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Xinliang An

Publications and source records attributed to Xinliang An.

At least 19 recordsLinked to original sources
arXiv2026

Naked Singularities beyond Spherical Symmetry: Instability of $\kappa$-Self-Similar Solutions via an Iteration Scheme

This paper provides the instability counterpart to our recent construction of nonspherically symmetric approximating $\kappa$-self-similar naked-singularity solutions for the Einstein--scalar field system. These singular solutions contain pervasive nonspherical borderline terms, and to prove instability the delicate renormalization procedure developed in [2] does not extend to the more singular setting considered here. To overcome these difficulties, we introduce a new iteration scheme adapted to singular backgrounds whose leading-order geometry depends on the angular variables. At each step, the nonlinear coefficients are frozen using the preceding double-null geometry, and the resulting equations are solved in a triangular order. In this way, the nonspherical borderline terms are incorporated into the approximate geometry rather than treated as perturbative errors, yielding successively sharper estimates. After sufficiently many iterations, the scheme controls the singular angular structure and produces an approximate spacetime whose difference from the exact solution satisfies the required bounds. In particular, these bounds provide an existence region large enough to carry out the instability argument. We further prove that anisotropic perturbations of the outgoing data, arbitrarily small in a scale-critical norm, lead to the formation of a trapped surface. We also formulate and verify a matter-focusing condition under which a parabolic flow argument guarantees the existence of a corresponding marginally outer trapped surface (MOTS). Together, these results establish the nonlinear instability of $\kappa$-self-similar naked singularities beyond spherical symmetry in the Einstein--scalar field system and introduce a framework for applications across Einstein systems.

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arXiv2026

Naked Singularities beyond Spherical Symmetry: Singular Inner Cauchy Horizons for the Einstein-Scalar Field System

In this work, we investigate the formation of naked singularities for the $3+1$-dimensional Einstein-scalar field system without symmetry assumptions. We generalize the spherically symmetric and self-similar naked-singularity solution constructed by Christodoulou in [4] by prescribing non-spherically symmetric initial data along both incoming and outgoing initial null hypersurfaces. We then establish global existence for the resulting solutions and analyze the singular structure of the inner Cauchy horizon. Our construction is based on employing a notion of four-type differences and designing a system of scale-invariant weighted norms to control the corresponding geometry. We show that the constructed spacetimes retain a global naked-singularity structure, characterized by an incomplete future null infinity and a singular inner Cauchy horizon. Moreover, we derive detailed asymptotics near the inner Cauchy horizon and prove the desired $C^{1, \frac{\kappa}{1-\kappa}+}$ inextendibility of these solutions, where $\kappa\in (0,1/3)$ is the self-similar parameter. This indicates a connection between weak and strong cosmic censorship: for the class of non-spherically symmetric solutions constructed here, the failure of weak cosmic censorship in its strict formulation is accompanied by a quantitative inextendibility mechanism at the inner Cauchy horizon.

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arXiv2026

Low regularity ill-posedness and shock formation for 3D ideal compressible MHD

The study of magnetohydrodynamics (MHD) significantly boosts the understanding and development of solar physics, planetary dynamics and controlled nuclear fusion. Dynamical properties of the MHD system involve nonlinear interactions of waves with multiple travelling speeds (the fast and slow magnetosonic waves, the Alfvén wave and the entropy wave). One intriguing topic is the shock phenomena accompanied by the magnetic field, which have been affirmed by astronomical observations. However, permitting the residence of all above multi-speed waves, mathematically, whether one can prove shock formation for three dimensional (3D) MHD is still open. The multiple-speed nature of the MHD system makes it fascinating and challenging. In this paper, we report our recent progress in answering the above question. For 3D ideal compressible MHD, we construct planar symmetric examples of shock formation allowing the presence of all characteristic waves with multiple wave speeds. This also answers a question raised by Majda on conservation law in 1984. Building on our construction, we further prove that the Cauchy problem for 3D ideal MHD is $H^2$ ill-posed. And this is caused by the shock formation. In particular, when the magnetic field is absent, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proof for 3D MHD is based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D ideal MHD equations into a $7\times 7$ non-strictly hyperbolic system. Via detailed calculations, we reveal its hidden subtle structures. With them we give a complete description of MHD dynamics up to the earliest singular event, when a shock forms.

math.AP↗
arXiv2025

Shock Formation for Compressible Euler Equations on $\mathbb{S}^2$

In this paper, we prove the finite-time shock formation for the compressible Euler equations on the two-dimensional sphere $\mathbb{S}^2$. In contrast to the flat Euclidean case $\mathbb{R}^2$, the geometry of $\mathbb S^2$ imposes new difficulties, and the fluid dynamics are affected by the curved background. To overcome these challenges, we modify the existing modulation method and employ a set of carefully constructed, time-dependent coordinates that precisely track the shock formation on $\mathbb{S}^2$. In particular, we first perform a time-dependent rotation of $\mathbb S^2$, then apply the stereographic projection to the sphere, straighten the steepening shock front, and finally construct shock-adapted coordinates. In the shock-adapted coordinates, the compressible Euler equations on $\mathbb{S}^2$ can be recast into a form suitable for self-similar analysis. Within this framework, we implement a detailed bootstrap argument and establish global well-posedness for the self-similar system. After transferring these results back to the original physical system, we thereby demonstrate the finite-time shock formation on $\mathbb{S}^2$.

