Mass-conserving weak solutions to the collision-induced breakage equation with and without mass transfer

Abstract: Breakage or fragmentation is a key process observed in many scientific and engineering fields, such as astrophysics (planet formation) and atmospheric science (raindrop breakup). Broadly, breakage processes can be categorized into two types based on particle dynamics: linear breakage and collision-induced (or nonlinear) breakage. Linear breakage typically arises spontaneously or due to external forces acting independently on particles, whereas collision-induced breakage results from interactions between pairs of particles. In certain scenarios, such collisions may involve mass transfer, potentially leading to the formation of a partic1le larger than the colliding particles. Consequently, the study of collision-induced breakage naturally divides into two cases: with and without mass transfer.

In this talk, we present results concerning the existence, uniqueness, and nonexistence of mass conserving weak solutions to the continuous collision-induced breakage equation. We begin with the case without mass transfer, in which we establish both global and local existence and uniqueness results for mass-conserving weak solutions under a class of collision kernels and non-integrable daughter distribution functions. Additionally, we identify conditions under which such solutions do not exist, even for arbitrarily short time intervals, particularly when the daughter distribution follows a power-law form. We then turn to the case with mass transfer, proving the existence and uniqueness of weak solutions for kernels exhibiting subquadratic growth and integrable daughter distributions.

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