arXiv subjects

Xinliang An

Publications and source records attributed to Xinliang An.

33 records · Page 2Linked to original sources

arXiv2021

Enhanced dissipation and nonlinear asymptotic stability of the Taylor-Couette flow for the 2D Navier-Stokes equations

In this paper, we study the nonlinear stability of a steady circular flow created between two rotating concentric cylinders. The dynamics of the viscous fluid are described by 2D Navier-Stokes equations. We adopt scaling variables. For the rescaled equations, we prove that the steady flow (Taylor-Couette flow) is asymptotically stable up to a large perturbation of initial data. Back to the original 2D Navier-Stokes equations, this implies an improved transition threshold for the Taylor-Couette flow. The improvement is due to enhanced dissipation and new observations and constructions of weighted $L^2$ norms, which capture a hidden structure between the viscosity constant $ν$ and (different) rotating speeds and locations of two coaxial cylinders. In particular, we allow the location of the outer cylinder to tend to infinity, which renders the initial fluid kinetic energy not uniformly bounded. Due to enhanced-dissipation effect, we also establish a sharp resolvent estimate, desired space-time bounds and optimal decaying estimates, which lead to the proof of nonlinear asymptotic stability of 2D Taylor-Couette flow.

math.AP↗

arXiv2021

Anisotropic Dynamical Horizons Arising in Gravitational Collapse

For the study of $3+1$ dimensional Einstein vacuum equations (EVEs), substantial progress has been made recently on the problem of trapped surface formation. However, very limited knowledge of existence and associated properties is acquired on the boundary of the emerged trapped region, i.e., the apparent horizon, which is composed of marginally outer trapped surfaces (MOTSs) and is of great physical importance. In this paper, concerning this emerged apparent horizon we prove a folklore conjecture relating to both cosmic censorship and black hole thermodynamics. In a framework set up by Christodoulou and under a general anisotropic condition introduced by Klainerman, Luk and Rodnianski, for $3+1$ EVEs we prove that in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This dynamical horizon censors singularities formed in gravitational collapse from non-trapped local observers near the center, and it also enables the extension of black hole thermodynamical theory along the apparent horizon to anisotropic scenarios. Our analysis builds on scale-critical hyperbolic method and non-perturbative elliptic techniques. New observations and equation structures are exploited. Geometrically, we furthermore construct explicit finger-type single and multi-valley anisotropic apparent horizons. They are the first mathematical examples of the anisotropic MOTS and the anisotropic apparent horizon formed in dynamics, which have potential applications in geometric analysis, black hole mechanics, numerical relativity and gravitational wave phenomenology.

gr-qc↗

arXiv2020

Emergence of Apparent Horizon in Gravitational Collapse

We solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smooth (except at that point) and spacelike. In the proof we employ a scale critical trapped surface formation criterion established by An and Luk and a new type of quasilinear elliptic equation is studied. The main conclusion of this paper proves a conjecture of Ashtekar on black hole thermodynamics. And the spacetimes constructed here could also be viewed as (non-spherically symmetric) generalizations of the well-known Vaidya spacetime.

gr-qc↗

arXiv2020

A scale-critical trapped surface formation criterion for the Einstein-Maxwell system

In this study we show that, from arbitrarily dispersed initial data, both the concentration of electromagnetic fields and the focusing of gravitational waves could lead to the formation of trapped surfaces. We establish a scale-critical semi-global existence result from the past null infinity for the Einstein-Maxwell system by assigning the signature for decay rates to geometric quantities and Maxwell components. This result generalizes an approach of the first author on studying Einstein vacuum equations by employing additional elliptic estimates and geometric renormalizations, and it also extends a result of Yu to the scale-critical regime.

math.AP↗

arXiv2020

Trapped surface formation for spherically symmetric Einstein-Maxwell-charged scalar field system with double null foliation

In this paper, under spherical symmetry we prove a trapped surface formation criterion for the Einstein-Maxwell-charged scalar field system. We generalize an approach introduced by Christodoulou for studying the Einstein-scalar field. In appendix, with double null foliation we also reprove Christodoulou's result and for Minkowskian incoming characteristic initial data, we improve Christodoulou's bound.

math.AP↗

arXiv2020

Curvature blow-up rates in spherically symmetric gravitational collapse to a Schwarzschild black hole

We study the black hole interiors of spacetimes arising from gravitational collapse in the spherically symmetric Einstein-scalar field setting, and we investigate the precise blow-up rates of curvature and mass at the spacelike singularity near timelike infinity. We show in particular that the Kretschmann scalar blows up faster than in the Schwarzschild setting, due to mass inflation. Moreover, the blow-up rate is not constant and converges to the Schwarzschild rate towards timelike infinity and it depends on the precise late-time polynomial behaviour of the scalar field along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism.

math.AP↗

arXiv2020

Polynomial Blow-up Upper Bounds for the Einstein-scalar field System Under Spherical Symmetry

For general gravitational collapse, inside the black-hole region, singularities $(r=0)$ may arise. In this article, we aim to answer how strong these singularities could be. We analyse the behaviours of various geometric quantities. In particular, we show that in the most singular scenario, the Kretschmann scalar obeys polynomial blow-up upper bounds $O(1/r^N)$. This improves previously best-known double-exponential upper bounds $O\big(\exp\exp(1/r)\big)$. Our result is sharp in the sense that there are known examples showing that no sub-polynomial upper bound could hold. Finally we do a case study on perturbations of the Schwarzschild solution.