math.AP↗
arXiv2025

Stability of Big Bang singularity for the Einstein-Maxwell-scalar field-Vlasov system in the full strong sub-critical regime

In $3+1$ dimensions, we study the stability of Kasner solutions for the Einstein-Maxwell-scalar field-Vlasov system. This system incorporates gravity, electromagnetic, weak and strong interactions for the initial stage of our universe. Due to the presence of the Vlasov field, various new challenges arise. By observing detailed mathematical structures and designing new delicate arguments, we identify a new strong sub-critical regime and prove the nonlinear stability with Kasner exponents lying in this full regime. This extends the result of Fournodavlos-Rodnianski-Speck [8] from the Einstein-scalar field system to the physically more complex system with the Vlasov field.

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arXiv2025

On Kerr black hole formation with complete apparent horizon and a new approach toward Penrose inequality

Arising from admissible extended scale-critical short-pulse initial data, we show that 3+1 dimensional Einstein vacuum equations admit dynamical Kerr black hole formation solutions. Our hyperbolic arguments combine the scale-critical gravitational-collapse result by An--Luk with the recent breakthrough by Klainerman--Szeftel on proving nonlinear Kerr stability with small angular momentum, which requires us to perform various specific coordinate changes and frame transformations. Furthermore, allowing large spacetime angular momentum, with new elliptic arguments and precise leading order calculations, we also solve the apparent horizon in Kerr black hole formation spacetimes (including Klainerman--Szeftel's Kerr stability spacetimes) and conduct an exploration, detailing the emergence, evolution, asymptotics and final state of the apparent horizon. Building on our analysis, without time symmetric assumption, we then put forward a new mathematical framework and prove both the dynamical Penrose inequality and the spacetime Penrose inequality in our black-hole formation spacetimes and in the perturbative regime of subextremal Kerr black holes. Collectively, without assuming any symmetry, we extend Christodoulou's celebrated trapped surface formation theorem to a black hole formation result.

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arXiv2024

Low-Regularity Local Well-Posedness for the Elastic Wave System

We study the elastic wave system in three spatial dimensions. For admissible harmonic elastic materials, we prove a desired low-regularity local well-posedness result for the corresponding elastic wave equations. For such materials, we can split the dynamics into the divergence-part and the curl-part, and each part satisfies a distinct coupled quasilinear wave system with respect to different acoustical metrics. Our main result is that the Sobolev norm $H^{3+}$ of the divergence-part (the faster-wave part) and the $H^{4+}$ of the curl-part (the slower-wave part) can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption $H^{3+}$ is optimal for the divergence-part. This marks the first favorable low-regularity local well-posedness result for a wave system with multiple wave speeds.

math.AP↗
arXiv2024

Angular Momentum Memory Effect

Utilizing recent mathematical advances in proving stability of Minkowski spacetime with minimal decay rates and nonlinear stability of Kerr black holes with small angular momentum, we investigate the detailed asymptotic behaviors of gravitational waves generated in these spacetimes. Here we report and propose a new angular momentum memory effect along future null infinity. This accompanies Christodoulou's nonlinear displacement memory effect and the spin memory effect. The connections and differences to these effects are also addressed.

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arXiv2024

A Proof of Weak Cosmic Censorship Conjecture for the Spherically Symmetric Einstein-Maxwell-Charged Scalar Field System

Under spherical symmetry, we show that the weak cosmic censorship holds for the gravitational collapse of the Einstein-Maxwell-charged scalar field system. Namely, for this system, with generic initial data, the formed spacetime singularities are concealed inside black-hole regions. This generalizes Christodoulou's celebrated results to the charged case. Due to the presence of charge $Q$ and the complexification of the scalar field $ϕ$, multiple delicate features and miraculous monotonic properties of the Einstein-(real) scalar field system are not present. We develop a systematical approach to incorporate $Q$ and the complex-valued $ϕ$ into the integrated arguments. For instance, we discover a new path, employing the reduced mass ratio, to establish the sharp trapped surface formation criterion for the charged case. Due to the complex structure and the absence of translational symmetry of $ϕ$, we also carry out detailed modified scale-critical BV area estimates with renormalized quantities to deal with $Q$ and $ϕ$. We present a new $C^1$ extension criterion by utilizing the Doppler exponent to elucidate the blueshift effect, analogous to the role of integrating vorticity in the Beale-Kato-Madja breakdown criterion for incompressible fluids. Furthermore, by utilizing only double-null foliations, we establish the desired first and second instability theorems for the charged scenarios and identify generic initial conditions for the non-appearance of naked singularities. Our instability argument requires intricate generalizations of the treatment for the uncharged case via analyzing the precise contribution of the charged terms and its connection to the reduced mass ratio.

