The continuous collision-induced nonlinear fragmentation equation with non-integrable fragment daughter distributions

2024/01/19 by Giri, Ankik Kumar, Jaiswal, Ram Gopal, Laurençot, Philippe · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.10575

Abstract

Existence, non-existence, and uniqueness of mass-conserving weak solutions to the continuous collision-induced nonlinear fragmentation equations are established for the collision kernels Φ satisfying Φ(x₁,x₂)=x₁^λ₁ x₂^λ₂+x₂^λ₁ x₁^λ₂, (x₁,x₂)∈(0,∞)², with λ₁ ≤ λ₂≤ 1, and non-integrable fragment daughter distributions. In particular, global existence of mass-conserving weak solutions is shown when 1≤λ:=λ₁+λ₂≤2 with λ₁≥ k₀, the parameter k₀∈(0,1) being related to the non-integrability of the fragment daughter distribution. The existence of at least one mass-conserving weak solution is also demonstrated when 2k₀ ≤ λ< 1 with λ₁≥ k₀ but its maximal existence time is shown to be finite. Uniqueness is also established in both cases. The last result deals with the non-existence of mass-conserving weak solutions, even on a small time interval, for power law fragment daughter distribution when λ₁

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