Singular perturbations of finite dimensional gradient flows

Chiara Zanini

doi:10.3934/dcds.2007.18.657

Abstract

In this paper we give a description of the asymptotic behavior, as $\varepsilon\to 0$, of the $\varepsilon$-gradient flow in the finite dimensional case. Under very general assumptions, we prove that it converges to an evolution obtained by connecting some smooth branches of solutions to the equilibrium equation (slow dynamics) through some heteroclinic solutions of the gradient flow (fast dynamics).

Highlights

The study of quasistatic rate-independent evolutionary models may lead to consider gradient flow-type problems
In this paper we study a model case in which the main simplifying assumption is that the dimension of the space X is finite
If ξ is a degenerate critical point of f (t, ·) satisfying all conditions considered above, we prove that there is a unique heteroclinic solution v(s) of v (s) = −∇xf (t, v(s)) issuing from the degenerate critical point ξ, and we suppose that v tends, as s → +∞, to a nondegenerate critical point y of f (t, ·), with ∇2xf (t, y) positive definite

Summary

Introduction

The study of quasistatic rate-independent evolutionary models may lead to consider gradient flow-type problems. For every τ ∈ [0, T ] and for every degenerate critical point ξ ∈ Rn for f (τ, ·), such that ∇2xf (τ, ξ) is positive semidefinite, there exists l ∈ Rn \ {0} such that the following conditions are satisfied:. Using differential inequalities (see, e.g., [5, Theorem 6.1]), we deduce that there exists ε0 = ε0(M) such that problem (2.10) admits a unique solution satisfying the limit condition (2.12), for every ε < ε0. For every (τ, ξ) ∈ [0, T ] × Rn such that ξ is a degenerate critical point for f (τ, ·), satisfying the assumptions of Lemma 2.5, let w be the unique solution of (2.5) corresponding to τ and ξ.

Preliminary results

Proof of the main result