Yan Guo, A. Shadi Tahvildar-Zadeh
We prove that for large enough data, the life span of smooth solutions to the Cauchy problem for the following two quasilinear hyperbolic systems is finite: (1) equations of relativistic compressible fluid dynamics, (2) equations of plasma dynamics (Euler-Maxwell equations with background charge) in the spherically symmetric case.
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By introducing a new averaged quantity with a fast decay weight to perform Sideris's argument (Commun Math Phys, 1985) developed for the Euler Equations, we extend the formation of singularities of classical solution to the 3D Euler Equations established in Sideris (1985) and Makino et al. (Jpn J Appl Math, 1986) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler-Poisson Equations, but remove the …
The (twice-contracted) second Bianchi identity is a differential curvature identity that holds on any smooth manifold with a metric. In the case when such a metric is Lorentzian and solves Einstein's equations with an (in this case inevitably smooth) energy-momentum-stress tensor of a "matter field" as the source of spacetime curvature, this identity implies the physical laws of energy and momentum conservation for the "matter field". The present work inquires into whether such a Bianchi identity can still hold in a weak sense for spacetimes with curvature singularities associated with timelik…
Kasner-like description of spacelike singularities in spherically symmetric spacetimes with scalar matter
本文证明球对称时空含标量场时，类空奇点附近的霍金质量等呈逆多项式爆发，并给出BKL型展开及Kasner指数，严格描述了宇宙学与引力坍缩奇点的对应。
In this work we investigate the stability and instability properties of a class of naked singularity spacetimes. The first rigorous study of naked singularity formation in the spherically symmetric Einstein-scalar field system was due to Christodoulou, who constructed a family $(\overline{g}_k, \overlineϕ_k)$ of $k$-self-similar solutions, for any $k^2 \in (0,\frac{1}{3})$. We extend the construction to produce examples of interior and exterior regions of naked singularity spacetimes locally modeled on the $(\overline{g}_k, \overlineϕ_k)$, without requiring exact self-similarity. The main resu…
This paper continues the investigation of the formation of naked singularities in the collapse of collisionless matter initiated in [RV]. There the existence of certain classes of non-smooth solutions of the Einstein-Vlasov system was proved. Those solutions are self-similar and hence not asymptotically flat. To obtain solutions which are more physically relevant it makes sense to attempt to cut off these solutions in a suitable way so as to make them asymptotically flat. This task, which turns out to be technically challenging, will be carried out in this paper. [RV] A. D. Rendall and J. J. L…
The coexistence of null and spacelike singularities inside spherically symmetric black holes
本研究通过分析黑洞内部局部初始数据，证明柯西视界破坏后剩余部分为类空奇点，给出零-类空过渡的尖锐估计，并发现Kasner指数为正且退化为(1,0,0)，首次定量刻画了黑洞内零-类空奇点过渡。
In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in $3+1$ dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Connected to this is a new geometric twisting phenomenon which plays the leading role in singularity formation. Prior to this work, the only known examples of naked singularities were the solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar-field syst…
We find a sharp condition on the density-dependent coefficient of damping of a one-dimensional repulsive Euler-Poisson system, which makes it possible to suppress the formation of singularities in the solution of the Cauchy problem with arbitrary smooth data. In the context of plasma physics, this means the possibility of suppressing the breakdown of arbitrary oscillations of cold plasma.
Scalar perturbations to naked singularities of perfect fluid
本文研究等温完美流体爱因斯坦方程中裸奇点的不稳定性，证明其球对称自相似裸奇点在标量场扰动下会形成俘获面，作为非球对称引力扰动研究的玩具模型。
This is a survey of recent studies of singularity formation in solutions of spherically symmetric Yang-Mills equations in higher dimensions. The main attention is focused on five space dimensions because this case exhibits interesting similarities with Einstein's equations in the physical dimension, in particular the dynamics at the threshold of singularity formation shares many features (such as universality, self-similarity, and scaling) with critical phenomena in gravitational collapse. The borderline case of four space dimensions is also analyzed and the formation of singularities is shown…
We consider the classical and relativistic Vlasov-Poisson systems with spherically-symmetric initial data and prove the optimal decay rates for all suitable $L^p$ norms of the charge density and electric field, as well as, the optimal growth rates for the largest particle position and momentum on the support of the distribution function. Though a previous work \cite{Horst} established upper bounds on the decay of the supremum of the charge density and electric field, we provide a slightly different proof, attain optimal rates, and extend this result to include all other norms. Additionally, we…
We present numerical evidence that singularities form in finite time during the evolution of 2+1 wave maps from spherically equivariant initial data of sufficient energy.
Naked Singularities beyond Spherical Symmetry: Singular Inner Cauchy Horizons for the Einstein-Scalar Field System
研究无对称性假设的3 + 1维爱因斯坦-标量场系统裸奇点形成，通过规定非球对称初始数据推广前人解，建立全局存在性并分析奇异结构，推导渐近性与不可延拓性，揭示弱强宇宙审查联系。
We study Wave Maps from R^{2+1} to the hyperbolic plane with smooth compactly supported initial data which are close to smooth spherically symmetric ones with respect to some H^{1+μ}, μ>0. We show that such Wave Maps don't develop singularities and stay close to the Wave Map extending the spherically symmetric data with respect to all H^{1+δ}, δ<μ_{0}(μ). We obtain a similar result for Wave Maps whose initial data are close to geodesic ones. This generalizes a theorem of Sideris for this context.
We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this time-periodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem.
Asymptotically flat black holes with a singular Cauchy horizon and a spacelike singularity
该研究通过类空-特征粘合等方法构建了满足相关场方程的单端、双端渐近平坦球对称黑洞时空，首次给出类光与类空奇点共存的黑洞示例，为引力坍缩研究提供了重要模型。
In this paper, we initiate the study of the instability of naked singularities without symmetries. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simplest case: a characteristic initial value problem of this coupled system with the initial data given on two intersecting null cones, the incoming one of which is assumed to be spherically symmetric and singular at its vertex, and the outgoing one of which has no symmetries. It is…
Instability of naked singularities of perfect fluid under $C^{1,α}$ gravitational perturbations
本文证明爱因斯坦-欧拉系统球对称自相似坍缩产生的裸奇点，在无对称性的微小 $C^{1,\alpha}$ 引力扰动下不稳定，因声速低于光速，扰动可形成俘获面。
We study the formation of singularity for the Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for $C^1$ blow-up of the Euler-Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the presureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can…
Splash-squeeze singularities and analytic breakdown in ideal incompressible MHD
研究构建了理想不可压缩磁流体力学中首个无Sobolev爆炸的解析破裂奇点，通过分析等离子体与真空边界相互作用，揭示了磁场在有限时间内被挤压形成无限阶奇点的机制。
We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighborhood of points of shock wave collision in General Relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier RSPA-publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to $C^{1,1}$, canno…
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