arXiv subjects

Jonathan Luk

Publications and source records attributed to Jonathan Luk.

At least 37 records · Page 2Linked to original sources

arXiv2021

Phase mixing for solutions to 1D transport equation in a confining potential

Consider the linear transport equation in $1$D under an external confining potential $Φ$: \begin{equation*} \partial_t f + v \partial_x f - \partial_x Φ\partial_v f = 0. \end{equation*} For $Φ= \frac{x^2}{2} + \frac{εx^4}{2}$ (with $ε>0$ small), we prove phase mixing and quantitative decay estimates for $\partial_t φ:= - Δ^{-1} \int_{\mathbb{R}} \partial_t f \, \mathrm{d} v$, with an inverse polynomial decay rate $O(\langle t\rangle^{-2})$. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov--Poisson system in $1$D under the external potential $Φ$.

math.AP↗

arXiv2021

Nonlinear interaction of three impulsive gravitational waves I: main result and the geometric estimates

Impulsive gravitational waves are (weak) solutions to the Einstein vacuum equations such that the Riemann curvature tensor admits a delta singularity along a null hypersurface. The interaction of impulsive gravitational waves is then represented by the transversal intersection of these singular null hypersurfaces. This is the first of a series of two papers in which we prove that for all suitable $\mathbb U(1)$-symmetric initial data representing three "small amplitude" impulsive gravitational waves propagating towards each other transversally, there exists a local solution to the Einstein vacuum equations featuring the interaction of these waves. Moreover, we show that the solution remains Lipschitz everywhere and is $H^2_{loc} \cap C_{loc}^{1, \frac 14-}$ away from the impulsive gravitational waves. This is the first construction of solutions to the Einstein vacuum equations featuring the interaction of three impulsive gravitational waves. In this paper, we focus on the geometric estimates, i.e. we control the metric and the null hypersurfaces assuming the wave estimates. The geometric estimates rely crucially on the features of the spacetime with three interacting impulsive gravitational waves, particularly that each wave is highly localized and that the waves are transversal to each other. In the second paper of the series, we will prove the wave estimates and complete the proof.

gr-qc↗

arXiv2020

High-frequency limits and null dust shell solutions in general relativity

Consider the characteristic initial value problem for the Einstein vacuum equations without any symmetry assumptions. Impose a sequence of data on two intersecting null hypersurfaces, each of which is foliated by spacelike $2$-spheres. Assume that the sequence of data is such that the derivatives of the metrics along null directions are only uniformly bounded in $L^2$ but the derivatives of the metrics along the directions tangential to the $2$-spheres obey higher regularity bounds uniformly. By the results in [J. Luk and I. Rodnianski, Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations, Camb. J. Math. 5(4), 2017], it follows that the sequence of characteristic initial value problems gives rise to a sequence of vacuum spacetimes $(\mathcal M, g_n)$ in a fixed double-null domain $\mathcal M$. Since the existence theorem requires only very low regularity, the sequence of solutions may exhibit both oscillations and concentrations, and the limit need not be vacuum. We prove nonetheless that, after passing to a subsequence, the metrics converge in $C^0$ and weakly in $W^{1,2}$ to a solution of the Einstein-null dust system with two families of (potentially measure-valued) null dust. We show moreover that all sufficiently regular solutions to the Einstein-null dust system (with potentially measure-valued null dust) adapted to a double null coordinate system arise locally as weak limits of solutions to the Einstein vacuum system in the manner described above. As a consequence, we also give the first general local existence and uniqueness result for solutions to the Einstein-null dust system for which the null dusts are only measures. This in particular includes as a special case solutions featuring propagating and interacting shells of null dust.

math.AP↗

arXiv2019

Asymptotic properties of linear field equations in anti-de Sitter space

We study the global dynamics of the wave equation, Maxwell's equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates "lose a derivative". We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

gr-qc↗

arXiv2019

Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. The interior of the black hole region

