Nonlinear “double porosity” type modelUn modèle non linéaire de type double porosité

We consider a variational problem in a bounded domain $\text{Ω=}\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$ with a microstructure $\text{F}{}^{\mathrm{(\epsilon )}}$ which is not in general periodic; a^ε=a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and as ε→0. A homogenized model is constructed. To cite this article: L. Pankratov, A. Piatnitski, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 435–440.     

Propagation of chaos for pressureless gas equations with viscosityPropagation du chaos pour un système de gaz sans pression avec viscosité

We use A.S. Sznitman ideas of probabilistic phenomenon of propagation of chaos for Burgers equation, and we derive the existence and uniqueness of a weak solution of the following system of pressureless gas equations with viscosity:

$\text{(}\text{S}\text{)}\text{?}\text{?t}\text{ρ+}\text{?}\text{?x}\text{(uρ)=}\text{1}\text{2}\text{?}{}^2\text{?}{}^2\text{x}\text{ρ,}\text{?}\text{?t}\text{(uρ)+}\text{?}\text{?x}\text{(u}{}^2\text{ρ)=}\text{1}\text{2}\text{?}{}^2\text{?}{}^2\text{x}\text{(uρ),}\text{ρ(}\text{d}\text{x,t)→ρ(}\text{d}\text{x,0),u(x,t)ρ(}\text{d}\text{x,t)→u}{}_0\text{(x)ρ(}\text{d}\text{x,0)}\text{weakly as}\text{t→0}{}^{\mathrm{+}}\text{.}$

To cite this article: A. Dermoune, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 935–940.     

A sharp inequality for Sobolev functionsUne inégalité dans un espace de Sobolev

Let N?5, a>0, $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$, $\text{2}{}^1\text{=}\text{2N}\text{N?2}$, $\text{2}{}^{\mathrm{\#}}\text{=}\text{2(N?1)}\text{N?2}$ and 6u6²=|?u|₂²+a|u|₂². We prove there exists an α₀>0 such that, for all $\text{u∈}\text{H}{}^1\text{(Ω)?{0}}$,

This inequality implies Cherrier's inequality. To cite this article: P.M. Girão, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 105–108     

A comparison result related to Gauss measureUn résultat de comparaison relatif à la mesure de Gauss

In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type

where $\text{Ω}$ is an open set of $\text{R}{}^{\mathrm{n}}$ (n?2), ?(x)=(2π)^?n/2exp(?|x|²/2), a_ij(x) are measurable functions such that a_ij(x)ξ_iξ_j??(x)|ξ|² a.e. $\text{x∈Ω}$, $\text{ξ∈}\text{R}{}^{\mathrm{n}}$ and f(x) is a measurable function taken in order to guarantee the existence of a solution $\text{u∈}\text{H}{}^1{}_0\text{(?,Ω)}$ of (1.1). We use the notion of rearrangement related to Gauss measure to compare u(x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.     

Le problème de Brill–Noether et le théorème de Teixidor

Let C be a smooth projective algebraic curve of genus g. The Brill–Noether problem is to determine at which conditions on n, d and k, there exists a stable vector bundle of rank n, degree d and with at least k independent global sections. Teixidor gave a rather general answer about the existence in her paper [Tei]. The result was available only for a generic curve, in a very imprecise sense, and the proof was really difficult. We show here, that the result is in fact true for all curves and a little more. Received: 5 September 1997 / Revised version: 15 June 1998     

Asymptotic properties of posterior distributions derived from misspecified modelsPropriétés asymptotiques des lois a posteriori sous un modèle incorrect

We investigate the asymptotic properties of posterior distributions when the model is misspecified, i.e. it is contemplated that the observations x₁,…,x_n might be drawn from a density in a family $\text{{h}{}_{\mathrm{\sigma }}\text{,}\text{σ∈Θ}}$ where $\text{Θ?}\text{R}{}^{\mathrm{d}}$, while the actual distribution of the observations may not correspond to any of the densities h_σ. A concentration property around a fixed value of the parameter is obtained as well as concentration properties around the maximum likelihood estimate. To cite this article: C. Abraham, B. Cadre, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 495–498.     

