The Hyperboloidal Foliation Method

LeFloch, Philippe G.; Ma, Yue

doi:10.1142/9427

Cited by 62 publications

(186 citation statements)

References 96 publications

(81 reference statements)

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“…For W-KG system of the form (1.1), we have established its global existence in [13] for quadratic nonlinearities satisfying the so-called "minimal null condition", i.e. we only demand null conditions on quadratic terms of wave components coupled in wave equation.…”

Section: Objectivementioning

confidence: 99%

Global solutions of nonlinear wave–Klein–Gordon system in one space dimension

2020

Nonlinear Analysis

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In this article we will develop some techniques aimed at the strong couplings in twodimensional wave-Klein-Gordon system. We distinguish the roles of different type of decay factors and develop a method which permits us to "exchange" one type of decay into the other. Then a global existence result of a model problem is established. We also give a sketch of the Klein-Gordon-Zakharov model system and establish the associate global existence result. * The present work belongs to a research project "Global stability of quasilinear wave-Klein-Gordon system in 2 + 1 space-time dimension" (11601414), supported by NSFC.

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Section: Objectivementioning

confidence: 99%

Global solutions of nonlinear wave–Klein–Gordon system in one space dimension

2020

Nonlinear Analysis

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“…The local existence theory with bounds that are uniform in κ is developed in Sections 6 to 8 and leads us in Section 9 to our main statement concerning the comparison between the modified and the classical theories. (9) Wave-Klein-Gordon systems have brought a lot of attention in mathematical analysis: see, for instance, Bachelot [1,2], Delort et al [7,8], Katayama [11], Lannes [12], and LeFloch and Ma [13] and the references therein. (10) specifically ρ = In this section, we formulate the initial value problem for the modified gravity system, by prescribing suitable initial data on a spacelike hypersurface.…”

Section: Theorem 11 (The Cauchy Developments Of Modified Gravity)mentioning

confidence: 99%

The Mathematical Validity of the f(R) Theory of Modified Gravity

LeFloch¹,

Ma²

2017

Mémoires Soc. Math. France

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Abstract. -We establish here a well-posedness theory for the f (R) theory of modified gravity, which is a generalization of Einstein's theory of gravitation. The scalar curvature R of the spacetime, which arises in the integrand of the Einstein-Hilbert functional, is replaced here by an arbitrary function f = f (R). The field equations involve up to fourth-order derivatives of the unknown spacetime metric, and the main challenge is to understand the structure of these high-order derivative terms. First of all, we propose a formulation of the initial value problem in modified gravity when the initial data are prescribed on a spacelike hypersurface. In addition to the induced metric and second fundamental form of the initial slice and the initial matter content, an initial data set for modified gravity must also provide the spacetime scalar curvature and its time-derivative. Next, in order to tackle the initial value problem, we introduce an augmented conformal formulation, as we call it, in which the spacetime scalar curvature is regarded as an independent variable. In particular, in the so-called wave gauge, we prove that the field equations of modified gravity reduce a coupled system of nonlinear wave-Klein-Gordon equations with defocusing potential, whose main unknowns are the conformally-transformed metric and the scalar curvature, as well as the matter fields. Based on this novel formulation, we are able to establish the existence of maximal globally hyperbolic developments of modified gravity when, for definiteness, the matter is represented by a scalar field. We analyze the so-called Jordan coupling and we work with the Einstein metric, which is conformally equivalent to the physical metric -the conformal factor depending upon the (unknown) scalar curvature. Our analysis of these conformal field equations in the Einstein metric leads us to a rigorous validation of the theory of modified gravity. We derive quantitative estimates in suitable functional spaces, which are uniform in terms of the nonlinearity f (R), and we prove that asymptotically flat spacetimes of modified gravity are 'close' to Einstein spacetimes, when the defining function f (R) in the action functional of modified gravity is 'close' to the Einstein-Hilbert integrand R.

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“…This makes decay estimates adapted to the standard t foliation more obviously compatible with the vector field method; in the relativistic case one may argue that the estimates are more adapted to hyperboloidal foliations (see e.g. [7]; and also [10,17] for some recent developments). We fully exploit this compatibility for our relatively short proofs given below.…”

Section: Proposition 1 Let ν(T Q) := R D ν(T Qmentioning

confidence: 99%

A commuting-vector-field approach to some dispersive estimates

Wong

2017

Arch. Math.

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Abstract. We prove the pointwise decay of solutions to three linear equations: (i) the transport equation in phase space generalizing the classical Vlasov equation, (ii) the linear Schrödinger equation, (iii) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain L 1 -L ∞ decay through directly estimating the fundamental solution in physical space, or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines "vector field" commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space-time weighted energy estimates. Combined with a simple version of Sobolev's inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrödinger equations we can recover sharp pointwise decay; in the Schrödinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on L 2 -L ∞ decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation.

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The Hyperboloidal Foliation Method

Cited by 62 publications

References 96 publications

Global solutions of nonlinear wave–Klein–Gordon system in one space dimension

Global solutions of nonlinear wave–Klein–Gordon system in one space dimension

The Mathematical Validity of the f(R) Theory of Modified Gravity

A commuting-vector-field approach to some dispersive estimates

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