Stabilité de solutions faibles globales pour les équations de Navier–Stokes compressible avec température

We present a global in time stability result for sequences of weak solutions ‘à la Leray’ to the Navier–Stokes equations modelling viscous compressible heat conducting fluids in the whole space ${\mathbf{R}}^3$ (or in the box ${\mathbf{T}}^3={[0,2\pi ]}^3$ with periodic boundary conditions) with arbitrary large initial data. Specific assumptions are made on the density and temperature dependence of the thermal conduction κ and the viscosity coefficients λ and μ in order to preserve a particular conservation property discovered by the authors. The underlying mathematical structure is the key ingredient to get additional information on the density which allows to define weak solutions and get strong compactness results needed on the temperature. The equation of state is assumed to be the perfect polytropic gas law with an additional zero isothermal component that plays a role only for small density. Our result extends the work of P.-L. Lions restricted to barotropic flows obtained in 1993. Note that approximate solutions construction process, i.e. sequences of suitable smooth approximate solutions, is explained elsewhere. To cite this article: D. Bresch, B. Desjardins, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Solutions faibles d'énergie infinie pour les équations de Navier—Stokes dans ℝ3

We prove existence of suitable weak solutions for the Navier-Stokes equations on (0, ∞) × ℝ³ when the initial value is uniformly locally square integrable and vanishes at infinity.     

Régularité conditionnelle des équations de Navier–Stokes

In this Note, we give sufficient conditions for the regularity of Leray–Hopf weak solutions to the Navier–Stokes equation. We prove that, if one of three conditions (i) $?u/?x_3\in L_t^{s_0}{L}_x^{r_0}$ where $2/s_0+3/r_0?2$ and $9/4?r_0?3$, (ii) $\mathrm{?}{u}_3\in L_t^{s_1}{L}_x^{r_1}$ where $2/s_1+3/r_1?11/6$ and $54/23?r_0?18/5$, or (iii) $u_3\in L_t^{s_0}{L}_x^{r_0}$ where $2/s_0+3/r_0?5/8$ and $24/5?r_0?\infty$, is satisfied, then the solution is regular. These conditions improve earlier results on the conditional regularity of the Navier–Stokes equations. To cite this article: I. Kukavica, M. Ziane, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Un modèle scalaire analogue aux équations de Navier–Stokes

We discuss on a new scalar model whose properties are similar to the NS equations (it preserves scaling, the antisymmetry of the bilinear term and is invariant by translations and dilations) but contains a singular integral operator. We construct a global-in-time weak solution when initial data is in a critical Morrey–Campanato space and show that it also satisfies a local energy inequality comparable to Scheffer?s one.     

Comportement asymptotique des solutions d'une classe d'équations paraboliques et de Hamilton-Jacobi

We study long-time behaviour of the solutions of a class of nonlinear parabolic equations, as well as their Hamilton-Jacobi version. The solutions are shown to converge to either periodic fronts, or to steady solutions, modulo a linear growth in time.     

Approximations filaires pour les équations de Poisson et de Maxwell harmoniques

We are interested in the approximation of the Poisson and time harmonic Maxwell equations around a circular cylinder with a small radius. Numerical approximation requires a typical meshsize comparable to the radius of the cylinder, thus increasing the computational cost. In many applications (spacecraft charging computation, wire antennas modelling) the mesh size becomes too large to perform realistic computations. A mathematical study proves the convergence of the sequence $u_{\mathrm{?}}$ of Poisson equation solutions when the radius ? goes to 0 toward a limit problem with a convergence rate of $1/\ln (\mathrm{?})$. A wire approximation is proposed exhibiting a rate of convergence larger that $\sqrt{\mathrm{?}}$ without constraint of mesh refinement related to the radius size. This method is extended to the case of arbitrary wire cross section and also to the time harmonic Maxwell equations. To cite this article: F. Rogier et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Eléments finis nodaux pour les équations de Maxwell

An original approach of the singular complement method for Maxwell's equations in bounded polygonal domains is presented. A splitting of the electric field à la Moussaoui is proposed: $\mathbf{E}={\mathbf{E}}^R+\lambda {\mathbf{x}}_P$, where ${\mathbf{E}}^R\in H^1{\left(\omega \right)}^2$, λ depends on the data and domain and ${\mathbf{x}}_P$ is known explicitly. The same splitting can be used for the magnetic field. No cut-off function is needed and improved error estimates are derived. To cite this article: E. Jamelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Un résultat de densité pour les équations de Maxwell

We prove the density of regular fielclx in the space of square-integrahle fields with square-integrable curls and square-integrable tangential traces on the boundary. This space is involved in one of the variational formulations of the electmmagnetism equations.     

Un schéma simple pour les équations de Saint-Venant

A Finite-Volume scheme to solve shallow-water equations is proposed herein. This scheme makes use of velocity-celerity variables. It enables computing dam-break waves on dry bottom and generation of dry beds.     

Existence et stabilité de roll-waves pour les équations de Saint Venant

Roll-waves are entropic travelling waves of the Saint Venant system with a source term. In this Note, we study two connected problems: on the one hand the existence of roll-waves in a channel with a periodic bottom, on the other hand the linear stability of roll-waves over a flat bottom. The main issue is due to the presence of an infinite number of shocks. With the help of a suitable change of variables, we can restrict our attention to C¹ functions on $\text{R}\text{?{iL,}\text{i∈}\text{Z}\text{}}$. Then we prove the existence of small amplitude roll-waves with wavespeeds oscillating around an average velocity in a channel with a periodic bottom. For channels with a flat bottom, we show the linear stability of roll-waves when the slope is small enough. To cite this article: P. Noble, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sur les équations fondamentales de l'électrodynamique pour les corps en mouvement

Sans résumé Traduit de l'allemand des Annales de Wiedemann.     

Sur l'ultraconvergence d'une classe de séries de Dirichlet

Sans résumé     

Existence globale pour une classe d'équations d'ondes perturbées

In this paper we prove a global well-posedness result for the following Cauchy problem:

where the initial data $\text{(f,g)∈}\text{H}\text{?}{}^1\text{(}\text{R}{}^3\text{)×L}{}^2\text{(}\text{R}{}^3\text{)}$ are compactly supported, 1?α<5, $\text{a}{}_{\mathrm{i}}\text{(t,x)∈L}{}^{\mathrm{\infty }}\text{(}\text{R}{}_{\mathrm{t}}\text{×}\text{R}{}_{\mathrm{x}}{}^3\text{)}$, $\text{V(t,x)∈L}{}^{\mathrm{\infty }}\text{(}\text{R}{}_{\mathrm{t}}\text{;L}{}^3\text{(}\text{R}{}^3{}_{\mathrm{x}}\text{))}$. To cite this article: N. Visciglia, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sur les racines de certaines équations

Sans résumé     

Sur les équations différentielles ordinaires et les équations de transport

We study in this Note ordinary differential equations for divergence-free vector-fields with a limited regularity. We first observe that it is equivalent to solve the associated transport equations (i.e. Liouville equations). Then, we show existence, uniqueness, and stability results for generic vector-fields in L¹ or for “piecewise” W^1.1 vector-fields.     

Solutions globales d'énergie infinie de l'équation de Navier–Stokes 2D

We study in this Note the solutions of the 2D Navier–Stokes equations with initial data in ?BMO. For $u{|}_{t=0}$ in the closure of the Schwartz class, we obtain the existence and uniqueness of a global solution, and besides an estimate on its norm in ?BMO. To cite this article: P. Germain, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Sur les racines de certaines équations

Ohne Zusammenfassung