Convergence of Metric Space-Valued Bv Functions

Fleischer, Isidore; Porter, John E.

doi:10.2307/44154127

Cited by 17 publications

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“…On the other hand, a sequence of Ê d -valued measures with uniformly bounded total variation, converging in Monge-Kantorovich norm, has as limit again a measure; see Theorem 3.2 in [51]. We now apply Helly's theorem in the form of Theorem 2.3 in [43] to obtain (6.12).…”

Section: Interpolation In Timementioning

confidence: 99%

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A variational time discretization for compressible Euler equations

Cavalletti

Sedjro²,

Westdickenberg

2019

Trans. Amer. Math. Soc.

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We introduce a variational time discretization for the multi-dimensional gas dynamics equations, in the spirit of minimizing movements for curves of maximal slope. Each timestep requires the minimization of a functional measuring the acceleration of fluid elements, over the cone of monotone transport maps. We prove convergence to measure-valued solutions for the pressureless gas dynamics and the compressible Euler equations. For one space dimension, we obtain sticky particle solutions for the pressureless case.Density and pressure define the specific thermodynamical entropy, given asin the case of polytropic gases. We assume thatso that the entropy density σ = ̺S is well-defined as a measure. Definition 1.1 (Internal Energy). Let U (r, S) := κe S r γ for all r 0 and S ∈ Ê, where κ > 0 and γ > 1 are constants. Then we define the internal energy U[̺, σ] :=   ˆÊ d U r(x), S(x) dx if ̺ = rL d and σ = ̺S, ∞ otherwise, for all pairs of measures (̺, σ) ∈ P(Ê d ) × M + (Ê d ).Since we are only interested in solutions with finite energy, the density must be absolutely continuous with respect to the Lebesgue measure, thus ̺(t, ·) = r(t, ·) L d . In this case, we define P (r, S) = U ′ (r, S)r − U (r, S) (here ′ denotes differentiation with respect to r), and the pressure term in (1.13) takes the form p(t, ·) = P r(t, ·), S(t, ·) L d for all t ∈ [0, ∞).(1.7)Moreover, combining (1.6) and (1.7) with (1.1), we obtain that(1.8)Equivalently, the specific entropy S must be constant along characteristics:(1.9)Formally, system (1.1) is equivalent to the one where the energy equation is replaced by (1.8) (or even (1.9)). But since the solutions to the compressible Euler equationsthe Lipschitz seminorm, with W 2 the Wasserstein distance; see (2.1).Definition 1.3. We denote by M Ent (Ê d ) the space of non-negative Borel measures with finite second moments and total variation equal to Ent ∈ [0, ∞), endowed with as suitably rescaled Wasserstein distance; see Definition 2.1.We will assume that the total momentum vanishes initially, which implies that the total momentum vanishes for all t > 0. This is not a restriction as the hyperbolic conservation law (1.1) is invariant under transformations to a moving reference frame in the absence of boundaries. The momentum map t → m t = ̺ t v t takes values in a convex set of Ê d -valued Borel measures whose total variations are uniformly bounded, as a consequence of a bound on the total energy. On this set, the narrow convergence of measures is metrized by the Monge-Kantorovich norm: Definition 1.4. We denote by Lip(Ê d ; Ê N ) the vector space of Lipschitz continuous maps ζ :We denote by BL(Ê d ; Ê N ) the subspace of bounded functions in Lip(Ê d ; Ê N ). It is a Banach space when equipped with the bounded Lipschitz normLet BL 1 (Ê d ; Ê N ) be the space of all ζ ∈ BL(Ê d ; Ê N ) with ζ BL(Ê d ) 1. FABIO CAVALLETTI, MARC SEDJRO, AND MICHAEL WESTDICKENBERGWe denote by M K (Ê d ; Ê N ) the space of Ê N -valued Borel measures m with zero mean and finite first moment, equipped with the Monge-Kanto...

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Section: Interpolation In Timementioning

confidence: 99%

“…. t m contained in E.We refer the reader to [43] for further information on metric space-valued functions of bounded variation. In particular, a version of Helly's compactness theorem for sequences of X-valued maps is proved there.…”

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confidence: 99%

A variational time discretization for compressible Euler equations

Cavalletti

Sedjro²,

Westdickenberg

2019

Trans. Amer. Math. Soc.

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“…As in the proof of Theorem 2.8, since the sequence v h : [0, T ] → H 1 (Ω) is such that (5.11) holds, t → v h (t) is non-increasing, and, for every t ∈ [0, T ], v h (t) takes values in [0, 1], applying a generalized version of Helly's Selection Theorem (see [16,Theorem 2.3], we find a non-increasing function v : [0, T ] → H 1 (Ω) such that, along a suitable subsequence h k → +∞, for every t ∈ [0, T ] v h k (t) converges to v(t) weakly in H 1 (Ω) and strongly in L p (Ω) for every p ∈ [1, +∞). In particular, v(0) = v 0 and 0 ≤ v(t) ≤ 1 in Ω for every t ∈ [0, T ], hence condition (i) of Definition 2.3 is satisfied.…”

Section: From Space-discrete To Space-continuous Evolutionmentioning

confidence: 99%

Consistent finite-dimensional approximation of phase-field models of fracture

Almi

Belz

2018

Annali di Matematica

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In this paper we focus on the finite-dimensional approximation of quasi-static evolutions of critical points of the phase-field model of brittle fracture. In a space discretized setting, we first discuss an alternating minimization scheme which, together with the usual time-discretization procedure, allows us to construct such finite-dimensional evolutions. Then, passing to the limit as the space discretization becomes finer and finer, we prove that any limit of a sequence of finite-dimensional evolutions is itself a quasi-static evolution of the phase-field model of fracture. In particular, our proof shows for the first time the consistency of numerical schemes related to the study of fracture mechanics and image processing.2000 Mathematics Subject Classification. 49M25, 49J40.

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