A Note on the Convergence of Singularly Perturbed Second Order Potential-Type Equations

Abstract

In this paper we study the limit as ε → 0 of the singularly perturbed second orderis a potential. We assume that u 0 (t) is one of its equilibrium points such that ∇ x V (t, u 0 (t)) = 0 and ∇ 2x V (t, u 0 (t)) > 0. We find that, under suitable initial data, the solutions u ε converge uniformly to u 0 , by imposing mild hypotheses on V . A counterexample shows that they cannot be weakened.

Existence and uniqueness of dynamic evolutions for a one-dimensional debonding model with damping

“…The issue of the quasistatic limit will be instead investigate in a future work. The choice of analysing such a damped problem is motivated by several works on different dynamic evolutions where the addition of a suitable friction term in the equation makes the convergence towards quasistatic solutions true, see for instance [15,18] for some damage models, or [1,17] for a finite dimensional setting.…”

A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension

“…Under suitable assumptions on F (t, x), it is proven that, when ε → 0, it holds (u ε (t), εBu ε (t)) → (u(t), 0), where u is a piecewise-continuous function solving ∇ x F (t, u(t)) = 0 (1.4) at every continuity time t. Moreover, the trajectories of the system at the jump times t i are described through the autonomous second order system It is worth noting that the presence of the damping term εBu ε is crucial for obtaining the above results, as it also will be in our setting. There are indeed examples of singularly perturbed second order potential-type equations (with vanishing inertial term), such that the dynamic solutions do not converge to equilibria, while the formal limit equation is (1.2) (see, e.g., [13]).…”

On the Approximation of Quasistatic Evolutions for the Debonding of a Thin Film via Vanishing Inertia and Viscosity

“…perfect plasticity and to [16], [17] for the undamped version of the debonding model we analyse in this work. The issue of quasistatic limit has also been studied in a finite-dimensional setting where, starting from the works [1], [28] and with the contribution of [22], an almost complete understanding on the topic has been reached in [26]. A common feature appearing both in finite both in infinite dimension is the validation of the quasistatic approximation only in presence of a damping term in the dynamic model.…”