math.AP↗

arXiv2020

Low regularity ill-posedness for elastic waves driven by shock formation

In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, is ill-posed in $H^3(\mathbb{R}^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case.

math.AP↗

arXiv2020

A Scale-Critical Trapped Surface Formation Criterion: A New Proof via Signature for Decay Rates

For gravitational collapse, we observe a correspondence between region close to past null infinity and region close to central singularity. In line with this philosophy, we construct a new ansatz, with which we first present a 40-page self-contained proof of trapped surface formation in a far-from-center region. A systematical approach with only signature for decay rates is used. Then by a rescaling argument, we convert this result into a reproof and an extension of a scale-critical theorem proved by An-Luk.

math.AP↗

arXiv2019

Fermi's Golden Rule and $H^1$ Scattering for Nonlinear Klein-Gordon Equations with Metastable States

In this paper, we explore the metastable states of nonlinear Klein-Gordon equations with potentials. These states come from the instability of a bound state under a nonlinear Fermi's golden rule. In [16], Soffer and Weinstein studied the instability mechanism and obtained an anomalously slow-decaying rate $1/(1+t)^{\frac{1}{4}}$. Here we develop a new method to study the $L^2_x$ norm of solutions to Klein-Gordon equations. With this method, we prove the first $H^1$ scattering result for Klein-Gordon equations with metastable states. By exploring the oscillations, we also give another more robust and more intuitive approach to derive the sharp decay rate $1/(1+t)^{\frac{1}{4}}$.

math.AP↗

arXiv2018

Examples of naked singularity formation in higher-dimensional Einstein-vacuum spacetimes

The vacuum Einstein equations in 5+1 dimensions are shown to admit solutions describing naked singularity formation in gravitational collapse from nonsingular asymptotically locally flat initial data that contain no trapped surface. We present a class of specific examples with topology $\mathbb{R}^{3+1} \times S^2$. Thanks to the Kaluza-Klein dimensional reduction, these examples are constructed by lifting continuously self-similar solutions of the 4-dimensional Einstein-scalar field system with a negative exponential potential. The latter solutions are obtained by solving a 3-dimensional autonomous system of first-order ordinary differential equations with a combined analytic and numerical approach. Their existence provides a new test-bed for weak cosmic censorship in higher-dimensional gravity. In addition, we point out that a similar attempt of lifting Christodoulou's naked singularity solutions of massless scalar fields fails to capture formation of naked singularities in 4+1 dimensions, due to a diverging Kretschmann scalar in the initial data.

gr-qc↗

arXiv2017

Warped Product Space-times

Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic presentation of the main geometric constructions, with emphasis on the Kodama vector field and the Hawking energy; the construction is signature independent. This leads to proofs of general Birkhoff-type theorems for warped product manifolds; our theorems in particular apply to situations where the warped product manifold is not necessarily Einstein, and thus can be applied to solutions with matter content in general relativity. Next we specialize to the Lorentzian case and study the propagation of null expansions under the assumption of the dominant energy condition. We prove several non-existence results relating to the Yamabe class of the fibers, in the spirit of the black-hole topology theorem of Hawking-Galloway-Schoen. Finally we discuss the effect of the warped product ansatz on matter models. In particular we construct several cosmological solutions to the Einstein-Euler equations whose spatial geometry is generally not isotropic.

gr-qc↗

arXiv2017

Trapped surfaces in vacuum arising dynamically from mild incoming radiation

In this paper, we study the "minimal requirement" on the incoming radiation that guarantees a trapped surface to form in vacuum. First, we extend the region of existence in Christodoulou's theorem on the formation of trapped surfaces and consequently show that the lower bound required to form a trapped surface can be relaxed. Second, we demonstrate that trapped surfaces form dynamically from a class of initial data which are large merely in a scaling-critical norm. This result is motivated in part by the scaling in Christodoulou's formation of trapped surfaces theorem for the Einstein-scalar field system in spherical symmetry.

gr-qc↗

arXiv2015

Naked singularity and black hole formation in self-similar Einstein-scalar fields with exponential potentials

Motivated by cosmic censorship in general relativity and string theory, we extend Christodoulou's celebrated examples of naked singularity formation in the Einstein-massless scalar field system to include a positive or negative scalar potential of exponential forms, i.e., $V(ϕ)=\pm\exp(2ϕ/κ)$ with a parameter $κ$. Under spherical symmetry and a self-similar ansatz depending on $κ$, we derive a 3-dimensional autonomous system of first-order ordinary differential equations, which incorporates the equations for massless scalar fields as a special case. Local behavior of the phase space is studied analytically with global solutions constructed numerically. Within the 3-dimensional solution manifold, we observe, for the negative potentials, naked singularity formation from nonsingular initial data for $κ^2<1$. Meanwhile, transitions between solutions containing naked singularities and black holes are also identified. However, when the potential is taken positive, numerical evolutions result in formation of black holes, but not naked singularities.

gr-qc↗

arXiv2014

Formation of Trapped Surfaces from Past Null Infinity

In this paper, we extend the results of Klainerman and Rodnianski in \cite{KR:Trapped}, which were obtained for a finite region, by showing similar results from past null infinity. This allows us to recover and extend the results from past null infinity in the work of Christodoulou \cite{Chr:book}.

gr-qc↗