On global discontinuous solutions of Hamilton–Jacobi equationsSur des solutions globales discontinues des équations d'Hamilton–Jacobi

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H=H(Du), provided the discontinuous initial value function ?(x) is continuous outside a set Γ of measure zero and satisfies

(1)$\text{?(x)??}{}_7\text{(x):=}\text{lim}\text{inf}\text{y→x,}\text{y∈}\text{R}{}^{\mathrm{d}}\text{?Γ}\text{?(y).}$

We prove that the discontinuous solutions with almost everywhere continuous initial data satisfying ($\text{1}$) become Lipschitz continuous after finite time for locally strictly convex Hamiltonians. The L¹-accessibility of initial data and a comparison principle for discontinuous solutions are shown for a general Hamiltonian. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L^∞-solutions is clarified. To cite this article: G.-Q. Chen, B. Su, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 113–118     

Sur un théorème de prolongement fonctionnel de Keldych concernant le problème de Dirichlet

Sans résumé Ces recherches ont fait l'objet d'exposés, d'abord au Séminaire de théorie du potential (Paris, javier 1960).     

Construction and stability of blowup solutions for a non-variational semilinear parabolic systemConstruction et stabilité de solutions explosives pour un système parabolique sémilinéaire non-variationel

We consider the following parabolic system whose nonlinearity has no gradient structure:

$\{\begin{array}{cc}{?}_{t}u=\mathrm{\Delta }u+|v{|}^{p?1}v,\hfill & {?}_{t}v=\mu \mathrm{\Delta }v+|u{|}^{q?1}u,\hfill \\ u(?,0)=u_0,\hfill & v(?,0)=v_0,\hfill \end{array}$

in the whole space ${\mathbb{R}}^N$, where $p,q>1$ and $\mu >0$. We show the existence of initial data such that the corresponding solution to this system blows up in finite time $T(u_0,v_0)$ simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics:

$\{\begin{array}{cc}\hfill & u(x,t)～\mathrm{\Gamma }{[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(p+1)}{pq?1}},\hfill \\ \hfill & v(x,t)～\gamma {[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(q+1)}{pq?1}},\hfill \end{array}$

with $b=b(p,q,\mu )>0$ and $(\mathrm{\Gamma },\gamma )=(\mathrm{\Gamma }(p,q),\gamma (p,q))$. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case $\mu =1$; and the fact that the case $\mu \ne 1$ breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.     

Problème de Monge pour n probabilitésMonge problem for n probabilities

In this Note, we generalize Gangbo–Swiech theorem for the Monge–Kantorovich problem. We study this problem for Orlicz and Köthe spaces when the function c has the form $\text{c(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)=h(∑x}{}_{\mathrm{i}}\text{),}\text{h}$ convex on $\text{R}{}^{\mathrm{d}}\text{.}$To cite this article: H. Heinich, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 793–795.     

On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical rangeSur les asymptotiques des solutions globales des équations paraboliques sémi-linéaires d'ordre supérieur dans le cas surcritique

We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear 2mth order parabolic equation u_t=?(?Δ)^mu+|u|^p in R^N×R₊, where m>1, p>1, with bounded integrable initial data u₀. We prove that in the supercritical Fujita range p>p_F=1+2m/N any small global solution with nonnegative initial mass, $\text{∫u}{}_0\text{dx?0}$, exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case $\text{p∈}\text{]1,p}{}_{\mathrm{F}}\text{]}$ where solutions can blow-up for any arbitrarily small initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents $\text{{p}{}_{\mathrm{l}}\text{=1+2m/(l+N),}\text{l=0,1,2,…}}$, where p₀=p_F, are discussed. To cite this article: Yu.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 805–810.     

Viscoélastodynamique monodimensionnelle avec conditions de SignoriniOne-dimensional viscoelastodynamics with Signorini boundary conditions

Let α be a positive number. The one-dimensional viscoelastic problem

$\text{u}{}_{\mathrm{tt}}\text{?u}{}_{\mathrm{xx}}\text{?αu}{}_{\mathrm{xxt}}\text{=f,}\text{x∈(?∞,0],}\text{t∈[0,+∞),}$

with unilateral boundary conditions

$\text{u(0,·)?0,}\text{(u}{}_{\mathrm{x}}\text{+αu}{}_{\mathrm{xt}}\text{)(0,·)?0,}\text{(u(u}{}_{\mathrm{x}}\text{+αu}{}_{\mathrm{xt}}\text{))(0,·)=0,}$

can be reduced to the following variational inequality:

