In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the <i>C</i>⁰-norm of the initial data is sufficiently small, while the <i>C</i>¹-norm of the initial data can be large. The analysis is based on <i>a priori</i> estimates, which are obtained by a generalised Lax transformation.

In this paper, we will introduce a precise classification of characteristic functions in the Fourier space according to the moment constraint in the physical space of any order. Based on this, we construct measure-valued solutions to the homogeneous Boltzmann equation with the exact moment condition as the initial data.

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic NavierStokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum states. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.

For systems of hyperbolic conservation laws, a new Glimm functional was recently constructed when the linearly degenerate manifold in each characteristic field is either the whole space or it consists of a finitely many smooth and transversal manifolds of codimension one. This new functional leads to the neat consistency and convergence rate estimation of the Glimm scheme. In this paper, by the motivation of the result in Ref. 2, we show that the corresponding new Glimm functional can be constructed for general systems only under the strict hyperbolicity assumption.

We give a generalized version of uncertainty principle, and apply it to the study of regularization properties of solutions to kinetic equations. In particular, both linearized and nonlinear space inhomogeneous Boltzmann equations without Grad's cutoff assumption are considered. <b><i>To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).</i></b>