This is the first and main paper of a two-part series, in which we prove the $C^{2}$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos-Rodnianski that the maximal globally hyperbolic future development of any admissible two-ended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, across which the spacetime is $C^{0}$-future-extendible (in particular, the $C^{0}$-formulation of the strong cosmic censorship conjecture is false). Nevertheless, the main conclusion of the present series of papers is that for a generic (in the sense of being open and dense relative to appropriate topologies) class of such data, the spacetime is future-inextendible with a Lorentzian metric of higher regularity (specifically, $C^{2}$). In this paper, we prove that the solution is $C^{2}$-future-inextendible under the condition that the scalar field obeys an $L^{2}$-averaged polynomial lower bound along each of the event horizons. This, in particular, improves upon a previous result of Dafermos, which required instead a pointwise lower bound. Key to the proof are appropriate stability and instability results in the interior of the black hole region, whose proofs are in turn based on ideas from the work of Dafermos-Luk on the stability of the Kerr Cauchy horizon (without symmetry) and from our previous paper on linear instability of Reissner-Nordström Cauchy horizon. In the second paper of the series (arXiv:1702.05716), which concerns analysis in the exterior of the black hole region, we show that the $L^2$-averaged polynomial lower bound needed for the instability result indeed holds for a generic class of admissible two-ended asymptotically flat Cauchy initial data.

gr-qc↗

arXiv2019

Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data II. The exterior of the black hole region

This is the second and last paper of a two-part series in which we prove the $C^2$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. In the first paper (arXiv:1702.05715), we showed that the maximal globally hyperbolic future development of an admissible asymptotically flat Cauchy initial data set is $C^2$-future-inextendible provided that an $L^2$-averaged (inverse) polynomial lower bound for the derivative of the scalar field holds along each horizon. In this paper, we show that this lower bound is indeed satisfied for solutions arising from a generic set of Cauchy initial data. Roughly speaking, the generic set is open with respect to a (weighted) $C^1$ topology and is dense with respect to a (weighted) $C^\infty$ topology. The proof of the theorem is based on extensions of the ideas in our previous work on the linear instability of Reissner-Nordström Cauchy horizon, as well as a new large data asymptotic stability result which gives good decay estimates for the difference of the radiation fields for small perturbations of an arbitrary solution.

gr-qc↗

arXiv2019

The interior of dynamical extremal black holes in spherical symmetry

We study the nonlinear stability of the Cauchy horizon in the interior of extremal Reissner-Nordström black holes under spherical symmetry. We consider the Einstein-Maxwell-Klein-Gordon system such that the charge of the scalar field is appropriately small in terms of the mass of the background extremal Reissner-Nordström black hole. Given spherically symmetric characteristic initial data which approach the event horizon of extremal Reissner-Nordström sufficiently fast, we prove that the solution extends beyond the Cauchy horizon in $C^{0,\frac{1}{2}}\cap W^{1,2}_{loc}$, in contrast to the subextremal case (where generically the solution is $C^0\setminus (C^{0,\frac{1}{2}}\cap W^{1,2}_{loc}))$. In particular, there exist non-unique spherically symmetric extensions which are moreover solutions to the Einstein-Maxwell-Klein-Gordon system. Finally, in the case that the scalar field is chargeless and massless, we additionally show that the extension can be chosen so that the scalar field remains Lipschitz.

gr-qc↗

arXiv2017

Weak null singularities in general relativity

We construct a class of spacetimes (without symmetry assumptions) satisfying the vacuum Einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2-sphere. The metric of these spacetimes extends continuously beyond the singularities while the Christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries. The construction shows moreover that the singularities are stable in a suitable sense. These singularities are stronger than the impulsive gravitational spacetimes considered by Luk-Rodnianski and conjecturally they are present in the interior of generic black holes arising from gravitational collapse.

