Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations

Brenier, Yann

doi:10.1002/(sici)1097-0312(199904)52:4<411::aid-cpa1>3.0.co;2-3

Cited by 146 publications

(215 citation statements)

References 21 publications

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“…The idea of a probabilistic representation is of course classical, and appears in many contexts (particularly for equations of diffusion type). To the best of our knowledge the first reference in the context of conservation laws and fluid mechanics is [41], where a similar approach is proposed for the incompressible Euler equation (see also [42][43][44]): in this case the compact (but neither metrizable, nor separable) space X [0,T ] , with X ⊂ R d compact, was considered.…”

Section: Open Problems Bibliographical Notes and Referencesmentioning

confidence: 99%

Continuity equations and ODE flows with non-smooth velocity

Ambrosio¹,

Crippa²

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

116

124

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In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable 'weak differentiability' assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

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Section: Open Problems Bibliographical Notes and Referencesmentioning

confidence: 99%

Continuity equations and ODE flows with non-smooth velocity

Ambrosio¹,

Crippa²

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

116

124

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“…In this work, we present a interesting family of stationary solutions for the Euler equations, which behaves in the same way that the approximated solutions presented in [6].…”

Section: Introductionmentioning

confidence: 79%

“…En este trabajo, presentamos una familia interesante de soluciones estacionarias para las ecuaciones de Euler, que se comportan de la misma manera que las soluciones aproximadas presentadas en [6].…”

Section: Introductionunclassified

“…

ABSTRACTIn this work, we present a interesting family of stationary solutions for the Euler equations, which behaves in the same way that the approximated solutions presented in [6].

RESUMENEn este trabajo, presentamos una familia interesante de soluciones estacionarias para las ecuaciones de Euler, que se comportan de la misma manera que las soluciones aproximadas presentadas en [6].

…”

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A Family of Stationary Solutions to the Euler Equations and Generalized Solutions

Precioso

2010

Cubo

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“…This point of view has been adopted and developed by several authors in their work on the Euler equations on fixed domains, such as D. G. Ebin and G. Marsden [EM70], A. Shnirelman [Sh85], and Y. Brenier [Br99] to mention a few. It is this point of view that we adopt to explain the motivation for our definition of energy.…”

Section: The Geometry Behind the Energymentioning

confidence: 99%