Monge–Kantorovich Norms on Spaces of Vector Measures

Chiţescu, Ion; Ioana, Loredana; Miculescu, Radu; Niţă, Lucian

doi:10.1007/s00025-016-0531-1

Cited by 9 publications

(7 citation statements)

References 20 publications

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“…We refer the reader to [18,27,38,51] for additional information. Notice that the integral in (1.11) is well-defined because m has finite first moment, by assumption.…”

Section: Introductionmentioning

confidence: 99%

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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“…We refer the reader to [18,27,38,51] for additional information. Notice that the integral in (1.11) is well-defined because m has finite first moment, by assumption.…”

Section: Introductionmentioning

confidence: 99%

A variational time discretization for compressible Euler equations

Cavalletti

Sedjro²,

Westdickenberg

2019

Trans. Amer. Math. Soc.

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“…This is sufficient because m s and m t have the same total momentum; testing their difference against a constant gives zero. We refer the reader to [35] for a related discussion. We will show in Lemma 1 that, with momentum flux We will be interested in dissipative solutions of (1) that minimize the metric derivative |m |.…”

Section: Minimal Accelerationmentioning

confidence: 99%

Minimal Acceleration for the Multi-dimensional Isentropic Euler Equations

Westdickenberg

2023

Arch Rational Mech Anal

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On the set of dissipative solutions to the multi-dimensional isentropic Euler equations, we introduce a quasi-order by comparing the acceleration at all time. This quasi-order is continuous with respect to a suitable notion of convergence of dissipative solutions. We establish the existence of minimal elements. Minimizing the acceleration amounts to selecting dissipative solutions that are as close to being a weak solution as possible.

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“…We continue introducing the basic facts from our previous papers [3] and [4]. Again (T, d) is a compact metric space and X is a Hilbert space.…”

Section: §1 Introductionmentioning

confidence: 99%

“…To this end, we developed a preliminary apparatus. The first part of this apparatus appears in detail in our previous papers [3] and [4]. Namely, in [3] we introduce a sesquilinear uniform integral, which is used in [4] to define various norms and distances in the space of vector measures of bounded variation.…”

Section: §1 Introductionmentioning

confidence: 99%

“…In the second paragraph ("Preliminary Facts") we introduce the notions and results which will be used throughout the paper. A special attention is given to the sesquilinear uniform integral introduced in [3] and to the generalizations of the Monge-Kantorovich and modified Monge-Kantorovich norms and distances introduced in [4], making the paper to be self-contained.…”

Section: §1 Introductionmentioning

confidence: 99%

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Operators on Spaces of Functions and Measures. Vector Invariant (Fractal) Measures

et al. 2018

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We consider a general schema involving measure spaces, contractions and linear and continuous operators. Within the framework of this schema we use our sesquilinear uniform integral and introduce some integral operators on continuous vector functions spaces, which lead us to operators on spaces of vector measures. Using these last operators, we generalize the Markov operators, obtaining via contractions vector invariant (fractal) measures. Concrete examples are provided.

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Monge–Kantorovich Norms on Spaces of Vector Measures

Cited by 9 publications

References 20 publications

A variational time discretization for compressible Euler equations

A variational time discretization for compressible Euler equations

Minimal Acceleration for the Multi-dimensional Isentropic Euler Equations

Operators on Spaces of Functions and Measures. Vector Invariant (Fractal) Measures

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