Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the 3D finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in 3D, which do exist for the same system in 2D.

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. Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\Phi=\Phi(u,v)$, equivalent to the $\L^1$ distance, which is almost decreasing i.e., $$ \Phi\big( u(t),~v(t)\big)-\Phi\big( u(s),~v(s)\big)\leq \O(\ve)\cdot (t-s)\quad\hbox{for all}~~t&gt;s\geq 0,$$ for every pair of <i></i>-approximate solutions <i>u</i>, <i>v</i> with small total variation, generated by a wave front tracking algorithm. The small parameter <i></i> here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in <i>u</i> and in <i>v</i>. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the ${\vec L}^1$ norm. This provides a new proof of the existence of the

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>

We study the vanishing viscosity of the Navier-Stokes equations for interacting shocks. Given an entropy solution to <i>p</i>-system which consists of two different families of shocks interacting at some positive time, we show that such entropy solution is the vanishing viscosity limit of a family of global smooth solutions to the isentropic Navier-Stokes equations. The key point of the proofs is to derive the estimates separately before and after the interaction time and connect the incoming and outgoing viscous shock profiles.