (s_i-6). \quad \end{aligned}$$

(94)

Here, we used the identity function $id:x \mapsto x$. Zero polyhedral defect for a trivalent planar network means that a grain has, on average, six sides, which follows from (36) for networks evolving on a torus.

Theorem 3

For the PDMP model with fixed parameters from Section 4, suppose we have initial polyhedral defect $P^N(0) = 0$ and total area $A^N(0) = A$, for all times t where the process is well-defined (i) $P^N(t) = 0$ and (ii) $A^N(t) = A.$

Proof

We consider a well-defined path $(\mathbf{s}(t), \mathbf{x}(t))$ for $t \in [0,T]$. In all realizations, this path will have a finite set of jump times $\tau _1\leqq \ldots \leqq \tau _n$.

To show conservation of zero polyhedral defect, suppose for a state $(\mathbf {s},\mathbf {x})$ that $(\mathbf{r}, \mathbf{y}) \in E^\partial _{(\mathbf {s},\mathbf {x})}$. We will directly show that defect does not change over jumps, or that

$$\begin{aligned} P^N[(\mathbf {s},\mathbf {x})]= P^N[(\mathbf{r}, \mathbf{y})] \end{aligned}$$

(95)

in the case of a three-sided grain vanishing (other critical events have similar proofs). Under a reindexing, we may write our state as

$$\begin{aligned} {(}\mathbf {s},\mathbf {x}) = ((3, s_2, \ldots ,s_{|\mathbf{s}|}), (0, x_2, \ldots , x_{|\mathbf{s}|})). \end{aligned}$$

(96)

We may assume, without loss of generality, From (38)–(42) three particles with indices 2,3, and 4 lose an edge from The mutated state then takes the form

$$\begin{aligned} {(}\mathbf{r}, \mathbf{y}) = (s_2-1, s_3-1, s_4-1, \ldots , s_{|\mathbf{s}|}), (x_2, \ldots , x_{|\mathbf{s}|})), \end{aligned}$$

(97)

from which (97) follows immediately. This implies that $\varDelta P^N[(\mathbf {s}(\tau _i),\mathbf {x}(\tau _i))] = 0$. Since $\mathbf{s}$ does not change between any jumps, if $P^N(\mathbf {s}(0),\mathbf {x}(0)) =0$, then $P^N(\mathbf {s}(t),\mathbf {x}(t))= 0$ for all times t in which the PDMP is well defined.

To show conservation of total area, again assume zero initial polyhedral defect. Then it is immediate that $\varDelta A^N[(\mathbf {s}(\tau _i),\mathbf {x}(\tau _i) ] = 0$, and for $t \in (\tau _i, \tau _{i+1})$ for $i = 1, \ldots , n-1$,

$$\begin{aligned} \frac{\partial A^N}{\partial t} =\sum _{i = 1}^{|\mathbf{s}|} \frac{\partial A^N}{\partial x_i} \frac{\partial x_i}{\partial t} = \sum _{i = 1}^{|\mathbf{s}|} (s_i(t)-6) = P^N[(\mathbf {s}(t),\mathbf {x}(t))] = 0. \end{aligned}$$

(98)

$\quad \square$

Proof of Well-Posedness

Fig. 20

Limiting equations for a two species example with $T^*<\infty$. The PDMP is characterized by $v_1 = -1, v_2 = 0, K^{(1)} = 1, R_{21}^{(1)} = 1,$ and weights $w_2^{(1)} = 1, w_1^{(1)} = 0$. Left: Initial densities on the two species with disjoint supports and $F_1 (0)= F_2(0) = 1/2.$ Center: As the initial density of species 1 is transported to the origin, species 2 mutates to species 1. At some time $t_1$, all of species 2 have mutated, so that $f_2(x,t_1) = 0$. Right: the density is transported until it reaches the origin at time $T^*$, at which point mutation probabilities are undefined