gr-qc↗

arXiv2017

The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon

We initiate a series of works where we study the interior of dynamical rotating vacuum black holes without symmetry. In the present paper, we take up the problem starting from appropriate Cauchy data for the Einstein vacuum equations defined on a hypersurface already within the black hole interior, representing the expected geometry just inside the event horizon. We prove that for all such data, the maximal Cauchy evolution can be extended across a non-trivial piece of Cauchy horizon as a Lorentzian manifold with continuous metric. In subsequent work, we will retrieve our assumptions on data assuming only that the black hole event horizon geometry suitably asymptotes to a rotating Kerr solution. In particular, if the exterior region of the Kerr family is proven to be dynamically stable---as is widely expected---then it will follow that the $C^0$-inextendibility formulation of Penrose's celebrated strong cosmic censorship conjecture is in fact false. The proof suggests, however, that the $C^0$-metric Cauchy horizons thus arising are generically singular in an essential way, representing so-called "weak null singularities", and thus that a revised version of strong cosmic censorship holds.

gr-qc↗

arXiv2017

Einstein equations under polarized $\mathbb U(1)$ symmetry in an elliptic gauge

We prove local existence of solutions to the Einstein--null dust system under polarized $\mathbb U(1)$ symmetry in an elliptic gauge. Using in particular the previous work of the first author on the constraint equations, we show that one can identify freely prescribable data, solve the constraints equations, and construct a unique local in time solution in an elliptic gauge. Our main motivation for this work, in addition to merely constructing solutions in an elliptic gauge, is to provide a setup for our companion paper in which we study high frequency backreaction for the Einstein equations. In that work, the elliptic gauge we consider here plays a crucial role to handle high frequency terms in the equations. The main technical difficulty in the present paper, in view of the application in our companion paper, is that we need to build a framework consistent with the solution being high frequency, and therefore having large higher order norms. This difficulty is handled by exploiting a reductive structure in the system of equations.

gr-qc↗

arXiv2017

High-frequency backreaction for the Einstein equations under polarized $\mathbb U(1)$ symmetry

Known examples in plane symmetry or Gowdy symmetry show that given a $1$-parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a non-trivial stress-energy-momentum tensor. We consider this phenomenon under polarized $\mathbb U(1)$ symmetry - a much weaker symmetry than most of the known examples - such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized-$\mathbb U(1)$-symmetric solution to the Einstein-multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large, and appears to be the first construction of such examples with more than two families.

gr-qc↗

arXiv2017

Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations

In this paper, we study the problem of the nonlinear interaction of impulsive gravitational waves for the Einstein vacuum equations. The problem is studied in the context of a characteristic initial value problem with data given on two null hypersurfaces and containing curvature delta singularities. We establish an existence and uniqueness result for the spacetime arising from such data and show that the resulting spacetime represents the interaction of two impulsive gravitational waves germinating from the initial singularities. In the spacetime, the curvature delta singularities propagate along 3-dimensional null hypersurfaces intersecting to the future of the data. To the past of the intersection, the spacetime can be thought of as containing two independent, non-interacting impulsive gravitational waves and the intersection represents the first instance of their nonlinear interaction. Our analysis extends to the region past their first interaction and shows that the spacetime still remains smooth away from the continuing propagating individual waves. The construction of these spacetimes are motivated in part by the celebrated explicit solutions of Khan-Penrose and Szekeres. The approach of this paper can be applied to an even larger class of characteristic data and in particular implies an extension of the theorem on formation of trapped surfaces by Christodoulou and Klainerman-Rodnianski, allowing non-trivial data on the initial incoming hypersurface.

