Regularity and approximation of strong solutions to rate-independent systems

Abstract

Rate-independent systems arise in a number of applications. Usually, weak solutions to such problems with potentially very low regularity are considered, requiring mathematical techniques capable of handling nonsmooth functions. In this work we prove the existence of Hölder-regular strong solutions for a class of rate-independent systems. We also establish additional higher regularity results that guarantee the uniqueness of strong solutions. The proof proceeds via a time-discrete Rothe approximation and careful elliptic regularity estimates depending in a quantitative way on the (local) convexity of the potential featuring in the model. In the second part of the paper we show that our strong solutions may be approximated by a fully discrete numerical scheme based on a spatial finite element discretization, whose rate of convergence is consistent with the regularity of strong solutions whose existence and uniqueness are established.MSC (2010): 49J40 (primary); 47J20, 47J40, 74H30.

Two-speed solutions to non-convex rate-independent systems

“…Remark 3.9 (Uniqueness). We note in passing that the estimate above also improves the uniqueness result obtained in [12]. Indeed, it implies that the uniqueness class for convex energies can be extended to weak solutions in BV(0, T ; L 1 (Ω; R m )) satisfying the stability estimate and an energy inequality.…”

“…We refer to [12,Remark 1.1] for some comments on the regularity classes in which we look for solutions. If (2.4) is satisfied at t ∈ (0, T ), we say that u is a strong solution at t to (2.3).…”

“…for all ψ ∈ W 1,2 0 (Ω; R m ), the existence and uniqueness of strong solutions is known (provided the given data satisfies the necessary regularity), see [12]. In the present paper we are concerned with non-convex W , in which case solutions might have jumps.…”

“…The functions θ → E (τ j +λ j θ , v j (θ )) converge uniformly on bounded sets to θ → E (t, v(θ )) as j → ∞, which is implied by the convergence v j v in W 1,2 loc (−∞, ∞; W 1,2 (Ω; R m )) together with the L ∞ loc (−∞, ∞; W 2,2 (Ω; R m ))-boundedness of v j , see (5.3). Indeed, the first convergence gives by the Aubin-Lions compactness lemma that v j → v in L ∞ loc (−∞, ∞; L 2 (Ω; R m )), which together with the boundedness in L ∞ loc (−∞, ∞; W 2,2 (Ω; R m )) yields by interpolation (see, e.g., [12,Remark 3.2] together with the Arzela-Ascoli theorem)…”

“…On the other hand, [12] developed a solution theory of strong solutions and derived essentially optimal regularity estimates in the case of "functionally convex" W 0 . For fully non-convex energies, the validity of the regularity results of [12] breaks down at jump points and a new analysis is necessary. We also refer to [9,10,14] for other regularity results in the theory of rate-independent systems.…”