Einstein vacuum equations with 𝕌(1) symmetry in an elliptic gauge: Local well-posedness and blow-up criterium

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Journal of Hyperbolic Differential Equations Pub Date : 2022-12-01 DOI:10.1142/s0219891622500187

Arthur Touati

引用次数: 1

Abstract

In this paper, we are interested in the Einstein vacuum equations on a Lorentzian manifold displaying [Formula: see text] symmetry. We identify some freely prescribable initial data, solve the constraint equations and prove the existence of a unique and local in time solution at the [Formula: see text] level. In addition, we prove a blow-up criterium at the [Formula: see text] level. By doing so, we improve a result of Huneau and Luk in [Einstein equations under polarized [Formula: see text] symmetry in an elliptic gauge, Commun. Math. Phys. 361(3) (2018) 873–949] on a similar system, and our main motivation is to provide a framework adapted to the study of high-frequency solutions to the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As a consequence we work in an elliptic gauge, particularly adapted to the handling of high-frequency solutions, which have large high-order norms.

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椭圆规范中具有(1)对称性的爱因斯坦真空方程:局部适定性和爆破准则

在本文中，我们对洛伦兹流形上显示[公式：见正文]对称性的爱因斯坦真空方程感兴趣。我们确定了一些可自由规定的初始数据，求解了约束方程，并证明了在[公式：见正文]级别存在唯一的局部时间解。此外，我们还证明了[公式：见正文]级别的爆破标准。通过这样做，我们改进了Huneau和Luk在椭圆规范Commun中极化[公式：见正文]对称性下的[爱因斯坦方程]中的结果。数学Phys。361（3）（2018）873–949]，我们的主要动机是提供一个适用于Huneau和Luk即将发表的论文中研究爱因斯坦真空方程高频解的框架。因此，我们在椭圆规范中工作，特别适用于处理具有大高阶范数的高频解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Hyperbolic Differential Equations 数学-物理：数学物理

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

24 months

期刊介绍： This journal publishes original research papers on nonlinear hyperbolic problems and related topics, of mathematical and/or physical interest. Specifically, it invites papers on the theory and numerical analysis of hyperbolic conservation laws and of hyperbolic partial differential equations arising in mathematical physics. The Journal welcomes contributions in: Theory of nonlinear hyperbolic systems of conservation laws, addressing the issues of well-posedness and qualitative behavior of solutions, in one or several space dimensions. Hyperbolic differential equations of mathematical physics, such as the Einstein equations of general relativity, Dirac equations, Maxwell equations, relativistic fluid models, etc. Lorentzian geometry, particularly global geometric and causal theoretic aspects of spacetimes satisfying the Einstein equations. Nonlinear hyperbolic systems arising in continuum physics such as: hyperbolic models of fluid dynamics, mixed models of transonic flows, etc. General problems that are dominated (but not exclusively driven) by finite speed phenomena, such as dissipative and dispersive perturbations of hyperbolic systems, and models from statistical mechanics and other probabilistic models relevant to the derivation of fluid dynamical equations. Convergence analysis of numerical methods for hyperbolic equations: finite difference schemes, finite volumes schemes, etc.