On the asymptotic regime of a model for friction mediated by transient elastic linkages

Tear-off versus global existence for a structured model of adhesion mediated by transient elastic linkages

“…We consider a microscopic model for friction mediated by transient elastic linkages introduced in [9,10]. In the present study we prove existence and uniqueness of a solution to the coupled system under weaker hypotheses.…”

“…The asymptotic limit is obtained assuming that the rate of linkage turnover becomes great as well as the stiffness of the bonds (typically as O(1/ε)). The rigorous justification of the limit ε → 0 is the ultimate goal of our investigations [9,10]. Nevertheless the highly non-linear nature of the delayed model leads to consider already the case of a fixed value of ε.…”

“…Their age distribution is denoted by ρ ε = ρ ε (t, a) where a ≥ 0 denotes the age of linkages and t ≥ 0 denotes time. Here we treat ε as a fixed constant, which we keep in our notation to maintain consistency with previous studies, in which we were analyzing the convergence with respect to ε [9,10].…”

“…In [10] we had shown that the system (1.3) is equivalent to an integral equation for the position of the binding site z ε = z ε (t) itself [9], namely…”

“…On the other hand, in [10] it has turned out to be beneficial to work on the system (1.1), (1.3) instead, since it allowed to derive powerful a-priori estimates on u ε . The analysis in the older studies [9] and [10] relied on the existence of an upper bound ζ max of the function ζ. In [10], for fixed ε, we prove existence and uniqueness of weak solutions of the coupled system (1.1), (1.2), (1.3).…”