$\text{λ}{}_1\text{1w=g+b,}\text{w?0,}\text{b?0,}\text{〈w,b〉=0.}$

Here $\text{λ}\text{?}{}_1\text{(ω)}$ is the causal determination of $\text{i}\text{ω}\text{1+}\text{i}\text{αω}$. We show that the energy losses are purely viscous; this result is a consequence of the relation $\text{〈}\text{w}\text{?}\text{,b〉=0}$; since a priori, b is a measure and $\text{w}\text{?}$ is defined only almost everywhere, this relation is not trivial. To cite this article: A. Petrov, M. Schatzman, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 983–988.     

Sur le problème d'équivalence de certaines structures infinitésimales

Résumé Après avoir rappelé la définition et quelques propriétés des pseudogroupes de Lie, on définit les structures infinitésimales régulières. On peut toujours associer des connexions affines à de telles structures; courbure et torsion de ces connexions. Equivalence locale de deux structures infinitésimales. Application à l'étude des structures presque complexes, presque paracomplexes, presque symplectiques, presque hermitiennes, presque parahermitiennes, presque quaternioniennes, presque quaternioniennes de deuxième espèce.     

Les espaces réguliers et le problème de métrisation

Sans résumé     

Critical exponents for the Pucci's extremal operatorsLes exposants critiques pour l'opérateur extrémal de Pucci

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

(1)$\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}\text{(D}{}^2\text{u)+u}{}^{\mathrm{p}}\text{=0,u?0}\text{in}\text{R}{}^{\mathrm{N}}\text{.}$

Here N?3, p>1 and $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}$ denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents $\text{1<p}{}^{\mathrm{s}}{}_{\mathrm{+}}\text{<p}{}^1{}_{\mathrm{+}}\text{<p}{}^{\mathrm{p}}{}_{\mathrm{+}}$, that satisfy: (i) If $\text{1<p<p}{}^1{}_{\mathrm{+}}$ then there is no nontrivial solution of ($\text{1}$). (ii) If $\text{p=p}{}^1{}_{\mathrm{+}}$ then there is a unique fast decaying solution of ($\text{1}$). (iii) If $\text{p}{}^1\text{<p?p}{}^{\mathrm{p}}{}_{\mathrm{+}}$ then there is a unique pseudo-slow decaying solution to ($\text{1}$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($\text{1}$). Similar results are obtained for the operator $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{?}}$. To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.     

Caractère bien posé du problème de Cauchy pour le système de Bérenger

The Bérenger perfectly matched layer is used in computational electromagnetism as an absorbing layer in scattering problems. It raises delicate mathematical issues. In this Note we show, for regular data, the existence and uniqueness of strong solutions to the Cauchy problem derived from the PML method. The result is presented in the 2-D case. The key to the proof is an appropriate control of a mixed H¹- L² norm of the solution by the same norm of the initial data. Beside a paper is in preparation about extensions of this results (L² estimates, 3-D case) (see also [5]).     

Analyse d'un élément mixte pour le problème de Stokes

Résumé Nous considérons quelques extensions des résultats classiques d'estimation d'erreur pour les problèmes mixtes. Nous étudions en particulier l'effet de l'emploi d'une approximation non conforme et celui de la décomposition de la formeb(·,·) enb ₁(·,·) etb ₂(·,·). Nous employons cette domposition pour affiner certaines estimations dans le cas où la formea(·,·) n'est pas coercive dans la bonne norme. Nous considérons aussi des résultats de dualité dans le cas non conforme.

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Analysis of a mixed finite element for the Stokes problemI. General results

Summary We consider some extensions of the classical error estimates for mixed problems. We take into account, in particular, nonconforming approximations and the splitting of the bilinear formb(·,·), intob ₁(·,·) andb ₂(·,·). We use this last fact to sharpen some estimates in a case where the forma(·,·) does not have coercivity in the right norm. We also consider duality results in the nonconforming case.

Ce travail a été réalisé grâce à des subventions du fonds FCAR et du CRSNG.     

Sur le problème de Cauchy pour des opérateurs faiblement hyperboliques

Sans résumé à la memoire de Lamberto Cattabriga