The multidimensional coagulation and collisional breakage equation with unbounded kernel

Abstract

Many scientific and technical fields have benefited from the understanding of the Coagulation and Collisional Breakage Equation (CCBE). In one-dimensional CCBE, only one property of the particle (like size or mass of the particle) is considered. However, there are other factors (like including volume, enthalpy, porosity, mole number, binder content, and more) that affect how particle ensembles behave. Therefore the study of multidimensional CCBE is more accurate. For the multi-dimensional (CCBE), we denote the property characteristics vector as x → = ( x 1 , x 2 , … , x d ) ∈ R + d . In this research work, the existence of a continuous solution is investigated where collision rate satisfies C ( x → , y → ) ≤ c ∏ i = 1 d ( 1 + x i ) ν ( 1 + y i ) ν , where c>0 is a constant, 0 ≤ ν ≤ θ and the coagulation kernel satisfies the following, K ( x → , y → ) ≤ k ∏ i = 1 d ( 1 + x i ) α ( 1 + y i ) α , where k is a positive constant, 0 ≤ α ≤ θ and 0 < θ ≤ 1 . Additionally, the uniqueness of the solution is shown. To prove the result, we have defined a new norm and examined the equicontinuity with respect to time and the property characteristics vector and Arzelà–Ascoli theorem is used.