Abstract

Shrinkage of large particles, either through depolymerisation (i.e. progressive shortening) or through fragmentation (breakage into smaller pieces) may be modelled by discrete equations, of Becker-Döring type, or by continuous ones. In many applications, the dynamic nature of the experiments, as well as their nanoscale,  makes it  challenging to estimate their features. In this talk, we review  inverse problems linked to the estimation of the initial size-distribution and of the fragmentation characteristics.

Departing from a model of discrete depolymerisation, we first evaluate the impact of using continuous approximations to solve the initial-state estimation problem. At second order, the asymptotic model becomes an advection-diffusion equation, where the diffusion is a corrective term. This approximation is much more accurate, but we face a classical accuracy versus stability trade-off: the inverse reconstruction reveals to be severely ill-posed. Thanks to Carleman inequalities and to log-convexity estimates, we prove observability results and error estimates for a Tikhonov regularisation. This is a joint work with P. Moireau, inspired by experiments by H. Rezaei.

To estimate the fragmentation kernel in experiments of polymer breakage, we propose several approaches based on the continuous fragmentation equation, studying and making use either of the long-term, the transient or the short-term dynamics. Error estimates in Bounded Lipshitz norm are obtained. This is a joint work with M. Escobedo and M. Tournus, based on biological questions and experiments of W.F. Xue.