Research Output from the Work Packages (explained below)
-Esenturk E., Mathematical theory of exchange-driven growth, Nonlinearity, (31), 3460-3483, (2018)
-Esenturk E., Connaughton C., Exchange-driven growth with source and sink, preprint
-Esenturk E., Velazquez J., Large time behavior of exchange-driven growth, submitted
-Esenturk E., Caginalp G., Asymptotic behavior of non-linear differential aggregation equations, in prep.
-Esenturk E., Connaughton C, Tse C., Stringer J., Smith P., Ta D., Coffee stain effect revisited, in prep.
Revised Work Package 2
In this work package (published), I obtained the first rigorous results for the fundamental mathematical properties of the mean field rate equations of this process. I showed that two different types of behavior arise depending on whether interaction kernel is symmetric or not. For the non-symmetric case, global existence of solutions and uniqueness of solutions were shown for kernels not growing faster than linear. On the other hand, for symmetric kernels it was proven that solutions exist locally for kernels growing faster than the 3/2th power (but slower than the 2nd power) of the cluster size. Hence, I identified the gelling regime for faster growing kernels while existence is completely lost for kernels growing faster than quadratic.
Revised Work Package 3
In this Work Package (submitted to journal) we studied the large time behavior of EDG equations. We rigorously proved, for a large class of kernels, that the EDG system approaches to equilibrium. For a class of kernels satisfying certain monotonicity conditions or for systems with small mass we were able to provide explicit convergence rates.
New Work Package 5
In this completed study (preprint), we presented results on exchange-driven process with the possibility of breakages non-zero clusters to form one-clusters (a cluster of single unit element). The effect is contingent and proportional to the available “active zero clusters”, that is the available volume to be occupied. We showed that for systems with active zero clusters in isolated systems, the classical indefinite growth trend is broken and the system approaches to an equilibrium even without the inclusion of a sink.
Work Package 4
This project is jointly done with an experimental group from the mechanical engineering department of Sheffield University (Patrick Smith’s group). I have been developing models for the clumping of ink particles during the process of evaporation of sessile droplet on a substrate. This is still a work in progress. The goal is to test the developed model against experimental measurements
Revised Work Package 1
This is the same work package proposed in the original program. I have had preliminary findings, but this is still a work in progress.
