Asymptotic behaviour of liquid–liquid dispersions

Abstract

Based on earlier results on existence, we study the asymptotic behaviour of solutions to the coalescence-breakage equations, including the volume-scattering phenomenon and high-energy collisions. The solutions are shown to converge towards one particular equilibrium, provided the kernels satisfy a kind of reversibility. We also derive stability of these equilibria in a suitable topology.

Local and global strong solutions to continuous coagulation–fragmentation equations with diffusion

“…From (i) of the proof of Theorem 7 we infer the existence of solutionsū = u(·; u 0 ) | [0,T ] andv = u(·; v 0 ) | [0,T ] , both belonging to X T , with initial value u 0 and v 0 , respectively, such that ū − Uu 0 X T 3R and v − Uv 0 X T 3R, whence ū X T + v X T 8R. Since, due to (26),…”

“…Then, since u X T 4R for each u ∈ B T , (26) and (27) imply that F : B T → B T is a contraction. Therefore, there exists a uniqueū ∈ B T with F (ū) =ū, that is,ū ∈ C ((0, T ], L 2p ) and…”

“…The daughter particles are then distributed according to s (y + y , y ) 0. For a more detailed physical, mathematical, and numerical discussion of this scattering phenomenon, we refer to [7,11,18,25,26,28].…”

On the constant kernel Smoluchowski equation: fast algorithm for solution with arbitrary initial conditions

“…The coagulation phenomena is of paramount importance for aerosol processes both industrial and atmospheric [2], polymer processes [3] and colloids processes [4]. In addition, the above equation has been used to describe the evolution of bubble/droplet size distribution in gas/liquid in liquid dispersions, respectively [5]. Several variants of the equation have been extensively studied from the fundamental point of view in physics literature [6][7][8].…”

On a new model for continuous coalescence and breakage processes with diffusion

“…Well-posedness of the spatially homogeneous version of ( * ) is studied in [41] by the author (there the case Q ≡ 1 − P is treated), whereas certain aspects of asymptotic behaviour are investigated in [43].…”