Un théorème de représentation des solutions de viscosité d'une équation d'Hamilton–Jacobi–Bellman ergodique dégénérée sur le tore

We consider an ergodic Hamilton–Jacobi–Bellman equation coming from a stochastic control problem in which there are exactly k points where the dynamics vanishes and the Lagrangian is minimal. Under a stabilizability assumption, we state that the solutions of the ergodic equation are uniquely determined by their value on these k points, and that the set of solutions is sup-norm isometric to a non-empty closed convex set whose dimension is less or equal to k. To cite this article: M. Akian et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Stabilized approximate solutions of the inverse time problem for a parabolic evolution equation

The set of all solutions of a composite fuzzy relation equation of Sanchez (Inform. and Control30 (1976)), defined on finite spaces, is studied by determining and characterizing all the lower solutions of such an equation.     

Existence and nonuniqueness of solutions to a functional-differential equation

We examine the functional-differential equation Δu(x) — div(u(H(x))f (x)) = 0 on a torus which is a generalization of the stationary Fokker-Planck equation. Under sufficiently general assumptions on the vector field f and the map H, we prove the existence of a nontrivial solution. In some cases the subspace of solutions is established to be multidimensional.     

Équation des ondes sur les espaces symétriques riemanniens

We prove that the solutions of the homogeneous wave equation on Riemannian symmetric spaces have dispersion properties and we deduce Strichartz type estimates for these solutions. To cite this article: A. Hassani, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Self-similar solutions with fat tails for a coagulation equation with nonlocal drift

We investigate the existence of self-similar solutions for a coagulation equation with nonlocal drift. In addition to explicitly given exponentially decaying solutions we establish the existence of self-similar profiles with algebraic decay. To cite this article: M. Herrmann et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Energy dissipation for weak solutions of incompressible MHD equations

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that stems from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.     

Résultats d'existence pour les écoulements réguliers de fluides viscoélastiques incompressibles à loi différentielle de type White–Metzner en dimension 3

This paper is concerned with incompressible viscoelastic fluids which obey a differential constitutive law of White–Metzner type. We establish the existence and uniqueness of local solutions in 3-D as well as the global existence of small solutions. We then deduce the existence and asymptotic stability of small periodic and stationary solutions. Finally, we prove that the 2-D results obtained in Hakim (J. Math. Anal. Appl. 185 (1994) 675–705) remain true without any restriction on the smallness of the retardation parameter which is the linking coefficient between the equation of velocity (Navier–Stokes equation) and the transport equation verified by the extra-stress tensor. To cite this article: L. Molinet, R. Talhouk, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Uniqueness of unbounded solutions of the Lagrangian mean curvature flow equation for graphs

We observe that the comparison result of Barles–Biton–Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow. To cite this article: J. Chen, C. Pang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Analytic solutions of a second-order nonautonomous iterative functional differential equation

In this paper a second-order nonautonomous iterative functional differential equation is considered. By reducing the equation with the Schröder transformation to another functional differential equation without iteration of the unknown function, we give existence of its local analytic solutions. We first discuss the case that the constant α given in the Schröder transformation does not lie on the unit circle in C and the case that the constant lies on the circle but fulfills the Diophantine condition. Then we further study the case that the constant is a unit root in C but the Diophantine condition is offended. Finally, we investigate analytic solutions of the form of power functions.     

Une redécouverte de l'équation de Korteweg–de Vries et de Kadomtsev–Petviashvili

We show that the flux of long waves of water surface, propagating in each characteristic direction of the equations for a vibrating string, to a first approximation, are close to the solutions of the Korteweg–de Vries equation. In a three dimensional flow, the phenomenon is of the same order as the Kadomtsev–Petviashvili equation. To cite this article: T. Kano, T. Nishida, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray-type solutions towards a vector field which satisfies the usual 2D Navier–Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat–type equation elsewhere. The method of proof uses weak compactness arguments. To cite this article: I. Gallagher, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Existence de solutions pour une classe de systèmes non linéaires de Boussinesq

We give a few existence results of solutions for a class of Boussinesq systems, with suitable conditions on the right-hand side of the momentum equation, the forcing term depending on temperature. To cite this article: A. Attaoui, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Solutions auto-similaires non radiales pour l'équation quasi-géostrophique dissipative critique

We prove the existence of self-similar solutions for the critical dissipative quasi-geostrophic equation by using the formalism of mild solutions in a space close to $L^{\infty }$. To cite this article: F. Marchand, P.G. Lemarié-Rieusset, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Étude d'un système non linéaire de Boussinesq–Stefan

We give a few existence results for solutions for a class of Boussinesq–Stefan systems, with suitable conditions on the forcing terms in the right-hand side of the momentum equation depending on the temperature. To cite this article: A. Attaoui, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Multiplicité des solutions d'équilibre d'une coque de Marguerre–von Kármán membranaireMultiplicity of solutions to the Marguerre–von Kármán membrane equations

We build explicitly an infinite number of equilibrium solutions of unloaded Marguerre–von Kármán membrane shells. This construction is based upon the existence of three elementary solutions, together with the solution of a Monge–Ampère equation associated with a partition of the reference configuration of the shell. To cite this article: A. Léger, B. Miara, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 649–654.     

Convergence of linear finite elements for diffusion equations with measure data

We show here the convergence of the linear finite element approximate solutions of a diffusion equation to a weak solution, with weak regularity assumptions on the data. To cite this article: T. Gallouët, R. Herbin, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Existence et unicité de la solutions faible-renormalisée pour un système non linéaire de Boussinesq

We give existence and uniqueness results of the weak-renormalized solution for a class of nonlinear Boussinesq systems. We establish regularity results for the heat equation which we combine with the usual techniques for Navier–Stokes equations mixed with the tools involved for renormalized solutions. To cite this article: N. Bruyère, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Transonic flow on an axially symmetric torus

On a Riemannian manifold the existence (and uniqueness) of subsonic gas flows with prescribed circulation has been previously established (Acta Math.125 1970, 57–73). If the manifold is a torus of revolution then the gas dynamics equation reduces to a nonlinear ordinary differential equation and the flow can be described explicitly. We show that, as the circulations are increased, one obtains a complete family of solutions: smooth subsonic, smooth transonic, transonic with shocks, and smooth supersonic flows.     

Existence of local solutions for the Boltzmann equation without angular cutoff

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. To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Solutions concentrating at curves for some singularly perturbed elliptic problems

We study positive solutions of the equation ?ε²Δu+u=u^p, where p>1 and ε>0 is small, with Neumann boundary conditions in a three-dimensional domain $\text{Ω}$. We prove the existence of solutions concentrating along some closed curve on $\text{?Ω}$. To cite this article: A. Malchiodi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).