Celebratio Mathematica

Y. Choquet-Bruhat and N. Noutchegueme: “Système hyperbolique pour les équations d’Einstein avec sources,” C. R. Acad. Sci. Paris Sér. I Math. 303 : 6 (1986), pp. 259–263. MR 860831 article

F. Cagnac, Y. Choquet-Bruhat, and N. Noutchegueme: “Solutions of the Einstein equations with data at past infinity,” pp. 35–54 in Seventh Italian conference on general relativity and gravitational physics (Rapallo, 1986). Edited by U. Bruzzo, R. Cianci, and E. Massa. World Sci. Publ. (Singapore), 1987. MR 892547 incollection

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Y. Choquet-Bruhat and N. Noutchegueme: “Solution des systèmes de Yang–Mills–Vlasov: Théorèmes d’existence locale [Solutions of the Yang–Mills–Vlasov systems: Local existence theorems],” C. R. Acad. Sci. Paris Sér. I Math. 311 : 10 (1990), pp. 611–616. MR 1081417 Zbl 0717.35072 article

Y. Choquet-Bruhat and N. Noutchegueme: “Solutions globales du système de Yang–Mills–Vlasov (masse nulle)” [Global solutions of the Yang–Mills–Vlasov system with zero rest mass], C. R. Acad. Sci. Paris Sér. I Math. 311 : 12 (1990), pp. 785–788. MR 1082633 Zbl 0715.53046 article

In a previous Note [Choquet-Bruhat 1990] we have considered the Yang–Mills–Vlasov system on a pseudo-Riemannian manifold \( (V,g) \) of hyperbolic signature \begin{align} & \hat{\nabla}_a F^{\alpha,\beta} = J^{\beta}, \quad J^{\beta}(x)= \int_{\mathcal{P}_x} f(x,p,q)p^{\beta}q\omega_p \omega_q, \tag{1}\\ & \mathcal{L}_Y f \equiv p^\alpha \frac{\partial f}{\partial x^{\alpha}} + (-\Gamma^{\alpha}_{\lambda\mu}p^{\lambda} p^{\mu} + q\cdot F^{\alpha}_{\lambda} p^{\lambda})\frac{\partial f}{\partial p^{\alpha}} - p^{\alpha}[A_{\alpha},q]\cdot \frac{\partial f}{\partial q} = 0 \tag{2} \end{align} where \( F = (F_{\alpha,\beta}) \) is a Yang–Mills field with potential \( A = (A_{\alpha}) \), \[ \hat{\nabla} = \nabla + [A,\,] \] is the gauge covariant derivative, \( [\ ,\,] \) the Lie bracket in the algebra \( \mathcal{G} \) where the fields take their values and a dot denotes an Ad invariant positive definite scalar product; \( J = (J_{\beta}) \) is the \( \mathcal{G} \) valued current generated by the distribution function f of particles with energy momentum p and Yang–Mills charge q: in the absence of collisions f satisfies the Vlasov equation (2). A possible physical situation modeled by (1), (2) is a plasma of quarks and gluons. In the above cited paper we have proved that when \( (V,g) \) is regularly hyperbolic, when the particles have a given rest mass m and their charges a given norm the Cauchy problem for (1), (2) has a local in time solution if:

The Cauchy data are in appropriate weighted Sobolev spaces.
These data satisfy the constraint \( \hat{\nabla}_{\alpha}F^{\alpha 0} = J^0 \).
In the case \( m = 0 \) the initial value \( \phi \) of f satisfies in addition Support \( \phi \subset \{p^0 > c > 0\} \).

In this Note we use this result together with the conformal invariance of the system in the case \( m = 0 \) to prove a global existence theorem on Minkowski space time \( M_4 \), for small initial data.

Y. Choquet-Bruhat and N. Noutchegueme: “Système de Yang–Mills–Vlasov en jauge temporelle,” Ann. Inst. H. Poincaré Phys. Théor. 55 : 3 (1991), pp. 759–787. MR 1141713 Zbl 0767.58045 article

Y. Choquet-Bruhat and N. Noutchegueme: “Yang–Mills Vlasov systems,” pp. 52–71 in Nonlinear hyperbolic equations and field theory: Papers from a workshop on nonlinear hyperbolic equations (Lake Como, 1990). Edited by M. K. V. Murthy and S. Spagnolo. Pitman Res. Notes Math. Ser. 253. Longman Sci. Tech.; Wiley (Harlow, UK; New York), 1992. MR 1175201 Zbl 0799.35188 incollection

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