Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion (Q963047)

The discrete diffusive coagulation-fragmentation equations read \[ \partial_t c_i - d_i \Delta c_i = Q_i + F_i , \quad (t,x)\in (0,\infty)\times\Omega, \quad i\geq 1, \] supplemented with homogeneous Neumann boundary conditions and initial data, the reaction terms being given by \[ Q_i = \frac{1}{2} \sum_{j=1}^{i-1} a_{j,i-j} c_j c_{i-j} - c_i \sum_{j=1}^\infty a_{i,j} c_j , \quad F_i = \sum_{j=1}^\infty B_{i+j} \beta_{i+j,i} c_{i+j} - B_i c_i. \] Here, \(\Omega\) is a bounded open subset of \(\mathbb R^N\), \(N\geq 1\), and the parameters \(B_i\), \(\beta_{i,j}\), \(a_{i,j}\), and \(d_i\) represent the total rate of fragmentation of particles of size \(i\), the average number of particles of size \(j\) resulting from the fragmentation of a particle of size \(i\), the coagulation rate of particles of size \(i\) and \(j\), and the diffusion coefficient of particles of size \(i\), respectively. The conservation of mass during fragmentation is assumed and requires \[ \sum_{j=1}^{i-1} j \beta_{i,j} = i , \quad i\geq 1. \] Assuming that \(0<d\leq d_i \leq D\) for some positive numbers \(d\) and \(D\), a duality technique is used to show that \(\rho=\sum i c_i\) is bounded in \(L^2((0,T)\times\Omega)\) for all \(T>0\). This estimate is then used to shown the expected mass conservation \(\|\rho(t)\|_1=\|\rho(0)\|_1\), \(t\geq 0\), for a large class of coagulation kernels satisfying \(a_{i,j}\leq K(i+j)\) including \(a_{i,j} =i^\alpha j^\beta + i^\beta j^\alpha\) for \(0<\alpha \leq \beta \leq 1-\alpha\), the kernel \(a_{i,j}=i+j\) being excluded. A similar \(L^2\)-estimate on \(\rho\) is valid for the discrete diffusive coagulation equation with collisional breakage and leads to an existence result in one space dimension \(N=1\).