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).     

Explicit limits for nonperiodic homogenization and reduction of dimensionLimites explicites pour l'homogénéisation non-périodique et la réduction de dimension

The aim of this Note is to give explicit limit expressions, for diffusion equations involving a small parameter ε, describing both nonperiodic homogenization and reduction of dimension. We consider two kinds of reduction of dimension: the case of plates and the case of thin cylinders. In particular, we give the limit diffusion equation for stratified plates. This is completely explicit and requires no special assumption, except stratification. In the case of thin cylinders, the formulae are less explicit, but we also indicate some simple applications to fibered materials. To cite this article: B. Gustafsson, J. Mossino, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 977–982.     

Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I

We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived.     

Generalized infinite-dimensional Fresnel integrals

A generalized infinite dimensional oscillatory integral with a polynomially growing phase function is defined and explicitly computed in terms of an absolutely convergent Gaussian integral. The results are applied to the Feynman path integral representation for the solution of the Schrödinger equation with an anharmonic oscillator potential. To cite this article: S. Albeverio, S. Mazzucchi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Applications of nonstandard analysis to partial differential equations—I. the diffusion equation

This paper begins by developing a basis for using ¹ finite difference equations to model physical phenomena. Under appropriate conditions the solution F of a ¹ finite difference equation has S-continuous “finite difference derivatives” up to order r. In these circumstances we can show that the standard function °F is a C^r-function and satisfies the differential equation corresponding to the original finite difference equation. The second part of the paper illustrates these techniques by applying them to the heat equation. In particular, we obtain a very nice model of the heat equation with initial conditions corresponding to all the heat concentrated at a single point.     

On the non existence of monotone linear schema for some linear parabolic equations

In this Note, we present a result concerning the non existence of linear monotone schema with fixed stencil on regular meshes for some linear parabolic equation in two dimensions. The parabolic equations of interest arise from non isotropic diffusion modelling. A corollary is that no linear monotone 9 points-schemes can be designed for the one-dimensional heat equation emerged in the plane with an arbitrary direction of diffusion. Some applications of this result are provided: for the Fokker–Planck–Lorentz model for electrons in the context of plasma physics; all linear monotone scheme for the one-dimensional hyperbolic heat equation treated as a two-dimensional problem are not consistent in the diffusion limit for an arbitrary direction of propagation. We also examine the case of the Landau equation. To cite this article: C. Buet, S. Cordier, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Propagation speed for reaction–diffusion equations in general domains

This Note is devoted to the analysis of some propagation phenomena for reaction–diffusion–advection equations with Fisher or Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearities. Some formulæ for the speed of propagation of pulsating fronts in periodic domains are given. These allow us to describe the influence of the various terms in the equation or of geometry on propagation. We also derive results for propagation speed in more general domains without periodicity. To cite this article: H. Berestycki et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Polynomial decay and control of a 1−d model for fluid–structure interaction

We consider a linearized and simplified 1?d model for fluid–structure interaction. The domain where the system evolves consists in two bounded intervals in which the wave and heat equations evolve respectively, with transmission conditions at the point of interface. First, we develop a careful spectral asymptotic analysis on high frequencies. Next, according to this spectral analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Finally, we prove the null-controllability of the system when the control acts on the boundary of the interval where the heat equation holds. The proof is based on a new Ingham-type inequality, which follows from the spectral analysis we develop and the null controllability result in Zuazua (in: J.L. Menaldi et al. (Eds.), Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 198–210) where the control acts on the wave component. To cite this article: X. Zhang, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Nulle contrôlabilité régionale d'une équation de type Crocco linéariséeRegional null controllability of a linearized Crocco type equation

We are interested in controllability problems of equations coming from a boundary layer model. This problem is described by a degenerate parabolic equation (a linearized Crocco type equation) where phenomena of diffusion and transport are coupled.First we give a geometric characterization of the influence domain of a locally distributed control. Then we prove regional null controllability results on this domain. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by decomposition of the space–time domain and Carleman type estimates along characteristics. To cite this article: P. Martinez et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 581–584.     

A fundamental solution for the heat equation which is supported in a strip

A fundamental solution is constructed for the heat operator which is defined on all of Rⁿ⁺¹ and vanishes for t ? 0 and for t ? ?. This solution is constructed so that it has as mild a growth as possible for $\text{¦ x ¦ → ∞}$. It is applied to the solution of the inhomogeneous heat equation with the right side supported in a strip.     

