Continuous-size coagulation-fragmentation weak model

Fix $a>0$ and $E=[a,\mathrm{\infty })$. A continuous-size coagulation-fragmentation model consists of a symmetric nonnegative aggregation kernel $K:E^2\to [0,K_{\ast }]$ and finite constants $L_K,L_{\beta }\ge 0$ such that $\mathrm{\mid }K(x,y)-K(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\mid }\le L_K(\mathrm{\mid }x-x^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }y-y^{\mathrm{\prime }}\mathrm{\mid }),\mathrm{\mid }\beta (x)-\beta (y)\mathrm{\mid }\le L_{\beta }\mathrm{\mid }x-y\mathrm{\mid },$ where $\beta :E\to [0,{\beta }_{\ast }]$ is the fragmentation rate. It also has a Borel kernel $x\mapsto B_x$ of finite nonnegative daughter-count measures on $E$. Fix a finite constant $D\ge 1$ and require $B_x(E)\le D$, $\mathrm{supp}(B_x)\subseteq [a,x]$, and ${\int }_{E}z{B}_x(dz)=x$. For $x<2a$, take $\beta (x)=0$ and $B_x={\delta }_x$. Assume $d_{BL}(B_x,B_y)\le L_B\mathrm{\mid }x-y\mathrm{\mid }$, where $d_{BL}$ is defined in Bounded-Lipschitz discrepancy for finite measures.

For signed $\mu ,\nu \in {\mathcal{M}}_w(E)$ define the symmetric bilinear coagulation operator and linear fragmentation operator by

$\mathcal{C}(\mu ,\nu )=\frac{1}{2}{\iint }_{E^2}K(x,y)({\delta }_{x+y}-{\delta }_x-{\delta }_y)\mu (dx)\nu (dy),\mathcal{F}\mu ={\int }_E\beta (x)(B_x-{\delta }_x)\mu (dx)\mathrm{.}$

On a time interval $[0,T]$, a weak solution is a path $\mu \in C([0,T];{\mathcal{M}}_w(E))$ of positive measures satisfying the following identity in integral form for each continuous $\phi$ with $\mathrm{\mid }\phi (x)\mathrm{\mid }\le C(1+x^2)$:

$\frac{d}{dt}{\int }_E\phi d{\mu }_t=\frac{1}{2}{\iint }_{E^2}K(x,y)[\phi (x+y)-\phi (x)-\phi (y)]{\mu }_t(dx){\mu }_t(dy)+{\int }_E\beta (x)[{\int }_E\phi (z)B_x(dz)-\phi (x)]{\mu }_t(dx)$

Equivalently, $\dot{\mu }=\mathcal{C}(\mu ,\mu )+\mathcal{F}\mu$ in ${\mathcal{M}}_w(E)$. The first term counts ordered coagulation pairs with factor $1\mathrm{/}2$. The second term removes one parent at each fragmentation event and adds its expected daughter measure; its first-moment normalization is the displayed identity. A binary split $x\mapsto (u,x-u)$ has $B_x={\delta }_u+{\delta }_{x-u}$.