Asymptotic Analysis

A new proof of the Darcy law is proposed in the framework of the homogenization theory. The starting point of the homogenization is an unsteady flow of compressible viscous liquid through a small‐scaled random porous domain. This non‐stationary microscopic flow is described by the Stokes equations supplemented by zero boundary conditions for the velocity field. The Darcy law is established for the leading term of the solution. The main geometric assumption on the structure is the connectedness of the porous domain. The set of assumptions contains neither a regularity of boundaries nor any quantitative properties appertaining to the connectedness.

We study the semi‐classical limit of the dynamics of electrons in a crystal under the influence of a self‐consistent Coulombian potential. To do this, we introduce two small parameters: $\varepsilon$ , the typical length of the crystal, and $h$ , the macroscopic Planck constant. When $\varepsilon$ and $h$ tend to zero, under special conditions, the electrons distribution limit follows a Vlasov equation where the electrons mass is replaced by an effective mass.

We study the limit behaviour near the origin of the nonnegative solutions of the semilinear elliptic system $ \cases{ -\Delta u+\vert x\vert ^{a}v^{p}=0,\cr \Delta v+\vert x\vert ^{b}u^{q}=0,\cr}\quad \hbox{in}\ \mathbb{R}^{N}\ (N\geqslant 3), $ where $p,q,a,b\in \mathbb{R},$ with $p,q>0$ , $pq\neq 1$ . We give a priori estimates without any restriction on the values of $p$ and $q$ .

The macroscopic dynamics of a kinetic equation involving a model wave–particle collision operator of plasma physics (see Degond and Peyrard, C. R. Acad. Sci. Paris 323, 1996) is investigated. Using relative entropy estimates about an absolute Maxwellian, it is shown, as in Bardos, Golse and Levermore (Comm. Pure Appl. Math. 46(5), 1993), that any properly scaled sequence of solutions has fluctuations that converge to a limiting density of the form $g=U(x,t).v +P_0 g(x,t)({\vert v\vert ^2/2})$ with fluid variables that satisfy the incompressibility and Boussinesq relations. For a convenient scaling of the initial fluctuations, the momentum densities globaly converge to a solution of the Stokes equation. Using an infinite number of entropies, we derive the diffusion equation for the energy distribution function of the particles. A similar discrete time version of this result holds for the Navier–Stokes equation coupled with a diffusion equation for the energy distribution function under an additional mild weak compactness assumption.