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- ItemBilinear coagulation equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Heydecker, Daniel; Patterson, Robert I.A.We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) · Aπ (x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time tg ∈ (0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.
- ItemWeak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Lasarzik, Robert; Rocca, Elisabetta; Schimperna, GiulioIn this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.