Size characteristics, positivity, and discrete number and size-moment balances

In [F2], the coefficient of $M_j^k$ outside the division source is $1-\mathrm{\Delta }t(g(x_j)\mathrm{/}{h}_j+d_j)$, which is nonnegative by the CFL. For $j\ge 2$ the incoming transport term has coefficient $\mathrm{\Delta }tg(x_{j-1})\mathrm{/}{h}_{j-1}\ge 0$; for $j=1$ the incoming flux is $F_0=0$. Every division term has coefficient $\mathrm{\Delta }t{A}_{j\mathrm{\ell }}\ge 0$. Hence each new mass is a sum of nonnegative terms when [A1] holds. Induction on $k$ proves $M_j^k\ge 0$.

Sum [F2] over $j$. The finite flux sum telescopes to $F_J-F_0=F_J$ because $F_0=0$. Interchanging the finite sums in the kernel source gives ${\sum }_{\mathrm{\ell }}({\sum }_j{A}_{j\mathrm{\ell }})M_{\mathrm{\ell }}={\sum }_{\mathrm{\ell }}{k}_{0,\mathrm{\ell }}^h{M}_{\mathrm{\ell }}$. Subtracting $N_h^k$ yields the number identity.

Multiply [F2] by $x_j^c$ and sum over $j$. For the flux term, finite summation by parts gives ${\sum }_j{x}_j^c(F_j-F_{j-1})=x_J^c{F}_J-{\sum }_{j=1}^{J-1}(x_{j+1}^c-x_j^c)F_j$ since $F_0=0$. The loss term is $-{\sum }_j{x}_j^c{d}_j{M}_j$, and the source term is ${\sum }_{\mathrm{\ell }}({\sum }_j{x}_j^c{A}_{j\mathrm{\ell }})M_{\mathrm{\ell }}={\sum }_{\mathrm{\ell }}{k}_{1,\mathrm{\ell }}^h{M}_{\mathrm{\ell }}$. Rearranging proves the weighted-moment identity and the stated condition for cancellation.

Let $M=\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}$ and let $L$ be a Lipschitz constant for $g$ on $[0,X]$. The projection ${\mathrm{\Pi }}_{[0,X]}$ is $1$-Lipschitz, so the extension $\tilde{g}=g\circ {\mathrm{\Pi }}_{[0,X]}$ is globally bounded by $M$ and globally Lipschitz with constant $L$ on the open domain $\mathbb{R}$. Since $g\ge g_{\min }>0$, choose $h_0>0$ with $h_{0}M\le 1$ and $L{h}_0<1$. For every starting time $t_0\in \mathbb{R}$ and state $z\in \mathbb{R}$, the cylinder $[t_0-h_0,t_0+h_0]\times [z-1,z+1]$ lies in $\mathbb{R}\times \mathbb{R}$, and [F3] gives a unique solution through $(t_0,z)$ on that interval; the same bounds apply after every restart. Restricting and concatenating at most $\lceil T\mathrm{/}{h}_0\rceil$ such forward segments gives a solution on $[0,T]$, and the uniqueness clause of [F3] makes it unique there. Along this solution ${\chi }_y^{\mathrm{\prime }}(s)=\tilde{g}({\chi }_y(s))\ge g_{\min }$, so it is strictly increasing. Starting from $y\in [0,X]$, it stays in $[0,X]$ through its first hitting of $X$, where $\tilde{g}=g$; that restricted segment is the model characteristic.