Local minimizers of one-dimensional variational problems and obstacle problems

In this Note we suggest a direct approach to study local minimizers of one-dimensional variational problems. To cite this article: M.A. Sychev, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A generalized existence theorem of BSDEs

In this Note, we deal with one-dimensional backward stochastic differential equations (BSDEs) where the coefficient is left-Lipschitz in y (may be discontinuous) and Lipschitz in z, but without explicit growth constraint. We prove, in this setting, an existence theorem for backward stochastic differential equations. To cite this article: G. Jia, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Derivatives of regularly varying functions in Rd and domains of attraction of stable distributions

In R² the integral of a regularly varying (RV) function f is regularly varying only if f is monotone. Generalization to R² of the one-dimensional result on regular variation of the derivative of an RV-function however is straightforward. Applications are given to limit theory for partial sums of i.i.d. positive random vectors in R²₊.     

Structure of A*-Fibrations over One-Dimensional Seminormal Semilocal Domains

In this paper we give a structure theorem for an A*-fibration over a one-dimensional noetherian seminormal semilocal domain and show that, in this situation, any A*-fibration whose spectrum occurs as an affine open subscheme of the spectrum of an A¹-fibration (equivalently, an affine line A¹) is actually A*. The structure theorem provides examples of A*-fibrations over one-dimensional noetherian seminormal semilocal domains whose spectra are not affine open subschemes of any affine line A¹ over the base ring. We also construct examples of nontrivial A*-fibrations over one-dimensional noetherian non-seminormal local domains whose spectra are open subschemes of A¹-fibrations over the base ring.     

An asymptotic-preserving well-balanced scheme for the hyperbolic heat equationsUn schema « équilibre » et « asymptotic-preserving » pour les equations de la chaleur hyperboliques

We propose here a well-balanced numerical scheme for the one-dimensional Goldstein–Taylor system which is endowed with all the stability properties inherent to the continuous problem and works in both rarefied and diffusive regimes. To cite this article: L. Gosse, G. Toscani, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 337–342.     

Large time behaviour of solutions of the Swift–Hohenberg equation

We study the limiting profiles v of solutions of the Swift–Hohenberg equation on a one-dimensional domain (0,L) for different values of L. We identify those values of L for which v=0, and discuss the size and the shape of v when it is nontrivial and a global minimiser of an associated energy functional. To cite this article: L.A. Peletier, V. Rottschäfer, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Non-homogeneous boundary conditions for a fourth-order diffusion equation

The existence of classical solutions to a one-dimensional non-linear fourth-order elliptic equation arising in quantum semiconductor modeling is proved for a class of non-homogeneous boundary conditions using degree theory. Furthermore, some non-existence results for other classes of boundary conditions are presented. To cite this article: P. Amster et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Discrétisation d'équations différentielles stochastiques unidimensionnelles à générateur sous forme divergence avec coefficient discontinu

In this Note, we discretize stochastic differential equations related to one-dimensional parabolic partial differential equations with a divergence form operator whose coefficient is discontinuous at 0. We establish the convergence rate in a weak sense. To cite this article: M. Martinez, D. Talay, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the boundary controllability of non-scalar parabolic systems

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

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).     

Branching one-dimensional periodic diffusion processes

Let X be a nonsingular conservative one-dimensional periodic diffusion process, λ₀ its principal eigenvalue and X a binary splitting branching diffusion process with nonbranching part X. For each α > λ₀ we have two positive martingales Wⁱ_t(α), i = 1, 2, of X attached to the two positive α-harmonic functions of X. The main purpose of this paper is to show that their limit random variables are positive for all α? (λ₀, α_i), where α_i are some constants greater than λ₀.     

Spectral properties of Schrödinger operators on radial N-dimensional infinite trees

We study the discreteness of the spectrum of Schrödinger operators which are defined on a class of radial N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.     

