On the discrete Safronov-Dubovskii coagulation equation: well-posedness, mass-conservation and asymptotic behaviour

Abstract

The existence and uniqueness of global classical solutions to the Safronov-Dubovskii coagulation equation are established for the coagulation kernels satisfying the linear growth condition i.e. γ i,j ≤ A(i δ + j δ ) for some δ ∈ [0, 1] which extends the results obtained in [6]. In particular, the initial condition may have an infinite second moment as long as δ < 1. Moreover, the solution constructed herein is shown to be mass-conserving. In the end, the continuous dependence on the initial data and the large-time behaviour of solutions are also addressed.

Global existence to the discrete Safronov-Dubovskiǐ coagulation equations and failure of mass-conservation

“…This, however, is not always the case. In [6,7,19,22], it is observed that for the discrete smoluchowski coagulation equations (1.1)-(1.2), the mass conservation property fails for coagulation kernels of the form γ i,j ≥ (ij) α 0 2 , where α 0 ∈ (1,2]. It is worth noting that gelation also occurs in the DSDC equations, and we get the same result as in the discrete Smoluchowski coagulation equation.…”

“…Unfortunately, the coagulation kernel Λ i,j = i + j is not included in either of the classes (2.2) or (2.3). Although the existence result for this kernel is available in the literature, see [1,8,15].…”

“…Additionally, for linear coagulation kernels and kernels fulfilling min{i, j}Λ i,j ≤ (i + j) for every i, j ≥ 1, the existence of weak solutions has been demonstrated in [15] and [16] respectively, whereas the uniqueness result is identical to [8]. The results of [8] have recently been improved, and various additional interesting results have been established in [1].…”