arXiv subjects

Pin Yu

Publications and source records attributed to Pin Yu.

26 records · Page 2Linked to original sources

arXiv2012

A Large Data Regime for non-linear Wave Equations

For semi-linear wave equations with null form non-linearities on $\mathbb{R}^{3+1}$, we exhibit an open set of initial data which are allowed to be large in energy spaces, yet we can still obtain global solutions in the future. We also exhibit a set of localized data for which the corresponding solutions are strongly focused, which in geometric terms means that a wave travels along an specific incoming null geodesic in such a way that almost all of the energy is confined in a tubular neighborhood of the geodesic and almost no energy radiating out of this tubular neighborhood.

math.AP↗

arXiv2012

Long time solutions for wave maps with large data

For 2 + 1 dimensional wave maps with $\mathbb{S}^2$ as the target, we show that for all positive numbers $T_0 > 0$ and $E_0 > 0$, there exist Cauchy initial data with energy at least $E_0$, so that the solution's life-span is at least $[0,T_0]$. We assume neither symmetry nor closeness to harmonic maps.

math.AP↗

arXiv2012

Construction of Cauchy Data of Vacuum Einstein field equations Evolving to Black Holes

We show the existence of complete, asymptotically flat Cauchy initial data for the vacuum Einstein field equations, free of trapped surfaces, whose future development must admit a trapped surface. Moreover, the datum is exactly a constant time slice in Minkowski space-time inside and exactly a constant time slice in Kerr space-time outside. The proof makes use of the full strength of Christodoulou's work on the dynamical formation of black holes and Corvino-Schoen's work on the constructions of initial data set.

gr-qc↗

arXiv2011

Dynamical formation of black holes due to the condensation of matter field

The purpose of the paper is to understand a mechanism of evolutionary formation of trapped surfaces when there is an electromagnetic field coupled to the background space-time. Based on the short pulse ansatz, on a given finite outgoing null hypersurface which is free of trapped surfaces, we exhibit an open set of initial data for Einstein equations coupled with a Maxwell field, so that a trapped surface forms along the Einstein-Maxwell flow. On one hand, this generalizes the black-hole-formation results of Christodoulou and Klainerman-Rodnianski. In fact, by switching off the electromagnetic field in our main theorem, we can retrieve their results in vacuum. On the other hand, this shows that the formation of black hole can be purely due to the condensation of Maxwell field on the initial null hypersurface where there is no incoming gravitational energy.

math.AP↗

arXiv2011

Energy Estimates and Gravitational Collapse

We study the cancelations in the energy estimates for Einstein vacuum equations in order to prove the formation of black holes along evolutions. The novelty of the paper is that, we completely avoid using rotation vector fields to establish the global existence theorem of the solution. More precisely, we use only canonical null directions as commutators to derive energy estimates at the level of one derivatives of null curvature components.

math.AP↗

arXiv2010

A Uniqueness Theorem for Free Waves on $\mathbb{R}^{n+1}$

In this short note, based on Carleman estimates and Holmgren's type theorems, we provide a converse theorem of the classical Huygens principle for free wave equations. Possible generalizations to other underlying space-times or other wave type equations are also discussed.

math.AP↗

arXiv2009

On strong unique continuation of coupled Einstein metrics

The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite order around a point, then they must coincide, up to a local diffeomorphism, in a neighborhood of the point. The novelty of our method lies in the use of a Carleman inequality and thus circumventing the use of analyticity; thus the method is robust under certain non-analytic perturbations. As an example, we also show the strong unique continuation property for the Riemannian Einstein-scalar-field system with cosmological constant.

math.AP↗

arXiv2009

On Hawking's Local Rigidity Theorems for Charged Black Holes

We show the existence of a Hawking vector field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth Einstein-Maxwell space-time without assuming the underlying space-time is analytic. It extends one result of Friedrich, Rácz and Wald, which was limited to the interior of the black hole region. Moreover, we also show, in the presence of an additional Killing vector field $T$ which tangent to the horizon and not vanishing on the bifurcate sphere, then space-time must be locally axially symmetric without the analyticity assumption. This axial symmetry plays a fundamental role in the classification theory of stationary black holes.

gr-qc↗