An asymptotically stable discretization for the Euler–Poisson system in the quasi-neutral limit

We are interested in the modeling of a plasma in the quasi-neutral limit using the Euler–Poisson system. When this system is discretized with a standard numerical scheme, it is subject to a severe numerical constraint related to the quasi-neutrality of the plasma. We propose an asymptotically stable discretization of this system in the quasi-neutral limit. We present numerical simulations for two different one-dimensional test cases that confirm the expected stability of the scheme in the quasi-neutral limit. To cite this article: P. Crispel et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well

We consider a quantum charged particle in a one-dimensional infinite square potential well moving along a line. We control the acceleration of the potential well. The local controllability in large time of this nonlinear control system along the ground state trajectory has been proved recently. We prove that this local controllability does not hold in small time, even if the Schrödinger equation has an infinite speed of propagation. To cite this article: J.-M. Coron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials

We address the analysis of a nonlinear and degenerating PDE system, proposed by M. Frémond for modelling phase transitions in viscoelastic materials subject to thermal effects. The system features an internal energy balance equation, governing the evolution of the absolute temperature ?, an evolution equation for the phase change parameter χ, and a stress-strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled.In this paper, we first prove a local (in time) well-posedness result for (a suitable initial-boundary value problem for) the above mentioned PDE system, in the (spatially) three-dimensional setting. Secondly, we restrict to the one-dimensional case, in which, for the same initial-boundary value problem, we indeed obtain a global well-posedness theorem.     

Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths

We study the asymptotic behavior of the solution of a diffusion problem posed in the union of a cylinder of small diameter and fixed length with another cylinder with much smaller diameter and length. The Dirichlet condition is assumed to hold at both extremities of this domain. Depending on the relative size of the parameters, we show that the boundary condition of the one-dimensional limit problem is a Dirichlet, Fourier or Neumann condition. We also prove a corrector result for every case. To cite this article: J. Casado-D??az et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Finite speed of propagation in porous media by mass transportation methods

In this Note we make use of mass transportation techniques to give a simple proof of the finite speed of propagation of the solution to the one-dimensional porous medium equation. The result follows by showing that the difference of support of any two solutions corresponding to different compactly supported initial data is a bounded in time function of a suitable Monge–Kantorovich related metric. To cite this article: J.A. Carrillo et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Zero-noise solutions of linear transport equations without uniqueness: an example

We consider a classical one-dimensional example of linear transport equation without uniqueness of weak solutions. Under a suitable multiplicative noise perturbation, the equation is well posed. We identify the two solutions of the deterministic equation obtained in the zero-noise limit. In addition, we prove that the zero-viscosity solution exists and is different from them. To cite this article: S. Attanasio, F. Flandoli, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Local smoothing effects for the water-wave problem with surface tension

The water-wave problem with a one-dimensional free surface of infinite depth is considered, based on the formulation as a second-order nonlinear dispersive equation. The local smoothing effects are established under the influence of surface tension, stating that on average in time solutions acquire locally 1/4 derivative of smoothness as compared to the initial state. The analysis combines energy methods with techniques of Fourier integral operators. To cite this article: H. Christianson et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Initial-Boundary Value Problem for a Class of Linear Relaxation Systems in Arbitrary Space Dimensions

In this paper, we consider the characteristic initial-boundary value problem (IBVP) for the multi-dimensional Jin-Xin relaxation model in a half-space with arbitrary space dimension n?2. As in the one-dimensional case (n=1, see (J. Differential Equations, 167 (2000), 388-437), our main interest is on the precise structural stability conditions on the relaxation system, particularly the formulation of boundary conditions, such that the relaxation IBVP is stiffly well posed, that is, uniformly well posed independent of the relaxation parameter ε>0, and the solution of the relaxation IBVP converges, as ε→0, to that of the corresponding limiting equilibrium system, except for a sharp transition layer near the boundary. Our main result can be roughly stated as Stiff Kreiss Condition=Uniform Kreiss Condition for the relaxation IBVP we consider in this paper, which is in sharp contrast to the one-dimensional case (Z. Xin and W.-Q. Xu, J. Differential Equations, 167 (2000), 388-437). More precisely, we show that the Uniform Kreiss Condition (which is necessary and sufficient for the well posedness of the relaxation IBVP for each fixed ε), together with the subcharacteristic condition (which is necessary and sufficient for the stiff well posedness of the corresponding Cauchy problem), also guarantees the stiff well posedness of our relaxation IBVP and the asymptotic convergence to the corresponding equilibrium system in the limit of small relaxation rate. Optimal convergence rates are obtained and various boundary layer behaviors are also rigorously justified.