THE DISCRETE AND CONTINUOUS REDNER-BENAVRAHAM- KAHNG COAGULATION MODELS: WELLPOSEDNESS, LARGE-TIME BEHAVIOUR AND THEIR CONNECTION

Abstract

This thesis is devoted to a cluster-eating model with a different type of coagulation. The basic mechanism of such models is binary cluster reactions, in which particles collide and merge to produce a single particle. However, the size of the resulting particle has been observed to decrease. To understand the modelling of this model, consider a closed system of particles colliding in binary collisions where the smaller particle totally annihilates and destroys the identical quantity of the larger particle. This coagulation model is known as the Redner-Ben-Avraham-Kahng (RBK) coagulation model. In the most fundamental coagulation models, cluster particles are distinguished by their cluster size (or volume), which can be continuous, or discrete, depending on the physical circumstances. In this thesis, we explore some mathematical study of both discrete and continuous RBK coagulation models as well as the connection between them. On the basis of this, we divide this thesis into three parts. The first part of the thesis is dedicated to the continuous RBK coagulation model. We demonstrate the well-posedness and large-time behaviour of the solutions under some specified conditions on the coagulation kernel. Furthermore, we look into the existence of solutions for vast classes of non-singular unbounded coagulation kernels. Next, we turn to the discrete RBK coagulation model. In this part, the global weak solution to the discrete RBK model is established for large classes of coagulation kernels. This research is the generalization of a portion of the work done by da Costa et al. in [31]. Finally, after studying discrete and continuous RBK models independently, we establish a connection between them. In the last part of the thesis, we show the convergence of weak solutions to the discrete RBK coagulation model towards the weak solution of the continuous RBK coagulation model by taking into account certain specific growth on the coagulation kernels.