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Bahouri, Hajer; Chemin, Jean-Yves

doi:10.1155/s107379289900063x

Cited by 49 publications

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“…) such that (u i , v i ) converges to (u 0 , v 0 ) weakly in L 2 and that div u i and curl v i are bounded 5 in L 2 , then d n=1 u n i v n i converges to d n=1 u n 0 v n 0 in the sense of distribution.…”

Section: Conjecture 11 and Compensated Compactnessmentioning

confidence: 99%

“…There was a long line of works that preceded theorem 1.4 regarding the low-regularity solutions to the Einstein vacuum equations; we refer the reader to [5,6,[68][69][70]107].…”

Section: Theorem 14 (Klainerman-rodnianski-szeftel [71]) the Time Of ...mentioning

confidence: 99%

“…Here, and from now on, we use the notation ⇀ to denote weak convergence 5. More generally, one only requires that div ui and curl vi are compact in H −1 loc 6.…”

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confidence: 99%

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High-frequency solutions to the Einstein equations

Huneau,

Luk

2024

Class. Quantum Grav.

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We survey recent mathematical results concerning the high-frequency solutions to the Einstein vacuum equations and the limits of these solutions. In particular, we focus on two conjectures of Burnett, which attempt to give a exact characterization of high-frequency limits of vacuum spacetimes as solutions to the Einstein–massless Vlasov system. Some open problems and future directions are discussed.

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Section: Conjecture 11 and Compensated Compactnessmentioning

confidence: 99%

“…There was a long line of works that preceded theorem 1.4 regarding the low-regularity solutions to the Einstein vacuum equations; we refer the reader to [5,6,[68][69][70]107].…”

Section: Theorem 14 (Klainerman-rodnianski-szeftel [71]) the Time Of ...mentioning

confidence: 99%

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High-frequency solutions to the Einstein equations

Huneau,

Luk

2024

Class. Quantum Grav.

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“…Moreover, it does not appeal to the semi-group theory used by Kato and his collaborators, but is simply based on energy estimates. Nevertheless, Majda demonstrates that the solution has regularity of the form (11). Finally, he proves a continuation criterion which essentially states that for any s > d/2 (where d is the spatial dimension), the maximal interval [0, T) on which a solution with regularity (11) exists is either such that = ∞ T or such that…”

Section: Local Existencementioning

confidence: 99%

“…Recent developments, the bounded L 2 -curvature theorem. In the late 1990s and early 2000s, a significant amount of progress was made on the problem of decreasing the degree of regularity required of the initial data for quasi-linear wave equations; see, e.g., [11,12,105,106,156,169] and references cited therein (see also [175,176] for related work in the case of constant mean curvature foliations). In particular, combining the results of [156] with the the counterexamples obtained in [1,117,118] yields the conclusion that for equations of the form…”

Section: Local Existencementioning

confidence: 99%

Origins and development of the Cauchy problem in general relativity

Ringström

2015

Class. Quantum Grav.

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The seminal work of Yvonne Choquet-Bruhat published in 1952 demonstrates that it is possible to formulate Einstein's equations as an initial value problem. The purpose of this article is to describe the background to and impact of this achievement, as well as the result itself. In some respects, the idea of viewing the field equations of general relativity as a system of evolution equations goes back to Einstein himself; in an argument justifying that gravitational waves propagate at the speed of light, Einstein used a special choice of coordinates to derive a system of wave equations for the linear perturbations on a Minkowski background. Over the following decades, Hilbert, de Donder, Lanczos, Darmois and many others worked to put Einstein's ideas on a more solid footing. In fact, the issue of local uniqueness (giving a rigorous justification for the statement that the speed of propagation of the gravitational field is bounded by that of light) was already settled in the 1930s by the work of Stellmacher. However, the first person to demonstrate both local existence and uniqueness in a setting in which the notion of finite speed of propagation makes sense was Yvonne Choquet-Bruhat. In this sense, her work lays the foundation for the formulation of Einstein's equations as an initial value problem. Following a description of the results of Choquet-Bruhat, we discuss the development of three research topics that have their origin in her work. The first one is local existence. One reason for addressing it is that it is at the heart of the original paper. Moreover, it is still an active and important research field, connected to the problem of characterizing the asymptotic behaviour of solutions that blow up in finite time. As a second topic, we turn to the questions of global uniqueness and strong cosmic censorship. These questions are of fundamental importance to anyone interested in justifying that the Cauchy problem makes sense globally. They are also closely related to the issue of singularities in general relativity. Finally, we discuss the topic of stability of solutions to Einstein's equations. This is not only an important and active area of research, it is also one that only became meaningful thanks to the work of Yvonne Choquet-Bruhat.

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Fluid Flow Modeling

Feireisl

Novotný

2017

Singular Limits in Thermodynamics of Viscous Fluids

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Cited by 49 publications

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High-frequency solutions to the Einstein equations

High-frequency solutions to the Einstein equations

Origins and development of the Cauchy problem in general relativity

Fluid Flow Modeling

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