Theorem 1 is proved in the following lemmas. The structure of the kinetic equations is a little more transparent when the flux is rewritten as a matrix vector product. Let $f=(f_1, \ldots , f_M)$ and ${\varvec{\jmath }}=(j_1, \ldots , j_M)$. We may then write

$$\begin{aligned} {\varvec{\jmath }}= \left( \sum _{l=1}^l A^{(l)}\dot{L}_l + \beta \gamma (t) \, A^{(0)}\right) f, \end{aligned}$$

(99)

where the matrices $A^{(l)}$ and $A^{(0)}$ have off-diagonal terms given by

$$\begin{aligned} A^{(l)}_{s,\sigma } = J^{(l)}_{s,\sigma } W^{(l)}_s, \quad A^{(0)}_{s,\sigma } = J^{(0)}_{s,\sigma } w^{(0)}_s, \quad \sigma \ne s, \end{aligned}$$

(100)

and diagonal terms given by

$$\begin{aligned} A^{(l)}_{\sigma ,\sigma } = - K^{(l)}W^{(l)}_\sigma , \quad A^{(0)}_{\sigma ,\sigma } = - K^{(0)}_{\sigma ,\sigma } w^{(0)}_\sigma . \end{aligned}$$

(101)

We first show that the flux ${\varvec{\jmath }}$, defined in (101), is a locally Lipschitz map. This allows us to obtain local existence of positive mild solutions by Picard’s method. We then extend the solutions to a maximal interval of existence by utilizing a more careful estimate of the flux.

Let $B_r(f_0) \subset X$ denote the ball of radius $r>0$ centered at $f_0 \in X$. As in (26) we denote

$$\begin{aligned} F_0 = \sum _{\sigma =1}^M \int _0^\infty f_{0,\sigma }(x). \end{aligned}$$

We adopt the following convention in the proof. The letter C denotes a universal, positive, finite constant depending only on the parameters of the model such as the number of species M, the constant velocities $v_\sigma$, the number of mutations $K^{(l)}$ and $K^{(0)}$, the mutation matrices $R^{(l)}$ and $R^{(0)}$, the weights $w^{(l)}$ and $w^{(0)}$. It does not depend on $f_0$.

Lemma 1

(Uniform bounds) Assume $f_0 \in X$ is positive and non-zero. There exists $r>0$, depending only on $f_0$, such that for each $f \in B_r(f_0)$.

$$\begin{aligned} \Vert {\varvec{\jmath }}(f) \Vert \leqq C \left( \beta + \frac{\Vert f_0 \Vert }{F_0} \right) \Vert f\Vert . \end{aligned}$$

(102)

Proof

Recall that the flux ${\varvec{\jmath }}(f)$ is defined by Eqs. (101)–(102). We will estimate each term in this expression in turn.

We first estimate $\dot{L}$. We find from (22) that for every $l \in S_-$

$$\begin{aligned} |\dot{L}_l| \leqq |v_l| |f_l(0)| \leqq \left( \max _\sigma |v_\sigma | \right) \Vert f_l \Vert _{L^\infty } \leqq C \Vert f_0\Vert . \end{aligned}$$

(103)

In order to estimate the weights $W^{(l)}_k$ defined by (18), we first establish a lower bound on the denominator $\sum _{n=1}^M w^{(l)}_n F_n$ for each $f \in B_r(f_0)$. Let

$$\begin{aligned} \underline{w} = \min _{\sigma ,l} \{w^{(l)}_\sigma : F_{0,\sigma }>0\}, \quad \overline{w} = \max _{\sigma ,l} w^{(l)}_\sigma . \end{aligned}$$

We then have

$$\begin{aligned}&{ \sum _{n=1}^M w^{(l)}_n F_n = \sum _{n=1}^M w^{(l)}_n (F_n - F_{0,n}+ F_{0,n}) \geqq \underline{w} F_0 - \sum _{n=1}^M w^{(l)}_n |F_n -F_{0,n}| } \nonumber \\&\quad \geqq \underline{w} F_0 - \sum _{n=1}^M w^{(l)}_n \Vert f_n -f_{0,n}\Vert _{L^1} \geqq \underline{w} F_0 - \overline{w} \Vert f-f_0\Vert \geqq \frac{1}{2}\underline{w} F_0, \end{aligned}$$