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arXiv2024

Naked Singularity Censoring with Anisotropic Apparent Horizon

Employing the Einstein-scalar field system, we demonstrate an approach for proving high co-dimensional nonlinear instability of naked-singularity solutions as constructed by Christodoulou in [18]. We further investigate the censorship of Christodoulou's naked singularity and show that a tiny anisotropic perturbation arising from the outgoing characteristic initial data would lead to the emergence of an anisotropic apparent horizon, which covers and censors the naked singularity. Our approach advances the hyperbolic short-pulse method by not requiring the aid of additional large parameters, by permitting the use of initial perturbations for the shear tensor and the derivative of scalar field to be with finite $BV$ and $C^0$ norms, and by allowing the initial perturbation to be arbitrarily small in scale-critical norms. New elliptic arguments based on non-perturbative methods are also developed.

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arXiv2023

Dynamics of Apparent Horizon and a Null Comparison Principle

This paper investigates the global dynamics of the apparent horizon. We present an approach to establish its existence and its long-term behaviors. Our apparent horizon is constructed by solving the marginally outer trapped surface (MOTS) along each incoming null hypersurface. Based on the nonlinear hyperbolic estimates established in [24] by Klainerman-Szeftel under polarized axial symmetry, we prove that the corresponding apparent horizon is smooth, asymptotically null and converging to the event horizon eventually. To further address the local achronality of the apparent horizon, a new concept, called the null comparison principle, is introduced in this paper. For three typical scenarios of gravitational collapse, our null comparison principle is tested and verified, which guarantees that the apparent horizon must be piecewise spacelike or piecewise null. In addition, we also validate and provide new proofs for several physical laws along the apparent horizon.

gr-qc↗
arXiv2023

Nonlinear asymptotic stability and transition threshold for 2D Taylor-Couette flows in Sobolev spaces

In this paper, we investigate the stability of the 2-dimensional (2D) Taylor-Couette (TC) flow for the incompressible Navier-Stokes equations. The explicit form of velocity for 2D TC flow is given by $u=(Ar+\frac{B}{r})(-\sin θ, \cos θ)^T$ with $(r, θ)\in [1, R]\times \mathbb{S}^1$ being an annulus and $A, B$ being constants. Here, $A, B$ encode the rotational effect and $R$ is the ratio of the outer and inner radii of the annular region. Our focus is the long-term behavior of solutions around the steady 2D TC flow. While the laminar solution is known to be a global attractor for 2D channel flows and plane flows, it is unclear whether this is still true for rotating flows with curved geometries. In this article, we prove that the 2D Taylor-Couette flow is asymptotically stable, even at high Reynolds number ($Re\sim ν^{-1}$), with a sharp exponential decay rate of $\exp(-ν^{\frac13}|B|^{\frac23}R^{-2}t)$ as long as the initial perturbation is less than or equal to $ν^\frac12 |B|^{\frac12}R^{-2}$ in Sobolev space. The powers of $ν$ and $B$ in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor-Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient $B$. The larger the $|B|$, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor-Couette flows.

math.AP↗
arXiv2022

$H^{\frac{11}{4}}(\mathbb{R}^2)$ ill-posedness for 2D Elastic Wave system

In this paper, we prove that for the 2D elastic wave equations, a physical system with multiple wave-speeds, its Cauchy problem fails to be locally well-posed in $H^{\frac{11}{4}}(\mathbb{R}^2)$. The ill-posedness here is driven by instantaneous shock formation. In 2D Smith-Tataru showed that the Cauchy problem for a single quasilinear wave equation is locally well-posed in $H^s$ with $s>\frac{11}{4}$. Hence our $H^{\frac{11}{4}}$ ill-posedness obtained here is a desired result. Our proof relies on combining a geometric method and an algebraic wave-decomposition approach, equipped with detailed analysis of the corresponding hyperbolic system.

math.AP↗
arXiv2022

Quantitative blow-up estimates for spacelike singularities in gravitational-collapse cosmological spacetimes

Under spherical symmetry, with double-null coordinates $(u,v)$, we study the gravitational collapse of the Einstein--scalar field system with a positive cosmological constant. The spacetime singularities arise when area radius $r$ vanishes and they are spacelike. We derive new quantitative estimates, obtain polynomial blow-up rates $O(1/r^N)$ for various quantities, and extend the results in [5] by the first author and Zhang and the arguments in [3] by the first author and Gajic to the cosmological settings. In particular, we sharpen the estimates of $r\partial_u r$ and $r\partial_v r$ in [5] and prove that the spacelike singularities where $r(u,v)=0$ are $C^{1,1/3}$ in $(u,v)$ coordinates. As an application, these estimates also give quantitative blow-up upper bounds of fluid velocity and density for the hard-phase model of the Einstein-Euler system under irrotational assumption. Near the timelike infinity, we also generalize the theorems in [3] by linking the precise blow-up rates of the Kretschmann scalar to the exponential Price's law along the event horizon. In cosmological settings, this further reveals the mass-inflation phenomena along the spacelike singularities for the first time.

gr-qc↗
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.

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