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↗

arXiv2017

Dynamical black holes with prescribed masses in spherical symmetry

We review our recent work on a construction of spherically symmetric global solutions to the Einstein--scalar field system with large bounded variation norms and large Bondi masses. We show that similar ideas, together with Christodoulou's short pulse method, allow us to prove the following result: Given $M_i \geq M_f>0$ and $ε>0$, there exists a spherically symmetric (black hole) solution to the Einstein--scalar field system such that up to an error of size $ε$, the initial Bondi mass is $M_i$ and the final Bondi mass is $M_f$. Moreover, if one assumes a continuity property of the final Bondi mass (which in principle follows from known techniques in the literature), then for $M_i>M_f>0$, the above result holds without an $ε$-error.

gr-qc↗

arXiv2016

Shock formation in solutions to the $2D$ compressible Euler equations in the presence of non-zero vorticity

We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple plane wave solutions, starting from smooth initial conditions, is stable under perturbations that break the plane symmetry. Moreover, we provide a sharp asymptotic description of the singularity formation. The new feature of our work is that the perturbed solutions are allowed to have small but non-zero vorticity, even at the location of the shock. Thus, our results provide the first constructive description of the vorticity near a singularity formed from compression: relative to a system of geometric coordinates adapted to the acoustic characteristics, the vorticity remains many times differentiable, all the way up to the shock. In addition, relative to the Cartesian coordinates, the vorticity remains bounded up to the shock. To control the vorticity, we rely on a coalition of new geometric and analytic insights that complement the ones used by Christodoulou in his groundbreaking, sharp proof of shock formation in vorticity-free regions. In particular, we rely on a new formulation of the compressible Euler equations exhibiting remarkable structures. To derive estimates, we construct an eikonal function adapted to the acoustic characteristics (which correspond to sound wave propagation) and a related set of geometric coordinates and differential operators. Thanks to the remarkable structure of the equations, the same set of coordinates and differential operators can be used to analyze the vorticity, whose characteristics are transversal to the acoustic characteristics. In particular, our work provides the first constructive description of shock formation without symmetry assumptions in a system with multiple speeds.

math.AP↗

arXiv2016

The hidden null structure of the compressible Euler equations and a prelude to applications

We derive a new formulation of the compressible Euler equations exhibiting remarkable structures, including surprisingly good null structures. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity, defined to be vorticity divided by density. The equations allow one to use the full power of the geometric vectorfield method in treating the "wave part" of the system. A crucial feature of the new formulation is that all derivative-quadratic inhomogeneous terms verify the strong null condition. The latter is a nonlinear condition signifying the complete absence of nonlinear interactions involving more than one differentiation in a direction transversal to the acoustic characteristics. Moreover, the same good structures are found in the equations verified by the Euclidean divergence and curl of the specific vorticity. This is important because one needs to combine estimates for the divergence and curl with elliptic estimates to obtain sufficient regularity for the specific vorticity, whose derivatives appears as inhomogeneous terms in the wave equations. The above structures collectively open the door for our forthcoming results: exhibiting a stable regime in which initially smooth solutions develop a shock singularity (in particular the first Cartesian coordinate partial derivatives of the velocity and density blow up) while, relative to a system of geometric coordinates adapted to the acoustic characteristics, the solution (including the vorticity) remains many times differentiable, all the way up to the shock. The good null structures, which are often associated with global solutions, are in fact key to proving that the shock singularity forms. Our secondary goal in this article is to overview the central role that the structures play in the proof.

math.AP↗

arXiv2016

Stable shock formation for nearly simple outgoing plane symmetric waves

In an influential 1964 article, P. Lax studied $2 \times 2$ genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax's data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou's framework for studying shock formation to treat a new solution regime in which wave dispersion is not present.

math.AP↗

arXiv2016

Instability results for the wave equation in the interior of Kerr black holes

We prove that a large class of smooth solutions $ψ$ to the linear wave equation $\Box_gψ=0$ on subextremal rotating Kerr spacetimes which are regular and decaying along the event horizon become singular at the Cauchy horizon. More precisely, we show that assuming appropriate upper and lower bounds on the energy along the event horizon, the solution has infinite (non-degenerate) energy on any spacelike hypersurfaces intersecting the Cauchy horizon transversally. Extrapolating from known results in the Reissner--Nordström case, the assumed upper and lower bounds required for our theorem are conjectured to hold for solutions arising from generic smooth and compactly supported initial data on a Cauchy hypersurface. This result is motivated by the strong cosmic censorship conjecture in general relativity.

gr-qc↗