(104)

provided the radius r satisfies

$$\begin{aligned} r < \frac{\underline{w}}{2\overline{w}}F_0. \end{aligned}$$

(105)

We assume that r is chosen as above. It then follows from (102) and (103) that each entry in the matrix $A^{(l)}$ is bounded above by

$$\begin{aligned} |A_{\sigma k}| \leqq \frac{C}{F_0}. \end{aligned}$$

(106)

Thus, the operator norm $\Vert A^{(l)}\Vert _o$ of the matrix $A^{(l)}$ satisfies the estimate²

$$\begin{aligned} \Vert A^{(l)}\Vert _o \leqq \frac{C}{F_0}. \end{aligned}$$

(107)

We combine (109) with (105) to see that the flux due to boundary events is bounded by

$$\begin{aligned} \left\| \left( \sum _{l=1}^{M_-} A^{(l)}\dot{L}_l \right) f \right\| \leqq C\frac{\Vert f_0\Vert \Vert f\Vert }{F_0}. \end{aligned}$$

(108)

The estimates for the interior events are simpler. We use the definition of $\gamma$ in (18) and the lower bound (106) to obtain the estimate

$$\begin{aligned} 0 \leqq \gamma \leqq C\frac{F}{F_0} \leqq C, \quad f \in B_r(f_0). \end{aligned}$$

(109)

It follows from the definition of $A^{(0)}$ in (102)–(103) that $\Vert A^{(0)}\Vert _o \leqq C$. Thus, the flux from interior events is bounded by

$$\begin{aligned} \Vert \beta \gamma A^{(0)}f \Vert \leqq C \beta \Vert f\Vert . \end{aligned}$$

(110)

We combine estimates (110) and (112) to complete the proof. $\quad \square$

Lemma 2

(Lipschitz estimate) Let $f_0$ and r be as in Lemma 1. Then for every $f,g \in B_r(f_0)$

$$\begin{aligned} \Vert {\varvec{\jmath }}(f) -{\varvec{\jmath }}(g)\Vert \leqq C \left( \beta + \frac{\Vert f_0\Vert }{F_0}\right) \left( 1+ \frac{\Vert f_0\Vert }{F_0} \right) \Vert f-g\Vert . \end{aligned}$$

(111)

Proof

We use the expression (101) to obtain the inequality

$$\begin{aligned}&{\Vert {\varvec{\jmath }}(f) -{\varvec{\jmath }}(g) \Vert } \\&\quad \leqq \sum _{l=1}^{M_-} \Vert A^{(l)}(f)\dot{L}_l(f) f - A^{(l)}(g)\dot{L}_l(g) g \Vert + \beta \Vert \gamma (f) A^{(0)}f - \gamma (g) A^{(0)}g\Vert . \nonumber \end{aligned}$$

(112)

Let l be fixed. It is clear that

$$\begin{aligned} | \dot{L}_l(f) - \dot{L}_l(g)| = |v_l| |f(0)-g(0)| \leqq C \Vert f-g\Vert . \end{aligned}$$

(113)

For each k, the difference $\left| W^{(l)}_k(f)-W^{(l)}_k(g)\right|$ is estimated as follows. Let $G_n = \int _0^\infty g_n(x) \, \mathrm{d}x$ and $G = \sum _{n=1}^M G_n$. Then

$$\begin{aligned} \left| \frac{w^{(l)}_k}{\sum _{n=1}^M w^{(l)}_n F_n} - \frac{w^{(l)}_k}{\sum _{n=1}^M w^{(l)}_n G_n}\right|&= \frac{w^{(l)}_k \left| \sum _{n=1}^M w^{(l)}_n (F_n -G_n)\right| }{\left| \sum _{n=1}^M w^{(l)}_n F_n\right| \left| \sum _{n=1}^M w^{(l)}_n G_n\right| } \\&\leqq \frac{C}{F_0^2} \sum _{n=1}^M \Vert f_n -g_n \Vert _{L^1} \leqq \frac{C}{F_0^2} \Vert f-g\Vert , \nonumber \end{aligned}$$

