We consider the spatially inhomogeneous Boltzmann equation without angular cutoff. We prove the existence and uniqueness of local classical solutions to the Cauchy problem, in the function space with Maxwellian type exponential decay with respect to the velocity variable. <b><i>To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).</i></b>

In this paper, we discuss the waiting time phenomena for a class of non-Newtonian polytropic filtration equation with convection. According to the influence of the convection on the waiting time property, the convection is classified into three classes, that is, the strong convection, the mild convection and the weak convection. For different cases, the sufficient and necessary conditions on the initial data are given for the existence of waiting time respectively.

The ellipsoidal BGK model was introduced in\cite {Ho} to fit the correct Prandtl number in the Navier-Stokes approximation of the classical BGK model. In the paper we establish the global existence of mild solutions to the Cauchy problem on the model for a class of initial data allowed to have large oscillations. The proof is motivated by a recent study of the same topic on the Boltzmann equation in\cite {DHWY}.

In the paper, we study the three-dimensional Prandtl equations, and prove that if one component of the tangential velocity field satisfies the monotonicity assumption in the normal direction, then the system is locally well-posed in the Gevrey function space with Gevrey index in ] 1, 2]. The proof relies on some new cancellation mechanism in the system in addition to those observed in the two-dimensional setting.

In this paper, we prove the global existence of solutions with analytic regularity to the 2D magnetohydrodynamic (MHD) boundary layer equations in the mixed Prandtl and Hartmann regime derived by formal multiscale expansion in [D. Gerard-Varet and M. Prestipino, <i>Z. Angew. Math. Phys.</i>, 68 (2017), 76]. The analysis shows that the combined effect of the magnetic diffusivity and transverse magnetic field on the boundary leads to a linear damping on the tangential velocity field near the boundary. And this damping effect yields the global-in-time analytic norm estimate in the tangential space variable on the perturbation of the classical steady Hartmann profile.