Global existence for the discrete diffusive coagulation-fragmentation equations in $L^1$

Abstract

Existence of global weak solutions to the discrete coagulation-fragmentation equations with diffusion is proved under general assumptions on the coagulation and fragmentation coefficients. Unlike previous works requiring L ∞ -estimates, an L 1 -approach is developed here which relies on weak and strong compactness methods in L 1 .

Moment Bounds for the Smoluchowski Equation and their Consequences

“…the global existence of a weak solution of (1.1) has been established in [10], which work includes fragmentation in the equations. From the physically reasonable assumptions that d(·) is uniformly bounded and r(n) = o(n), we see that (1.3) is satisfied in dimension d = 3 by choices of α satisfying (1.2).…”

Smoothness of moments of the solutions of discrete coagulation equations with diffusion

“…This ensures thatṽ, and therefore v, are nonnegative. Multiplying (19) by the solution v of the dual problem and integrating on T , we end up with…”

“…The following result, which is a direct application of [19,Theorem 3], states that we can obtain weak solution of (1), (2) from the truncated systems (9), (10). We also refer to [29,30].…”

“…The conditions (3) are sufficient to provide the existence of global L 1 -weak solutions (with nonnegative concentrations) to system (1), (2) (for which the below estimate (8) on the mass holds), when suitable nonnegative initial data are considered, see [19]. Thanks to the symmetry assumption on the coagulation coefficients, we can write (at a formal level) the following weak formulation of the coagulation operator: for any test-sequence (ϕ i ) i∈N * , we have…”

“…Then, using some compactness arguments, one extracts a solution of the limiting system (1), (2), again see for instance [19]. In this work, for any n ∈ N * , we define c n = c n 1 , .…”