(114)

using (106). It then follows from (102) and (103) that each term in the matrix $A^{(l)}(f)-A^{(l)}(g)$ satisfies an estimate as above, so that

$$\begin{aligned} \Vert A^{(l)}(f) - A^{(l)}(g) \Vert _o \leqq \frac{C}{F_0^2} \Vert f-g\Vert . \end{aligned}$$

(115)

Finally, we use the estimates (105), (109), (115) and (116) to obtain the Lipschitz bound:

$$\begin{aligned}&{\Vert A^{(l)}(f)\dot{L}_l(f) f - A^{(l)}(g)\dot{L}_l(g) g \Vert \leqq \Vert A^{(l)}(f) - A^{(l)}(g) \Vert _o |\dot{L}_l(f)| \Vert f\Vert } \\&\quad + \Vert A^{(l)}(g)\Vert _0 |\dot{L}_l(f) - \dot{L}_l(g)| \Vert f\Vert + \Vert A^{(l)}(g)\Vert _o|L_l(g)| \Vert f-g \Vert \\&\quad \leqq C \left( \frac{\Vert f_0\Vert ^2}{F_0^2} + \frac{\Vert f_0\Vert }{F_0} \right) \Vert f-g\Vert . \end{aligned}$$

A calculation similar to (111) yields the estimate

$$\begin{aligned} \Vert \gamma (f)-\gamma (g)\Vert \leqq \frac{C}{F_0}|F-G| \leqq \frac{C}{F_0}\Vert f-g\Vert . \end{aligned}$$

(116)

Thus, we find (also using the fact that $A^{(0)}$ is a constant)

$$\begin{aligned} \beta \Vert \gamma (f)A^{(0)}f - \gamma (g)A^{(0)}g \Vert \leqq C\beta \left( 1+ \frac{\Vert f_0\Vert }{F_0} \right) \Vert f-g\Vert . \end{aligned}$$

(117)

$\quad \square$

Lemma 3

(Local existence) Assume $f_0 \in X$ is positive and non-zero. There exists a time $T_0>0$ and a map $f \in C([0,T];X)$ such that f is the unique mild solution to (19) on the time interval [0, T] that satisfies the initial condition $f(0)=f_0$.

Further, f(t) is positive for each $t \in [0,T]$.

Proof

Let $r(f_0)$ be chosen as in Lemma 1. It then follows from Lemma 2 that the flux ${\varvec{\jmath }}(f)$ is locally Lipschitz. The existence of a unique mild solution now follows by a standard application of the contraction mapping theorem.

The fact that the solution preserves positivity is seen as follows. We note that the loss term in (21), may be rewritten as $j_\sigma ^-(x,t) = \alpha _\sigma (t) f_\sigma (x,t)$ where

$$\begin{aligned} \alpha _\sigma (t) = \sum _{l = 1}^{M_-} \dot{L_l} K^{(l)}W_\sigma ^{(l)}(t) + \beta \gamma (t) K^{(0)}w^{(0)}_\sigma . \end{aligned}$$

(118)

We now rewrite the kinetic equation (19) in the form

$$\begin{aligned} \partial _t f_\sigma + v_\sigma \partial _x f_\sigma + \alpha _\sigma (t) f_\sigma = j_\sigma ^+, \end{aligned}$$

(119)

and observe that integration along characteristics yields

$$\begin{aligned} f_\sigma (x,t)&= e^{-\int _0^t \alpha (s) \, \mathrm{d}s} f_\sigma (x-v_\sigma t, 0) \nonumber \\&+ \int _0^t e^{-\int _\tau ^t \alpha (s) \, \mathrm{d}s} j_\sigma ^+ \left( x-v_\sigma (t-\tau ),\tau \right) \, \mathrm{d}\tau , \end{aligned}$$

