Weak Null Condition Research Articles

The weak null condition on Kerr backgrounds

The weak null condition on Kerr backgrounds

Asymptotic Completeness for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in Yu (Modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition, 2021). Starting from a global solution u u to the quasilinear wave equation, we rigorously show that well chosen asymptotic variables solve the same reduced system with small error terms. This allows us to recover the scattering data for our system, as well as to construct a matching exact solution to the reduced system.

Scattering from infinity for semilinear wave equations satisfying the null condition or the weak null condition

We show global existence backward from scattering data at infinity for semilinear wave equations satisfying the null condition or the weak null condition. Semilinear terms satisfying the weak null condition appear in many equations in physics. The scattering data is given in terms of the radiation field, although in the case of the weak null condition there is an additional logarithmic term in the asymptotic behavior that has to be taken into account. Our results are sharp in the sense that the solution has the same spatial decay as the radiation field does along null infinity, which is assumed to decay at a rate that is consistent with the forward problem. The proof uses a higher order asymptotic expansion together with a new fractional Morawetz estimate with strong weights at infinity.

Global existence for systems of nonlinear wave and Klein–Gordon equations in two space dimensions under a kind of the weak null condition

We consider coupled systems of nonlinear wave and Klein–Gordon equations in two space dimensions with cubic nonlinearities. For these kinds of systems, the small data global existence is already known if the cubic nonlinearity satisfies a certain condition related to the null condition. In this article, our aim is to investigate the small data global existence under a condition related to the weak null condition. We also make a remark on the asymptotic behavior of global solutions.

Global existence and blow up for systems of nonlinear wave equations related to the weak null condition

<p style='text-indent:20px;'>We discuss how the higher-order term <inline-formula><tex-math id="M1">\begin{document}$ |u|^q $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (q>1+2/(n-1)) $\end{document}</tex-math></inline-formula> has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_t^2 u-\Delta u = |v|^p, \qquad \partial_t^2 v-\Delta v = |\partial_t u|^{(n+1)/(n-1)} +|u|^q $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in <inline-formula><tex-math id="M3">\begin{document}$ n\,(\geq 2) $\end{document}</tex-math></inline-formula> space dimensions. We show the existence of a certain "critical curve" in the <inline-formula><tex-math id="M4">\begin{document}$ pq $\end{document}</tex-math></inline-formula>-plane such that for any <inline-formula><tex-math id="M5">\begin{document}$ (p,q) $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M6">\begin{document}$ (p,q>1) $\end{document}</tex-math></inline-formula> lying below the curve, nonexistence of global solutions occurs, whereas for any <inline-formula><tex-math id="M7">\begin{document}$ (p,q) $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M8">\begin{document}$ (p>1+3/(n-1),\,q>1+2/(n-1)) $\end{document}</tex-math></inline-formula> lying exactly on it, this system admits a unique global solution for small data. When <inline-formula><tex-math id="M9">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula>, the discussion for the above system with <inline-formula><tex-math id="M10">\begin{document}$ (p,q) = (3,3) $\end{document}</tex-math></inline-formula>, which lies on the critical curve, has relevance to the study on systems satisfying the weak null condition, and we obtain a new result of global existence for such systems. Moreover, in the particular case of <inline-formula><tex-math id="M11">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ p = 4 $\end{document}</tex-math></inline-formula> it is observed that no matter how large <inline-formula><tex-math id="M13">\begin{document}$ q $\end{document}</tex-math></inline-formula> is, the higher-order term <inline-formula><tex-math id="M14">\begin{document}$ |u|^q $\end{document}</tex-math></inline-formula> never becomes negligible and it essentially affects the lifespan of small solutions.</p>

A Fuchsian viewpoint on the weak null condition

We analyze systems of semilinear wave equations in 3 + 1 dimensions whose associated asymptotic equation admit bounded solutions for suitably small choices of initial data. Under this special case of the weak null condition, which we refer to as the bounded weak null condition , we prove the existence of solutions to these systems of wave equations on neighborhoods of spatial infinity under a small initial data assumption. Existence is established using the Fuchsian method. This method involves transforming the wave equations into a Fuchsian equation defined on a bounded spacetime region. The existence of solutions to the Fuchsian equation then follows from an application of the existence theory developed in [11] . This, in turn, yields, by construction, solutions to the original system of wave equations on a neighborhood of spatial infinity.

