arXiv subjects

Jonathan Luk

Publications and source records attributed to Jonathan Luk.

47 records · Page 3Linked to original sources

arXiv2016

Solutions to the Einstein-scalar-field system in spherical symmetry with large bounded variation norms

It is well-known that small, regular, spherically symmetric characteristic initial data to the Einstein-scalar-field system which are decaying towards (future null) infinity give rise to solutions which are foward-in-time global (in the sense of future causal geodesic completeness). We construct a class of spherically symmetric solutions which are global but the initial norms are consistent with initial data not decaying towards infinity. This gives the following consequences: (1) We prove that there exist foward-in-time global solutions with arbitrarily large (and in fact infinite) initial bounded variation (BV) norms and initial Bondi masses. (2) While general solutions with non-decaying data do not approach Minkowski spacetime, we show using the results of Luk--Oh that if a sufficiently strong asymptotic flatness condition is imposed on the initial data, then the solutions we construct (with large BV norms) approach Minkowski spacetime with a sharp inverse polynomial rate. (3) Our construction can be easily extended so that data are posed at past null infinity and we obtain solutions with large BV norms which are causally geodesically complete both to the past and to the future. Finally, we discuss applications of our method to construct global solutions for other nonlinear wave equations with infinite critical norms.

gr-qc↗

arXiv2015

Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry

In this paper, we study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field $ϕ$ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that $ϕ$ and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up.

gr-qc↗

arXiv2015

Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations

It has long been suggested that solutions to linear scalar wave equation $$\Box_gϕ=0$$ on a fixed subextremal Reissner-Nordström spacetime with non-vanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W^{1,2}_{loc}$. This instability is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein-Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordström spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price's law decay is generically sharp along the event horizon.

gr-qc↗

arXiv2014

Local Propagation of Impulsive Gravitational Waves

In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of non-regular characteristic data.

gr-qc↗

arXiv2014

Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system I. The $2$-D and $2\frac 12$-D cases

Consider the relativistic Vlasov-Maxwell system with initial data of unrestricted size. In the two dimensional and the two and a half dimensional cases, Glassey-Schaeffer (1997, 1998, 1998) proved that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the $L^\infty_x$ norms of the electromagnetic fields compared to Glassey-Schaeffer.

math.AP↗

arXiv2014

A new continuation criterion for the relativistic Vlasov-Maxwell system

The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem. The main result of Glassey-Strauss (1986) shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded. Alternate proofs were later given by Bouchut-Golse-Pallard (2003) and Klainerman-Staffilani (2002). We show that only the boundedness of the momentum support of $f$ after projecting to any two dimensional plane is needed for $(f, E, B)$ to remain $C^1$.

math.AP↗

arXiv2013

A fully anisotropic mechanism for formation of trapped surfaces in vacuum

We present a new, fully anisotropic, criterion for formation of trapped surfaces in vacuum. More precisely we provide conditions on null data, concentrated in a neighborhood of a short null geodesic segment (possibly flat everywhere else) whose future development contains a trapped surface. This extends considerably the previous result of Christodoulou \cite{Chr:book} which required instead a uniform condition along all null geodesic generators. To obtain our result we combine Christodoulou's mechanism for the formation of a trapped surface with a new deformation process which takes place along incoming null hypersurfaces.

gr-qc↗

arXiv2011

On the Local Existence for the Characteristic Initial Value Problem in General Relativity

Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere $S$ with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of $S$ in the future of $S$. We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy type estimates and relies heavily on the null structure of the Einstein equations in the double null foliation.

gr-qc↗

arXiv2011

A Vector Field Method Approach to Improved Decay for Solutions to the Wave Equation on a Slowly Rotating Kerr Black Hole

We prove that sufficiently regular solutions to the wave equation $\Box_{g_K}Φ=0$ on the exterior of a sufficiently slowly rotating Kerr black hole obey the estimates $|Φ|\leq C (t^*)^{-3/2+η}$ on a compact region of $r$. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski and Schwarzschild spacetime. This result improves the known robust decay rates that are proved using the vector field method in the region of finite $r$ and along the event horizon.

gr-qc↗

arXiv2010

The Null Condition and Global Existence for Nonlinear Wave Equations on Slowly Rotating Kerr Spacetimes

We study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the improved decay rates in the region $\{r\leq \frac{t}{4}\}$.

gr-qc↗

arXiv2009

Improved decay for solutions to the linear wave equation on a Schwarzschild black hole

We prove that sufficiently regular solutions to the wave equation $\Box_gϕ=0$ on the exterior of the Schwarzschild black hole obey the estimates $|ϕ|\leq C_δv_+^{-{3/2}+δ}$ and $|\partial_tϕ|\leq C_δ v_+^{-2+δ}$ on a compact region of $r$ and along the event horizon. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite $r$ and along the event horizon.

gr-qc↗