(120)

which clearly preserves positivity. $\quad \square$

Lemma 4

(Maximal existence) Let $f \in C([0,T];X)$ be a positive, mild solution. Then

$$\begin{aligned} F(t) + \sum _{l=1}^{M_-} L_l(t) = F(0), \quad L_l(t) {:=} |v_l| \int _0^t f_l(0,s) \, \mathrm{d}s, \quad t \in [0,T]. \end{aligned}$$

(121)

There also exists a universal constant $C >0$ such that

$$\begin{aligned} \Vert f (t) \Vert _{L^\infty } \leqq \Vert f(0)\Vert _{L^\infty } \exp \left( C\int _0^t\varPhi (\tau )\mathrm{d}\tau \right) , \quad t \in [0,T], \end{aligned}$$

(122)

where $\varPhi (t) = t+\max _{l \leqq M_-}\sum W_k^{(l)}(t)$.

Equation (123) expresses conservation of the total number density of the system. The bound (124) degenerates if and only $F(t) \rightarrow 0$, i.e. if and only if $F_p(t) \rightarrow 0$ for each $p=1, \ldots , M$, as t approaches a critical time, say $T_*$. It is well-known that continuous mild solutions on an interval [0, T] can be uniquely continued onto a maximal interval of existence $[0,T_*)$, such that $\lim _{t \rightarrow T_*} \Vert f(t)\Vert _X = +\infty$. Thus, the above estimates suffice to complete the proof of Theorem 1.

Proof

• 1. The conservation of number for the kinetic equations is a consequence of the switching rules for the particle system. We use the identity (13), and the definition of the fluxes in Eqs. (20) and (21) to obtain the identity

  $$\begin{aligned} \sum _{\sigma =1}^M j_\sigma =0. \end{aligned}$$

  (123)

  It follows from (19) and (125) that

  $$\begin{aligned} \sum _{\sigma =1}^M \partial _t f_\sigma + v_\sigma \partial _x f_\sigma =0. \end{aligned}$$

  (124)

  We integrate over $x \in [0,\infty )$ to obtain the identity

  $$\begin{aligned} \frac{\mathrm{d}F}{\mathrm{d}t} = \sum _{\sigma =1}^M v_\sigma f_\sigma (0,t) = -\sum _{l=1}^{M_-} \dot{L}_l. \end{aligned}$$

  (125)

  The integral form of this identity is (123).

• 2. In order to prove (124) we combine equations (102) and (103) to obtain the pointwise estimate

  $$\begin{aligned} \Vert A^{(l)}(t) \Vert _o \leqq C\sum _{k = 1}^MW^{(l)}_k(t) , \quad t \in [0,T]. \end{aligned}$$

  (126)

  Consequently, the flux due to boundary events satisfies the $L^\infty$ estimate

  $$\begin{aligned} \left\| \sum _{l=1}^{M_-} A^{(l)}\dot{L}_l f(t)\right\| _\infty&\leqq \sum _{l=1}^{M_-} \Vert A^{(l)}(t) \Vert _o \dot{L}_l \Vert f(t)\Vert _\infty \\&\leqq C\left( \max _{l \leqq M_-}\sum _{k = 1}^{M} W_k^{(l)}(t) \right) \Vert f(t)\Vert _\infty . \end{aligned}$$

  (127)

The flux due to interior events is controlled in a similar manner. As in (111) we find $\gamma (t) \leqq C$, $t \in [0,T]$. Since $\Vert A^{(0)}\Vert _o \leqq C$, we find

$$\begin{aligned} \Vert \beta \gamma (t) A^{(0)}f (t)\Vert _\infty \leqq C \beta \Vert f(t) \Vert _\infty . \end{aligned}$$

(128)

We combine (129) and (131) to obtain

$$\begin{aligned} \Vert {\varvec{\jmath }}(t) \Vert _\infty \leqq C \varPhi (t)\Vert f(t)\Vert _\infty . \end{aligned}$$

(129)