Scattering from Infinity of the Maxwell Klein Gordon Equations in Lorenz Gauge

We prove global existence backwards from the scattering data posed at infinity for the Maxwell Klein Gordon equations in Lorenz gauge satisfying the weak null condition. The asymptotics of the solutions to the Maxwell Klein Gordon equations in Lorenz gauge were shown to be wave like at null infinity and homogeneous towards timelike infinity in arXiv:1803.11086 and expressed in terms of radiation fields, and thus our scattering data will be given in the form of radiation fields in the backward problem. We give a refinement of the asymptotics results in arXiv:1803.11086, and then making use of this refinement, we find a global solution which attains the prescribed scattering data at infinity. Our work starts from the approach in [22] and is more delicate since it involves nonlinearities with fewer derivatives. Our result corresponds to "existence of scattering states" in the scattering theory. The method of proof relies on a suitable construction of the approximate solution from the scattering data, a weighted conformal Morawetz energy estimate and a spacetime version of Hardy inequality.

Modified Wave Operators for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

We prove the existence of the modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition. This is accomplished in three steps. First, we derive a new reduced asymptotic system for the quasilinear wave equation by modifying Hörmander’s method. Next, we construct an approximate solution, by solving our new reduced system given some scattering data at infinite time. Finally, we prove that the quasilinear wave equation has a global solution which agrees with the approximate solution at infinite time.

Necessary and Sufficient Null Space Condition for Nuclear Norm Minimization in Low-Rank Matrix Recovery

Low-rank matrix recovery has found many applications in science and engineering such as machine learning, system identification, and Euclidean embedding. However, the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is the nuclear norm minimization. Recently, some authors established the necessary and sufficient null space conditions for nuclear norm minimization to recover every possible low-rank matrix with rank at most $r$ (the strong null space condition). Oymak et al. established a null space condition for successful recovery of a given low-rank matrix (the weak null space condition) using nuclear norm minimization, and derived the phase transition for the nuclear norm minimization. In this paper, we show that the weak null space condition proposed by Oymak et al. is only a sufficient condition for successful matrix recovery using nuclear norm minimization, and is not a necessary condition as claimed. We further give a weak null space condition for low-rank matrix recovery, which is both necessary and sufficient for the success of nuclear norm minimization. At the core of our derivation are an inequality for characterizing the nuclear norms of block matrices, and the conditions for equality to hold in that inequality.

Global smooth solutions to 3D irrotational Euler equations for Chaplygin gases

We study here the Cauchy problem associated with the isentropic and compressible Euler equations for Chaplygin gases. Based on the new formulation of the compressible Euler equations in J. Luk and J. Speck [The hidden null structure of the compressible Euler equations and a prelude to applications, J. Hyperbolic Differ. Equ. 17 (2020) 1–60] we show that the wave system satisfied by the modified density and the velocity for Chaplygin gases satisfies the weak null condition. We then prove the global existence of smooth solutions to the irrotational and isentropic Chaplygin gases without introducing a potential function, when the initial data are small perturbations to a constant state.

Stability of Minkowski space and polyhomogeneity of the metric

We study the nonlinear stability of the $$(3+1)$$ -dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the nonlinear initial value problem using an iteration scheme in which we solve a linearized equation globally at each step; we use a generalized harmonic gauge and implement constraint damping to fix the geometry of null infinity. The linear analysis is largely based on energy and vector field methods originating in work by Klainerman. The weak null condition of Lindblad and Rodnianski arises naturally as a nilpotent coupling of certain metric components in a linear model operator at null infinity. Upon compactifying $$\mathbb {R}^4$$ to a manifold with corners, with boundary hypersurfaces corresponding to spacelike, null, and timelike infinity, we show, using the framework of Melrose’s b-analysis, that polyhomogeneous initial data produce a polyhomogeneous spacetime metric. Finally, we relate the Bondi mass to a logarithmic term in the expansion of the metric at null infinity and prove the Bondi mass loss formula.

Possibility of traversable wormhole formation in the dark matter halo with istropic pressure

We explore the possibility of traversable wormhole formation in the dark matter halos. We obtain the exact solutions of the spherical symmetry traversable wormhole with isotropic pressure condition, based on the Navarro–Frenk–White (NFW), Thomas–Fermi (TF) and Pseudo Isothermal (PI) matter density profiles. The derived traversable wormhole solution satisfies the flare-out condition for a specific dark matter center density and equation of state. We extend the spherical symmetry traversable wormhole solutions to an axisymmetric one. The weak energy condition (WEC) and null energy condition (NEC) are then checked near the wormhole throat, and we find that these traversable wormholes violate the WEC and NEC. Our traversable wormhole solutions show that the dark matter at the center of wormhole spacetime will be redistributed by the presence of a traversable wormhole, and the behavior of dark matter density is similar to a black hole spike.

On the Global Behavior of Weak Null Quasilinear Wave Equations

AbstractWe consider a subclass of those quasilinear wave equations in 3 + 1 space‐time dimensions that satisfy the “weak null condition” as defined by Lindblad and Rodnianski , and study the large‐time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is . Global solutions for such equations have been constructed by Lindblad and Alinhac. Our main results are the derivation of a precise asymptotic system with good error bounds, and a detailed description of the behavior of solutions close to the light cone, including the blowup at infinity. © 2019 Wiley Periodicals, Inc.

