Équation de Klein-Gordon

Résumé

L'équation de Klein-Gordon, parfois également appelée équation de Klein-Gordon-Fock, est une version relativiste de l'équation de Schrödinger décrivant des particules massives de spin nul, sans ou avec charge électrique, établie indépendamment en 1926 par les physiciens Oskar Klein et Walter Gordon. C'est un exemple d'équation aux dérivées partielles dispersive. L'équation de Klein-Gordon Dérivation L'équation de Klein-Gordon standard (sans champs électromagnétique) peut être obtenue de plusieurs façons. Une méthode consiste à écrire une formulation covariante de l'équation d'Euler-Lagrange, et une autre consiste à partir de l'invariant relativiste donnant l'énergie d'une particule isolée telle que : E^2 \ = \ p^2 \ c^2 \ + \ m^2 \ c^4 = \ \vec{p} . \vec{p} \ c^2 \ + \ m^2 \ c^4 où E est l'énergie totale de la particule, p sa quantité de mouvement, m sa masse propre, et c la célérité de la lumière dans le vide. On applique alors à cette équation

Source officielle

https://fr.wikipedia.org/wiki/%C3%89quation_de_Klein-Gordon

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Stable Manifold for the Critical Non-Linear Wave Equation: A Fourier Theory Approach

Gérôme Graf

I will try to explain, without going into too much detail, how one can consider a non-linear wave equation as a dynamical system and what it brings to the study of its solutions. We begin by considering our model case, the non-linear Klein-Gordon equation and state its basic properties. We will then see what happens for solutions with energies below that of the ground state. After that, we place ourselves energetically around the ground state and we show the apparition of the so-called invariant manifolds. Finally, we consider the critical "pure" (without the mass term) wave equation and describe some of its interesting solutions. The last part will be concerned with an attempt to rely what we have learn so far with the critical case.

EPFL2019

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Global dynamics above the ground state energy for the one-dimensional NLKG equation

Joachim Krieger

In this paperwe obtain a global characterization of the dynamics of even solutions to the one-dimensional nonlinear Klein–Gordon (NLKG) equation on the line with focusing nonlinearity |u|p−1u, p > 5, provided their energy exceeds that of the ground state only sightly. The method is the same as in the three-dimensional case (Nakanishi and Schlag in Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, preprint, 2010), the major difference being in the construction of the center-stable manifold. The difficulty there lies with the weak dispersive decay of 1-dimensional NLKG. In order to address this specific issue, we establish local dispersive estimates for the perturbed linear Klein–Gordon equation, similar to those of Mizumachi (J Math Kyoto Univ 48(3):471–497, 2008). The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator.

Springer Verlag2012

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We give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19–37, 2015), which can loosely speaking, be formulated as follows: consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law. Can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one does not in general have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

2019

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