We now substitute these $L^\infty$ estimates in the solution formula (24) to obtain

$$\begin{aligned}&{\Vert f(t) \Vert _\infty = \sum _{\sigma =1}^M \Vert f_\sigma (\cdot ,t) \Vert _\infty \leqq \sum _{\sigma =1}^M \Vert f_\sigma (\cdot ,0)\Vert _\infty + \int _0^t \Vert j_\sigma (\cdot , \tau ) \Vert _\infty \, \mathrm{d}\tau } \nonumber \\&\quad {\mathop {\leqq }\limits ^{(132)}} \Vert f(0)\Vert _\infty + C \int _0^t \varPhi (\tau ) \Vert f(\tau )\Vert _\infty \, \mathrm{d}\tau . \end{aligned}$$

(130)

An application of Gronwall’s lemma yields (124). $\quad \square$

From here, it is straightforward to obtain several regularity properties for mild solutions.

Theorem 4

The following hold for mild solutions:

• 1. Let $f(x,t; f_0)$ denote the unique mild solution with initial condition $f_0$ for $t <T_*(f_0)$. Then, the map defined by $(f_0,t) \mapsto f(x,t;f_0)$ is continuous in both time and space at all points where the map is well-defined.

• 2. The maximum time of existence $T^*$ for mild solutions depends continuously on initial conditions.

• 3. Let $f_0 \in X \cap C^1([0,\infty )^M)$ with $f'_0 \in X$. Then the mild solution $f(x,t;f_0)$ is differentiable in both space and time for $t \in [0,T^*)$, and satisfies (19).

Proof

We have already shown continuity in t from a contraction mapping theorem. We now show continuous dependence on initial conditions. To show continuity in space, let $f_0^1, f_0^2 \in X$ with mild solutions $f_1(x,t)$ and $f_2(x,t)$. For $f_0^2 \in X$ sufficiently close to $f_0^1$, we may use (24) and (113) to show there exists $t^*>0$ such that for $y(t) = \Vert f^1(t) - f^2(t)\Vert$,

$$\begin{aligned} y(t) \leqq y(0) +C(f_0)\int _0^t y(s)\,\mathrm{d}s, \quad t \in [0,t^*). \end{aligned}$$

(131)

An application of Gronwall’s inequality implies continuous dependence up to time $t^*$. By a standard compactness argument, this may be extended to any [0, T] with $T<T_*(f_0)$. This shows part (1), from which (2) follows immediately.

To prove (3), we show differentiability in space, using the (without loss of generality) right difference quotients

$$\begin{aligned} y_h(t) = \Vert D_h^+(f(x,t))\Vert _\infty =\left\| \frac{f(x+ h,t)-f(x)}{h}\right\| _\infty \end{aligned}$$

(132)

for $h>0$. Similar to Part (1), through (24) and (113) we may show that for $t \in [0,T]$ with $T<T^*$, there exists a finite constant $C(T,f_0) <\infty$

$$\begin{aligned} y_h(t) \leqq C(T, f_0)\Vert D_h f_0(x)\Vert _\infty . \end{aligned}$$

(133)

Since $f'_0 \in L^\infty$, we may take $h\rightarrow 0$ to obtain $\Vert \partial _x f(x,t)\Vert _\infty < \infty .$ The argument for time derivatives is identical. $\quad \square$

For the purposes of demonstrating global existence for PDMPs related to grain coarsening in Section B.1, we mention that characteristic speeds are bounded by $\bar{v} = \max _{s \leqq M} v_s$.

Theorem 5

(Finite speed of propagation) If $f_0$ has support contained in [0, L], then for $0<t<T_*$, the support f(x, t) is contained in $[0, L+\bar{v}t]$.

Proof

The argument is similar to Theorem 4, but now using (24) and Gronwall’s inequality applied to

$$\begin{aligned} g_\sigma (t) = \sup _{x>L+(t-T)\bar{v}} f_\sigma (x,t), \quad \sigma = 1, \ldots , M. \end{aligned}$$

(134)

$\quad \square$

1.1 Properties of Kinetic Equations for Grain Coarsening

1.1.1 Conserved Quantities

Conservation of area and zero polyhedral defect for kinetic equations, defined similarly from (96), are conserved for the kinetic equations when considering initial data which is compactly supported.

