We present a unique global solvability and flocking estimate of an entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation in all-to-all coupling setting. This model appears naturally as a quasi-equilibrium model for hydrodynamic description of CuckerSmale flocking. For the unique global solvability, we adopt the variation approach from Ding and Huang's work [19] on the inhomogeneous pressureless gas dynamic model. When initial mass and velocity are locally integrable and bounded measurable functions, respectively, we give explicit representations for the global entropic weak solutions. Our results do not require any smallness of initial data except that initial mass density is almost everywhere positive.

We prove that classical solution of the spatially inhomogeneous and angular non-cutoff Boltzmann equation is C with respect to all variables, locally in the space and time variables. The proof relies on a generalized uncertainty principle, some improved upper bound and coercivity estimates on the nonlinear collision operator, and some subtle analysis on the commutators between the collision operators and some appropriately chosen pseudo-differential operators. To cite this article: R. Alexandre et al., CR Acad. Sci. Paris, Ser. I 347 (2009).

It is well-known in the combustion community that curvature effect in general slows down ame propagation speeds because it smooths out wrinkled ames. However, such a folklore has never been justied rigorously. In this paper, as the rst theoretical result in this direction, we prove that the turbulent ame speed (an eective burning velocity) is decreasing with respect to the curvature diusivity (Markstein number) for shear flows in the well-known G-equation model. Our proof involves several novel and rather sophisticated inequalities arising from the nonlinear structure of the equation. On a related fundamental issue, we solve the selection problem of weak solutions or nd the "physical fluctuations" when the Markstein number goes to zero and solutions approach those of the inviscid G-equation model. The limiting solution is given by a closed form analytical formula.

We studied coupled systems of the Fokker--Planck equation and the Navier--Stokes equation modeling the Hookean and the finitely extensible nonlinear elastic (FENE)-type polymeric flows. We proved the continuous embedding and compact embedding theorems in weighted spaces that naturally arise from related entropy estimates. These embedding estimates are shown to be sharp. For the Hookean polymeric system with a center-of-mass diffusion and a superlinear spring potential, we proved the existence of a global weak solution. Moreover, we were able to tackle the FENE model with $L^2$ initial data for the polymer density instead of the $L^\infty$ counterpart in the literature.

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