Asymptotic Stability of Minkowski Space-Time with Non-compactly Supported Massless Vlasov Matter

Bigorgne, Léo; Fajman, David; Joudioux, Jérémie; Smulevici, Jacques; Thaller, Maximilian

doi:10.1007/s00205-021-01639-2

Cited by 34 publications

(23 citation statements)

References 41 publications

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“…If Λ > 0, θ 0 is additionally required to satisfy θ 0 > [2Λ/(n − 1)] 1/2 . Let K be the expansion normalised Weingarten map of the leaves of the foliation and ȟ be defined by (14). Then the following correspondence between the solution and the asymptotic data is required to hold.…”

Section: 2mentioning

confidence: 99%

“…Moreover, θ is assumed to diverge uniformly to ∞ as t → 0+. Let ȟ be defined by (14), K be the expansion normalised Weingarten map of the leaves of the foliation and Û be the future directed unit normal with respect to ĝ := θ 2 g. Assume (18a) and (18b) to hold. Finally, assume that there is a constant C rel such that…”

Section: 2mentioning

confidence: 99%

“…Even though the BKL proposal dates back more than 50 years, there are only results concerning oscillatory behaviour in the spatially homogeneous setting; cf., e.g., [59,77,60,35,48,8,49,15,25]. This is unsatisfactory, and can be contrasted with a growing number of results concerning the future/past (or both) global non-linear stability of solutions to Einstein's equations; cf., e.g., [31,32,33,18,78,4,52,61,62,12,13,51,5,76,72,64,42,54,68,73,34,7,56,20,39,53,40,3,45,14,26,27,28,19]. In fact, there are even stability results in the direction of the big bang singularity; cf.…”

Section: Oscillatory Big Bang Singularitiesmentioning

confidence: 99%

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Initial data on big bang singularities

Ringström¹

2022

Preprint

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The goal of this article is to parametrise solutions to Einstein's equations with big bang singularities and quiescent asymptotics, disregarding issues such as spikes. To this end, we introduce a notion of initial data on big bang singularities and conjecture that it can be used to parametrise quiescent solutions. A mathematical statement of the conjecture presupposes a precise definition of the class of quiescent solutions as well as a proof of existence and uniqueness of developments corresponding to initial data on a big bang singularity. Needless to say, there many possible definitions. Nevertheless, we provide one definition of quiescence which is satisfied by many examples and that leads to initial data of the type considered here. We also appeal to existing results in order to illustrate that, in certain cases, there are unique developments corresponding to initial data on the singularity. However, the perspective developed here naturally leads to a large class of open problems corresponding to the general conjecture. Moreover, there are many versions of the conjecture in that one can consider different matter models as well as different foliations of the spacetimes in the vicinity of a big bang singularity. An additional benefit of the notion of initial data developed here is that it can be used to give a unified perspective on the existing results concerning quiescent singularities. In fact, we provide several examples of how existing results can be considered to be special cases of the framework developed here. A second, potential, application is to oscillatory and spatially inhomogeneous big bang singularities. Considering the existing arguments in the spatially homogeneous setting, a crucial first step in the study of oscillatory behaviour is to understand how solutions approach the Kasner circle along a stable manifold, and then depart via an unstable manifold. In order to carry out a similar analysis in the spatially inhomogeneous setting, it is of central importance to first identify the stable manifold. Building on the work of Fournodavlos and Luk, we here propose such an identification.

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Oscillatory Big Bang Singularitiesmentioning

confidence: 99%

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Initial data on big bang singularities

Ringström¹

2022

Preprint

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“…As mentioned in Remark 1.5, our proof is based on a commutating vector field method. In the context of kinetic theory, the commutating vector field method has been most successful in capturing dispersion, see [6,7,18,19,28,32,33,36] for some results for collisionless models and [10,11,29] for some results on collisional models.…”

Section: Introduction Consider the Linear Transport Equation In 1dmentioning

confidence: 99%

Phase mixing for solutions to 1D transport equation in a confining potential

Chaturvedi¹,

Luk²

2022

KRM

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Consider the linear transport equation in 1D under an external confining potential <inline-formula><tex-math id="M1">\begin{document}$ \Phi $\end{document}</tex-math></inline-formula>:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $\end{document} </tex-math></disp-formula>For <inline-formula><tex-math id="M2">\begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document}</tex-math></inline-formula> (with <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon >0 $\end{document}</tex-math></inline-formula> small), we prove phase mixing and quantitative decay estimates for <inline-formula><tex-math id="M4">\begin{document}$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document}</tex-math></inline-formula>, with an inverse polynomial decay rate <inline-formula><tex-math id="M5">\begin{document}$ O({\langle} t{\rangle}^{-2}) $\end{document}</tex-math></inline-formula>. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in <inline-formula><tex-math id="M6">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>D under the external potential <inline-formula><tex-math id="M7">\begin{document}$ \Phi $\end{document}</tex-math></inline-formula>.

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“…In the work of Fajman-Joudioux-Smulevici [10], the vector field method was also developed for relativistic transport equations. There has been recently a lot of activities concerning the study of small data Vlasov systems using such methods, see for instance [11,12,4,5,6,7,16,2,3,8]. See also [21] where Fourier techniques and vector field type techniques are used.…”

mentioning

confidence: 99%

Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data

Duan¹

2022

KRM

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In this paper, we present sharp decay estimates for small data solutions to the following two systems: the Vlasov-Poisson (V-P) system in dimension 3 or higher and the Vlasov-Yukawa (V-Y) system in dimension 2 or higher. We rely on a modification of the vector field method for transport equation as developed by Smulevici in 2016 for the Vlasov-Poisson system. Using the Green's function in particular to estimate the bilinear terms, we improve Smulevici's result by removing the requirement of some <inline-formula><tex-math id="M1">\begin{document}$ v $\end{document}</tex-math></inline-formula>-weighted <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> integrability for the initial data and extend the result to the Vlasov-Yukawa system.

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Asymptotic Stability of Minkowski Space-Time with Non-compactly Supported Massless Vlasov Matter

Cited by 34 publications

References 41 publications

Initial data on big bang singularities

Initial data on big bang singularities

Phase mixing for solutions to 1D transport equation in a confining potential

Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data

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