Global stability of solutions to nonlinear wave equations

Yang, Shiwu

doi:10.1007/s00029-014-0165-7

Cited by 18 publications

(27 citation statements)

References 33 publications

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“…We remark explicitly that the decay estimates of the above Corollary are indeed sufficient for applications to quasilinear problems with quadratic non-linearities. See [50,69,68,67]. We note also that the non-quantitative fixed-azimuthal mode statements of [35,36] are of course implied a fortiori by the above Corollary.…”

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confidence: 74%

Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M

2016

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This paper concludes the series begun in [M. Dafermos and I. Rodnianski Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases a ≪ M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal a < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al (ed.), World Scientific, Singapore, , pp. 132189, arXiv:1010]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincaré], together with a streamlined continuity argument in the parameter a, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notations so that it can be read independently of previous work.

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confidence: 74%

Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M

2016

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“…The energy decay estimates derived in the previous section are sufficient to obtain pointwise bound for the Maxwell field F after commuting the equation with Z = {∂ t , Ω} for sufficiently many times, e.g., in [29], four derivatives were used to show the pointwise bound for the solution. The aim of this section is to derive the pointwise bound for the Maxwell field F merely assuming M 2 is finite, that is, we commute the equation with Z for only twice.…”

Section: Pointwise Bound For the Maxwell Fieldmentioning

confidence: 99%

“…Then using the pigeon hole argument similar to the proof of Proposition 5 for the energy flux decay of the Maxwell field in the interior, the energy decay estimate (64) for the scala field follows from the energy estimate (59), the integral of the energy flux estimate (60) and the r-weighted energy estimate (63). For a detailed proof for this, we refer to Proposition 2 of [29].…”

Section: Proposition 14mentioning

confidence: 99%

“…The integral of the first term in the above inequality can be controlled by using Gronwall's inequality both in (29) and (30). In particular this shows that estimate (28) follows from (30).…”

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confidence: 91%

“…The inhomogeneous term can be bounded by using Cauchy-Schwarz inequality and the integrated local energy estimates. Once we have bound for the boundary terms on {r = R}, the r-weighted energy estimate (27) follows from the identity (29) and Gronwall's inequality.…”

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Decay of solutions of Maxwell–Klein–Gordon equations with arbitrary Maxwell field

Yang

2016

Anal. PDE

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In the author's work [30], it has been shown that solutions of Maxwell-Klein-Gordon equations in R 3+1 possess some form of global strong decay properties with data bounded in some weighted energy space. In this paper, we prove pointwise decay estimates for the solutions for the case when the initial data are merely small on the scalar field but can be arbitrarily large on the Maxwell field. This extends the previous result of Lindblad-Sterbenz [16], in which smallness was assumed both for the scalar field and the Maxwell field.We make several remarks:Remark 1. The second order derivatives of the initial data is the minimum regularity we need to derive the above pointwise decay of the solution. Similar decay estimates hold for the higher order derivatives of the solution if higher order weighted Sobolev norms of the initial data are known.Remark 2. The restriction on γ 0 , that is 0 < γ 0 < 1, is merely for the sake of brevity. If γ 0 ≥ 1, then the decay property of the solutions propagates in the exterior region (t + 2 ≤ r). In other words, we have the same decay estimates as in the theorem for τ ≤ 0. However in the interior region where τ > 0, the maximal decay rate is τ −2 + (corresponding to γ 0 = 1), that is, the decay rate in the interior region for γ 0 ≥ 1 in general can not be better than that of γ 0 = 1.Remark 3. Since we assume the scalar field is small, the charge is also small by definition. Combined with the techniques in [30], our approach can be adapted to the case with large charge. There are two ways of generalizations. The first one is to relax the assumption on the scalar field as in [28] so that the charge can be large. Secondly we can consider the following unphysical equations:for some constant λ.Compared to the previous result of Lindblad-Sterbenz [16], we have made the following improvements: first of all, we obtain pointwise decay estimates for solutions of (MKG) for a class of large initial data. We only require smallness on the scalar field. In particular our initial data for (MKG) can be arbitrarily large. Combining the method in [28], we can even make the data on the scalar field large in the energy space. Secondly we have lower regularity on the initial data. In [16], it was assumed that the derivative of the initial data decays one order better, that is, ∇ I (E df , H), D I (Dφ 0 , φ 1 ) belong to the weighted Sobolev space with weights (1 + r) 1+γ0+2|I| , while in this paper we only assume that the angular derivatives of the data obey this improved decay (see the definition of M, E). For the other derivatives, the weights is merely (1 + r) 1+γ0 . This makes the analysis more delicate. Moreover, as the solution decays weaker initially, our decay rate is weaker than that in [16] (only decay rate in τ , the decay in r is the same). However if we assume the same decay of the initial data as in [16], then we are able to obtain the same decay for the solution.We use a new approach developed in [30] to study the asymptotic behavior of solutions of (MKG). This new method was originally int...

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A Local Energy Estimate for Wave Equations on Metrics Asymptotically Close to Kerr

Lindblad

Tohaneanu

2020

Ann. Henri Poincaré

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Global stability of solutions to nonlinear wave equations

Cited by 18 publications

References 33 publications

Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M

Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M

Decay of solutions of Maxwell–Klein–Gordon equations with arbitrary Maxwell field

A Local Energy Estimate for Wave Equations on Metrics Asymptotically Close to Kerr

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