The weak null condition in free-evolution schemes for numerical relativity: dual foliation GHG with constraint damping

All strategies for the treatment of future null-infinity in numerical relativity involve some form of regularization of the field equations. In a recent proposal that relies on the dual foliation formalism this is achieved by the use of an asymptotically Minkowskian generalized harmonic tensor basis. For the scheme to work however, derivatives of certain coordinate light-speeds must decay fast enough. Presently, we generalize the method of asymptotic expansions for nonlinear wave equations to treat first order symmetric hyperbolic systems. We then use this heuristic tool to extract the expected rates of decay of the metric near null-infinity in a free-evolution setting. We show, within the asymptotic expansion, that by carefully modifying the non-principal part of the field equations by the addition of constraints, we are able to obtain optimal decay rates even when the constraints are violated. The light-speed condition can hence be satisfied, which paves the way for the explicit numerical treatment of future null-infinity. We then study the behavior of the Trautman–Bondi mass under the decay results predicted by the asymptotic expansion. Naively the mass seems to be unbounded, but we see first that the divergent terms can be replaced with a combination of the constraints and the Einstein field equations, and second that the Bondi mass loss formula is recovered within the framework. Both of the latter results hold in the presence of small constraint violations.

Global Stability of Minkowski Space for the Einstein\u2013Vlasov System in the Harmonic Gauge

Minkowski space is shown to be globally stable as a solution to the massive Einstein–Vlasov system. The proof is based on a harmonic gauge in which the equations reduce to a system of quasilinear wave equations for the metric, satisfying the weak null condition, coupled to a transport equation for the Vlasov particle distribution function. Central to the proof is a collection of vector fields used to control the particle distribution function, a function of both spacetime and momentum variables. The vector fields are derived using a general procedure, are adapted to the geometry of the solution and reduce to the generators of the symmetries of Minkowski space when restricted to acting on spacetime functions. Moreover, when specialising to the case of vacuum, the proof provides a simplification of previous stability works.

Asymptotic Behavior of the Maxwell–Klein–Gordon System

In previous work on the Maxwell-Klein-Gordon system first existence and then decay estimates have been shown. Here we show that the Maxwell-Klein-Gordon in the Lorentz gauge satisfy the "weak null condition" and we give the detailed asymptotics for the scalar field and the potential. These asymptotics have two parts, one wave like along outgoing light cones at null infinity, and one homogeneous inside the light cone at time like infinity. Here the charge plays a crucial role in imposing an oscillating factor in the asymptotic system for the field, and in the null asymptotics for the potential. Similar results have previously been shown for Einstein's equations in wave coordinates, and the Maxwell-Klein-Gordon system apart from being interesting in itself also provides a simpler semilinear model of the quasilinear Enstein's equations.

Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition

We give an alternative proof of the global existence result originally due to Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave equations in three space dimensions satisfying the weak null condition. The feature of the new proof lies in that it never uses the Lorentz boost operator in the energy integral argument. The proof presented here has an advantage over the former one in that the assumption of compactness of the support of data can be eliminated and the amount of regularity of data can be lowered in a straightforward manner. A recent result of Zha for the scalar unknowns is also refined.

The weak null condition and Kaluza–Klein spacetimes

We prove the non-linear stability of a system of non-linear wave equations satisfying the weak null condition. In particular, this includes the case of the non-linear stability of Minkowski spacetime times a [Formula: see text]-torus subject to perturbations depending only on the non-compact coordinates. Our argument closely follows the proof of the non-linear stability of Minkowski spacetime by Lindblad and Rodnianski.

Global Existence for a System of Quasi-Linear Wave Equations in 3D Satisfying the Weak Null Condition

Abstract We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration. The original theorem of Alinhac for the scalar unknowns is also refined.

The small data solutions of general 3-D quasilinear wave equations. II

This paper is a continuation of the work in [8], where the authors established the global existence of smooth small data solutions to the general 3-D quasilinear wave equation ∑i,j=03gij(u,∂u)∂ij2u=0 when the weak null condition holds. In the present paper, we show that the smooth small data solutions of equation ∑i,j=03gij(u,∂u)∂ij2u=0 will blow up in finite time when the weak null condition does not hold and a generic nondegenerate condition on the initial data is satisfied, moreover, a precise blowup time is completely determined. Therefore, collecting the main results in this paper and [8], we have given a basically complete study on the blowup or global existence of small data solutions to the 3-D quasilinear wave equation ∑i,j=03gij(u,∂u)∂ij2u=0.