Solutions globales des systèmes de Dirac-Klein-Gordon

Bachelot, Alain

doi:10.5802/jedp.336

Cited by 40 publications

(91 citation statements)

References 4 publications

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“…Such an estimate will be crucial for the proof of the global existence of solution to some important nonlinear systems of wave and Klein-Gordon equations, when the initial data are not compactly supported (see the approach developed in [1][2][3][4] Combining the above estimates with the estimates due to W. von Wahl [11], one can treat also the case when the initial data of the solution to (1.2) are nonzero. Namely, assuming u(0, x) = ef(x) and dtu(0, x) = eg(x), f, g G S^(R3), for any positive real number ô > 0 we have (2.2) (I + t+ 1x1)1^^1, x)\< Ce+ C J2 sup \\Yßf(s)\\y2+s.…”

Section: Introductionmentioning

confidence: 99%

“…Essential results concerning the global existence of solution to the (3 + 1)-dimensional nonlinear wave equation have been obtained recently by Klainerman [9,10], Hörmander [6,7] (see also [1][2][3][4] Here LQ = tdt + 2j,=i x¡dj is the radial vector field.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, some important coupled systems of nonlinear wave and Klein-Gordon equations have been investigated in [1][2][3][4]; these show that the use of the radial vector field L0 is not convenient for the proof of a global existence theorem. To overcome this difficulty, in [2,3] another L°°-L weighted estimate is proposed, when the inclusion (1.4) supp/(5,y)ç{|v|<S + JR} is fulfilled for some constant R > 0.…”

Section: Introductionmentioning

confidence: 99%

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Weighted decay estimate for the wave equation

Covachev

Georgiev

1991

Proc. Amer. Math. Soc.

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Abstract.The work is devoted to the proof of a new L°°-L weighted estimate for the solution to the nonhomogeneous wave equation in (3 + l)-dimensional space-time. The weighted Sobolev spaces are associated with the generators of the Poincaré group. The estimate obtained is applied to prove the global existence of a solution to a nonlinear system of wave and Klein-Gordon equations with small initial data.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weighted decay estimate for the wave equation

Covachev

Georgiev

1991

Proc. Amer. Math. Soc.

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“…This was performed by using a technique introduced by Klainerman (see [100,101,102]) to obtain L ∞ a priori estimates via the Lorentz invariance of the equations and a generalized version of the energy inequalities. The same method was used by Bachelot [8] to obtain a similar result for the Klein-Gordon-Dirac equation. Finally, more recent efforts have been directed toward proving existence of solutions for the time-dependent Klein-Gordon-Dirac and Maxwell-Dirac equations in the energy space, namely…”

Section: Nonlinear Dirac Evolution Problemsmentioning

confidence: 67%

Variational methods in relativistic quantum mechanics: new approach to the computation of Dirac eigenvalues

Dolbeault

Esteban

Séré

2000

Lecture Notes in Chemistry

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Abstract. This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R 3 , the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems.In the first part, we consider the fixed eigenvalue problem for models of a free self-interacting relativistic particle. They allow us to describe the localized state of a spin-1/2 particle (a fermion) which propagates without changing its shape. This includes the Soler models, and the Maxwell-Dirac or KleinGordon-Dirac equations.The second part is devoted to the presentation of min-max principles allowing us to characterize and compute the eigenvalues of linear Dirac operators with an external potential in the gap of their essential spectrum. Many consequences of these min-max characterizations are presented, among them are new kinds of Hardy-like inequalities and a stable algorithm to compute the eigenvalues.In the third part we look for normalized solutions of nonlinear eigenvalue problems. The eigenvalues are Lagrange multipliers lying in a spectral gap. We review the results that have been obtained on the Dirac-Fock model which is a nonlinear theory describing the behavior of N interacting electrons in an external electrostatic field. In particular we focus on the problematic definition of the ground state and its nonrelativistic limit.In the last part, we present a more involved relativistic model from Quantum Electrodynamics in which the behavior of the vacuum is taken into account, it being coupled to the real particles. The main interesting feature of this model is that the energy functional is now bounded from below, providing us with a good definition of a ground state.

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“…Among these theories, the so-called f (R)-theory of modified gravity (associated with a prescribed function f (R) of the scalar curvature R) was recognized as a physically viable alternative to Einstein's theory. Despite the important role played by this theory in physics (1) , the corresponding field equations have not been investigated by mathematicians yet. This is due to the fact that the modified gravity equations are significantly more involved than the Einstein equations: they contain up to fourth-order derivatives of the unknown metric, rather than solely second-order derivatives.…”

Section: Introductionmentioning

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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Solutions globales des systèmes de Dirac-Klein-Gordon

Cited by 40 publications

References 4 publications

Weighted decay estimate for the wave equation

Weighted decay estimate for the wave equation

Variational methods in relativistic quantum mechanics: new approach to the computation of Dirac eigenvalues

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

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