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

Abstract

Abstract. 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 theory we present covers the case where the off-rate of linkages is unbounded but increasing at most linearly with respect to the mechanical load.The time of existence is typically bounded and corresponds to tear-off where the moving binding site does not have any bonds with the substrate. However, under additional assumptions on the external force we prove global in time existence of a solution that consequently stays attached to the substrate.

Space dependent adhesion forces mediated by transient elastic linkages: New convergence and global existence results

“…This parameter is assumed to be small and the fact that the elasticity is O(ε −1 ), is a scaling assumption required for a non-vanishing effect of adhesions in the limit ε → 0. Our works construct various tools in order to handle rigorously this asymptotic [14,15,16,17]. In addition, concerning adhesion forces, a similar Ansatz was performed formally in a somehow different mechanical setting in [23,8].…”

“…In a second step we consider, for a fixed ε, existence and uniqueness of the fully coupled problem where ζ is a Lipschitz function of u ε , the elongation. In [16], this model was considered at a single point. Here the presence of the space variable greatly complexifies the mathematical setting.…”

“…Proof. We proceed as in Theorem 3.2 [16]. Indeed, for a given w, w solves (7.24) with ζ(w) as the death rate.…”

“…The second estimate is due to the Sobolev embedding H 1 0 (Ω) ⊂ C(Ω) holding when n = 1, while the third one is the consequence of the Lax-Milgram Theorem. In order to obtain the last estimate above, we follow the steps in part b) of the proof of Theorem 3.2 [16], the constraction follows up to a time T small enough. Then it is possbile to show (see part c) in the proof of Theorem 3.1 [16]) that the contraction time does not depend on the initial data but on k, one concludes the global existence result.…”

“…In order to obtain the last estimate above, we follow the steps in part b) of the proof of Theorem 3.2 [16], the constraction follows up to a time T small enough. Then it is possbile to show (see part c) in the proof of Theorem 3.1 [16]) that the contraction time does not depend on the initial data but on k, one concludes the global existence result. Theorem 4.3 holds as well for ( w ,w), the solution of the truncated problem.…”