Theorem 6

Let $f_0 \in X \cap C_c([0,\infty ))$. Then for $t \in [0,T^*)$,

$$\begin{aligned} P(t){:=}\sum _{n = 2}^M(n-6)F_n(t),\quad A(t) {:=} \sum _{n = 2}^M\int _0^\infty a\cdot f_n(a,t)\,\mathrm{d}a, \end{aligned}$$

(135)

if $P(0) = 0$ and $A(0) = A$, then $P(t) = 0$ and $A(t) = A$ for all $t>0$.

Proof

Suppose $f_0^j \in X$ are differentiable with compact support for $j\in \mathbb N$, and $f_0^j\rightarrow f_0$ in X as $j\rightarrow \infty$. By continuous dependence of parameters, for all $t>0$, solutions $f^j(a,t) \rightarrow f(a,t)$ in X. Thus it is sufficient to show (138) for classical solutions. To do so, we integrate (43) and sum over all species to obtain

$$\begin{aligned} \sum _{n = 2}^M (n-6)F_n(t) = -\sum _{n = 2}^{M_-} (n-6)^2f_n(0,t)+\sum _{n = 2}^M (n-6) \int _0^\infty j_n(a,t)\,\mathrm{d}a. \end{aligned}$$

(136)

The left hand side of (139) is the polyhedral defect, and a straightforward computation using (44)–(47) shows that

$$\begin{aligned} \sum _{n = 2}^M (n-6) \int _0^\infty j_n(a,t)\,\mathrm{d}a = \sum _{n = 2}^{M_-} (n-6)^2f_n(0,t), \end{aligned}$$

(137)

which shows the conservation of polyhedral defect.

To show the conservation of area, we find through an integration by parts of (43) with (125) that

$$\begin{aligned} \frac{\mathrm{d}A(t)}{\mathrm{d}t}&= \sum _{n = 2}^M (n-6)\int _0^\infty a \cdot \partial _x f_n(a,t)\,\mathrm{d}a+ \int _0^\infty a\sum _{n=2}^M j_n(a,t)\,\mathrm{d}a \end{aligned}$$

(138)

$$\begin{aligned}&= P(t). \end{aligned}$$

(139)

For initial conditions with zero polyhedral defect, conservation of area then follows. $\quad \square$

The conservation of total area is sufficient to show global existence under a wide choice of weights.

Theorem 7

(Global existence) Suppose $M \ne 10$, and $w_k^{(l)}>0$, for $(k,l) \in \{2, \ldots M\} \times \{2\times M_-\}\backslash \{\cup _{i \in \{0,2,3,4,5\}} (2,i), (3,2), (M,5), (M,0)\}.$ For nonzero initial conditions with zero polyhedral defect, the maximum interval of existence for mild solutions is infinite.

Remark 1

We require $M\ne 10$ due to the impossibility of edge deletion in the pathological case of initial conditions $F_2(0) = F_{10} (0) = 1/2$, and $F_\sigma = 0$ for $\sigma \notin \{2,10\}$.

Proof

Suppose for the sake of contradiction, that a finite maximum interval of existence $T_*<\infty$. From Theorem 1, $\sum _{\sigma =2}^M w^{(l)}_\sigma F_\sigma (T_*^-) = 0$ for some $l = 1, \ldots , M^-$. We can check directly that from the conditions on $w_k^{(l)}$, using zero polyhedral defect, that the stronger condition of $F_\sigma (T_*^-) = 0$ must hold for all $\sigma = 2, \ldots , M$. Since $f(x,t) \in C_c^1(\mathbb {R}_+)$ from Theorem 5, there exists $L>0$ such that $f(x,t) = 0$ for $x>L$ and $0<t<T_*$. This implies, however, that

$$\begin{aligned} A(T_*^-) =\sum _{\sigma = 2}^M \int _0^\infty a\cdot f_\sigma (a,T_*^-)\,\mathrm{d}a \leqq L\sum _{\sigma = 2}^M F_\sigma (T_*^-) = 0, \end{aligned}$$

(140)

which is in contradiction to the conservation of total area. $\quad \square$