On a 3D isothermal model for nematic liquid crystals accounting for stretching terms

In the present contribution, we study a PDE system describing the evolution of a nematic liquid crystals flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the velocity field u is ruled by the Navier–Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamics of the director field d is described by a variation of a parabolic Ginzburg–Landau equation with a suitable penalization of the physical constraint |d| = 1. Such equation accounts for both the kinematic transport by the flow field and the internal relaxation due to the elastic energy. The main aim of this contribution is to overcome the lack of a maximum principle for the director equation and prove (without any restriction on the data and on the physical constants of the problem) the existence of global in time weak solutions under physically meaningful boundary conditions on d and u.     

Résolution en temps court d'une équation de Hamilton–Jacobi non locale décrivant la dynamique d'une dislocation

This Note studies a nonlocal geometric Hamilton–Jacobi equation that models the motion of a planar dislocation in a crystal. Within the framework of viscosity solutions and of the level-set approach, we show that the equation has a unique solution on a small time interval when the initial curve is the graph of a Lipschitz bounded function. To cite this article: O. Alvarez et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Dead-core rates for the fast diffusion equation with a spatially dependent strong absorption

This paper deals with the dead-core rates problem for the fast diffusion equation with a spatially dependent strong absorption $$u_t=(u^{m})_{xx}-x^{q}u^p, \quad(x,t)\in(0,1)\times(0,\infty),$$ where 0 < p < m < 1 and ?1 < q < 0. By using the self-similar transformation technique and the Zelenyak method, we proved that the temporal dead-core rate is non-self-similar.     

Derivation of the Schrödinger–Poisson equation from the quantum N-body problemJustification de l'équation de Schrödinger–Poisson à partir du problème quantique à N corps

We derive the time-dependent Schrödinger–Poisson equation as the weak coupling limit of the N-body linear Schrödinger equation with Coulomb potential. To cite this article: C. Bardos et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 515–520.     

A finite volume scheme for anisotropic diffusion problems

A new finite volume for the discretization of anisotropic diffusion problems on general unstructured meshes in any space dimension is presented. The convergence of the approximate solution and its discrete gradient is proven. The efficiency of the scheme is illustrated by numerical results. To cite this article: R. Eymard et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

The elliptic K3 surfaces with a maximal singular fibre

We give the defining equation of complex elliptic K3 surfaces with a maximal singular fibre. Then we study the reduction modulo p at a particularly interesting prime p. To cite this article: T. Shioda, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Biharmonic problem in exterior domains

We study here a biharmonic equation in an exterior domain of $\text{R}{}^{\mathrm{n}}$. We give in L^p theory, with 1<p<∞ existence, uniqueness and regularity results. To cite this article: C. Amrouche, M. Fontes, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Large deviations for invariant measures of general stochastic reaction–diffusion systems

In this paper we prove a large deviations principle for the invariant measures of a class of reaction–diffusion systems in bounded domains of $\text{R}{}^{\mathrm{d}}\text{,}\text{d?1}$, perturbed by a noise of multiplicative type. We consider reaction terms which are not Lipschitz-continuous and diffusion coefficients in front of the noise which are not bounded and may be degenerate. To cite this article: S. Cerrai, M. Röckner, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Harnack estimates for non-negative weak solutions of a class of singular parabolic equations

We prove forward, backward and elliptic Harnack type inequalities for non-negative local weak solutions of singular parabolic differential equations of type $$u_t={\rm div}{\bf A}(x, t, u, Du)$$ where A satisfies suitable structure conditions and a monotonicity assumption. The prototype is the parabolic p?Laplacian with 1 < p < 2. By using only the structure of the equation and the comparison principle, we generalize to a larger class of equations the estimates first proved by Bonforte et al. (Adv. Math. 224, 2151–2215, 2010) for the model equation.     

Hypocoercivity for kinetic equations with linear relaxation terms

This Note is devoted to a simple method for proving the hypocoercivity associated to a kinetic equation involving a linear time relaxation operator. It is based on the construction of an adapted Lyapunov functional satisfying a Gronwall-type inequality. The method clearly distinguishes the coercivity at microscopic level, which directly arises from the properties of the relaxation operator, and a spectral gap inequality at the macroscopic level for the spatial density, which is connected to the diffusion limit. It improves on previously known results. Our approach is illustrated by the linear BGK model and a relaxation operator which corresponds at macroscopic level to the linearized fast diffusion. To cite this article: J. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).