Global existence of smooth solution to 2D relativistic membrane equation in asymptotically flat FLRW space‐time

Wei, Changhua

doi:10.1002/mma.5610

Cited by 4 publications

(2 citation statements)

References 19 publications

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“…Fortunately, these borderline terms exhibit linearizable feature. Therefore, motivated by [13] (see also [29,[40][41][42]), we resolve these difficulties by the method of linearization and hierarchies of energy estimates. In particular, we use the L ∞ − L ∞ estimate for the KG equation [27] in the procedure of linearization.…”

Section: The Ekg Systemmentioning

confidence: 99%

Future stability of the 1 + 3 Milne model for the Einstein–Klein–Gordon system

Wang

2019

Class. Quantum Grav.

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We study the small perturbations of the 1 + 3-dimensional Milne model for the Einstein-Klein-Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on the 1 + 3 splitting of the Bianchi-Klein-Gordon equations. Moreover, we treat the Bianchi-Klein-Gordon equations as evolution equations and establish the energy scheme in the sense that we only commute the Bianchi-Klein-Gordon equations with the spatially covariant derivatives while the normal derivative is not allowed. Contents1 2 J. WANG 7.1. Closing the bootstrap argument 53 Appendix A. Local in time development of CMC data 58 Appendix B. ODE estimate 61 Appendix C. Basic geometric conventions and identities 61 References 68 the future direction is locally isometric to the k = −1 vacuum Friedmann-Lemîatre-Robertson-Walker (FLRW) model. Andersson and Moncrief [6] first consider the nonlinear stability of this model (called hyperbolic cone spacetimes there), and show that for the constant mean curvature (CMC) Cauchy data for the vacuum Einstein equations close to the standard data for the hyperbolic cone spacetimes, the maximal future Cauchy development is globally foliated by the CMC Cauchy surfaces and causally geodesically complete to the future. This also motivates the current project. We refer to [1, 2, 3, 17] and references therein for more backgrounds on the existence of CMC foliations. There are several results concerning the decay of the KG field [19, 20, 30, 31]. In contrast to the wave equation, the KG equation is not conformal invariant and hence does not commute well with the symmetry of scaling. Due to this fact, Klainerman [20] employs the hyperbloid foliations which respect the Lorentz invariance of the KG operator and derives the asymptotic behavior of the KG field by the energy method alone. Furthermore, Katayama [19] somehow overcomes the incompatibility between the wave and KG field. Other method including Fourier analysis for the wave-Klein-Gordon system is established by Ionescu-Pausader [18]. Global solutions for the semilinear KG equations in the FLRW spacetimes are studied by Galstian-Yagdjian [16]. Let us review some nonlinear stability results of the Minkowski spacetime. Christodoulou and Klainerman [13] initiate a covariant proof based on the Bel-Robinson energy. Their proof relies upon the geometric foliations of spacetime, including maximal t = const slices and a family outgoing null cones, and null condition hidden inside the Bianchi equations. Lindblad and Rodnianski [21,22] devise another proof based on the wave coordinate gauge, under which the Einstein equations satisfy the weak null condition. This method is extended to the case of massive Einstein-Vlasov system by Lindblad and Taylor [23]. All of these works require the full symmetries of the Minkowski spacetime, including the conformal symmetries, suggesting that these methods can not apply to the EKG system st...

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Section: The Ekg Systemmentioning

confidence: 99%

Future stability of the 1 + 3 Milne model for the Einstein–Klein–Gordon system

Wang

2019

Class. Quantum Grav.

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“…Luo and Wei [19] proved the global existence of smooth solutions to 2D isentropic Chaplygin gases without vorticity in curved space when the initial data is small and has compact support. Recently, in order to investigate the influence of the background metric to the stability of the large solution of relativistic membrane equations, Wei [24] studied a class of time-dependent Lorentzian metric and showed that when the decay rate of the derivative of the given metric is larger than 3/2, then a class of time-dependent large solution to the relativistic membrane is globally stable, while, therein, the highest-order energy is still growing with respect to time.…”

Section: Background Materials and Outline Of This Papermentioning

confidence: 99%

Boundedness of the total energy of relativistic membranes evolving in a curved spacetime

LeFloch¹,

Wei²

2018

Journal of Differential Equations

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We establish a global existence theory for the equation governing the evolution of a relativistic membrane in a (possibly curved) Lorentzian manifold, when the spacetime metric is a perturbation of the Minkowski metric. Relying on the Hyperboloidal Foliation Method introduced by LeFloch and Ma in 2014, we revisit a theorem established earlier by Lindblad (who treated membranes in the flat Minkowski spacetime) and we provide a simpler proof of existence, which is also valid in a curved spacetime and, most importantly, leads to the important property that the total energy of the membrane is globally bounded in time. 1Our proof uses LeFloch and Ma's Hyperboloidal Foliation Method [15]- [18], which allows one to treat coupled systems of wave and Klein-Gordon equations. This method was built upon earlier work by Klainerman [12] and Hormander [9] on the quasilinear Klein-Gordon equation. The use of the hyperboloidal foliation of Minkowski spacetime to study coupled systems of wave and Klein-Gordon equation was investigated in [15] and several classes of such systems were then treated by this method, including the Einstein equations of general relativity [15,17,18]. Estimates in the hyperboloidal foliation are often more precise and it was observed in [15] that, for certain classes of equations, the energy of the solution is bounded globally in time. See LeFloch-Ma's theorem in [15, Chap. 6]. Our aim in the present paper is to extend this observation to the membrane equation (1.1). Our proof will, in addition, rely on a "double null" property of the membrane equation which was first observed by Lindblad [20]. Statement of the theoremBy the property of finite speed of propagation of the relativistic membranes, we at first specify the domain of the nonvanishing functions, which is the interior of the future light cone from the point (1, 0, 0) K := {(t, x) | r < t − 1}, and the following domain limited by two hyperboloids (with s 0 < s 1 )

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Stability of traveling wave for the relativistic string equation in de Sitter spacetime

Huang

Wei

2020

Journal of Mathematical Physics

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In this paper, we are concerned with the relativistic string equations in de Sitter spacetime. Since the background spacetime is nonflat, the governing system of equations is semilinear under the usual isothermal coordinates, contrary to the linear wave equations in the Minkowski spacetime. Based on the null condition enjoyed by the system and the method of [Luli et al., Adv. Math 329, 174–188 (2018)], we succeeded in proving the stability of the traveling wave solution to the relativistic string equations. We note that the traveling wave solution should be small in the sense of weighted energy.

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Global existence of smooth solution to 2D relativistic membrane equation in asymptotically flat FLRW space‐time

Cited by 4 publications

References 19 publications

Future stability of the 1 + 3 Milne model for the Einstein–Klein–Gordon system

Future stability of the 1 + 3 Milne model for the Einstein–Klein–Gordon system

Boundedness of the total energy of relativistic membranes evolving in a curved spacetime

Stability of traveling wave for the relativistic string equation in de Sitter spacetime

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