Un problème d'optimisation de forme pour la contrôlabilité exacte de l'équation des ondes 2D

We consider the wave equation defined on $\Omega ?{\mathbb{R}}^2$ and $\omega ?\Omega$. We designate by $v_{\omega }$ the distributed control of minimal $L^2(\omega \times (0,T))$ norm obtained with the Hilbert Uniqueness Method which stabilizes the system at time $T>0$. This Note addresses the question of the optimal position of ω in order to minimize $J:\omega \to 6v_{\omega }{6}_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^{1,1}(\Omega )$, we express the shape derivative of J as a curvilinear integral on ?ω (independently of any adjoint solution) leading to a descent algorithm. A numerical application is given. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Estimées Ck pour l'équation ∂̄b sur un convexe de type fini de Cn

Let q=1,…,n?1 and D be a bounded convex domain in $\text{C}{}^{\mathrm{n}}$ of finite type m. We construct two integral operators T_q and $\text{T}{}_{\mathrm{q}}$ such that for all $\text{p∈}\text{N}\text{,}\text{T}{}_{\mathrm{q}}\text{,}\text{T}{}_{\mathrm{q}}\text{:C}{}^{\mathrm{p}}{}_{\mathrm{0,q}}\text{(bD)→C}{}^{\mathrm{p+1/m}}{}_{\mathrm{0,q?1}}\text{(bD)}$ are continuous, and for all (0,q)-forms h continuous on bD with $\text{?}{}_{\mathrm{b}}\text{h}$ continuous on bD too, with the additional hypothesis when q=n?1 that ∫_bDh∧φ=0 for all φ∈C^∞_n,0(bD) $\text{?}\text{?}{}_{\mathrm{b}}$-fermée, we show $\text{h=}\text{?}\text{?}{}_{\mathrm{b}}\text{(T}{}_{\mathrm{q}}\text{?}\text{T}{}_{\mathrm{q}}\text{)h+(T}{}_{\mathrm{q+1}}\text{?}\text{T}{}_{\mathrm{q+1}}\text{)}\text{?}\text{?}{}_{\mathrm{b}}\text{h}$. For this construction, we use the Diederich–Fornæss support function of Alexandre (Publ. IRMA Lille 54 (III) (2001)). To prove the continuity of T_q, we integrate by parts and take care of the tangential derivatives. The normal component in z of the kernel of $\text{T}{}_{\mathrm{q}}$ will have a bad behaviour, so, in order to find a good representative of its equivalence class, we isolate the tangential component of the kernel and then integrate by parts again. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Valuations invariantes pour l'action des groupes de Galois différentiels

Let $(F/K\text{,}?)$ be a differential field extension with differential Galois group $G={Gal}_{?}(F/K)$. For the natural action of G on the Riemann–Zariski variety $S^{?}=S^{?}(F/K)$ of the field extension $F/K$, we study the invariant valuations $\nu \in {\left(S^{?}\right)}^G$ when they do exist. We show close relations between these invariant valuations and the elements of F holonomic over K. Next, we study the continuity of the derivation ? with respect to these ν-adic topologies. We give a geometric structure property of G-invariant valuation inspired by Zariski. Finally, we give an answer for the existence problem of invariant valuations in the context of Picard–Vessiot extension. To cite this article: G. Duval, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

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).     

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).     

Paradoxe de Klein pour l'équation de Klein–Gordon chargée : superradiance et opérateur de diffusion

We develop the scattering theory for the charged Klein–Gordon equation on ${\mathbb{R}}_t\times {\mathbb{R}}_x$, when the electrostatic potential $A(x)$ has different asymptotics $a^{\pm }$ as $x\to \pm \infty$. In this case, the conserved energy is not positive definite (Klein Paradox). We construct the spectral representation for the harmonic equation, and we establish the existence of a Scattering Operator the symbol of which has a norm strictly larger than 1, for the frequencies in $(a^{?}\text{,}{a}^{+})$. These results can be applied to the DeSitter–Reissner–Nordstrøm metric, to justify the notion of superradiance of the charged black-holes. To cite this article: A. Bachelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Stabilisation Neumann pour l'équation des ondes dans un domaine extérieur

In odd dimension space, we prove, under a microlocal geometric condition, the exponential decay of the local energy for solutions of the wave equation on exterior domains, with Neumann damping ∂_nu+a(x)∂_tu=0.     

Inégalités d'oracle pour l'estimation d'une densité de probabilité

We study the problem of the nonparametric estimation of a probability density in $L_2(\mathbb{R})$. Expressing the mean integrated squared error in the Fourier domain, we show that it is close to the mean squared error in the Gaussian sequence model. Then, applying a modified version of Stein's blockwise method, we obtain a linear monotone oracle inequality and a kernel oracle inequality. As a consequence, the proposed estimator is sharp minimax adaptive (i.e. up to a constant) on a scale of Sobolev classes of densities. To cite this article: Ph. Rigollet, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Optimalité d'estimations d'énergie pour une équation des ondes amortie

We consider a wave equation damped by a nonlinear feedback. When the feedback has a polynomial growth, M. Nakao, A. Haraux, F. Conrad et al, E. Zuazua, V. Komornik obtained explicit estimates of energy decay rate. In the case of a boundary feedback and in one space dimension, we prove that these estimates are in fact optimal. © Académie des Sciences/Elsevier, Paris     

Formule d'Itô pour des Diffusions Uniformément Elliptiques,et Processus de Dirichlet

Soit X une diffusion uniformément elliptique sur R ^d ,F une fonction dans H _loc ¹(R ^d ) et la loi initiale de la diffusion. On montre que si l'intégrale |F|²(x)U(x)dx est finie, oùU désigne le potentiel de la mesure , alors F(X) est un processus de Dirichlet. Si de plus, F appartient àH ² _loc(R ^d ) et si les intégrales |F|²(x)U(x)dx et |f _k |²(x)U(x)dx sont finies, pour les dérivées faibles f _k de F, alors on peut écrire une formule d'Itô. En particulier, on définit l'intégrale progressive F(X)dX et on prouve l'existence des covariations quadratiques [f _k (X),X ^k ].     

Spectres pour l'approximation d'un nombre réel et de son carré

Let $\alpha (\xi )$ be the exponent that measures how a non-quadratic real number ξ and its square can be simultaneously approximated by rational numbers with the same denominator. Davenport and Schmidt have proved that $\alpha (\xi )$ is always between the golden ratio γ and 2. Roy, and after him Bugeaud and Laurent, have constructed numbers ξ such that $\alpha (\xi )<2$. Their method involves infinite words with many palindrome prefixes. In this text, we define new exponents of approximation that allow us to obtain, to some extent, a characterization of the values $\alpha (\xi )$ obtained by these authors. To cite this article: S. Fischler, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

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).     

Contrôlabilité de l'équation des ondes à densité rapidement oscillante à une dimension d'espace

We consider the one-dimensional wave equation with periodic density of period ε → 0. By a counterexample due to Avellaneda, Bardos, and Rauch, we know that the boundary controllability property does not hold uniformly as ε → 0. We prove that the control remains uniformly bounded if we control the projection of the solution over the subspace generated by the eigenfunctions associated with the eigenvalues λ ≤ C_ε⁻², C > 0 being small enough. This result is sharp in the sense that the control diverges when the projection over the eigenfunctions such that λ ~ C_ε⁻², with C large, is controlled. We use the classical WKB asymptotic development that provides sharp results on the convergence of the spectrum and the theory of non-harmonic Fourier series.     

Pavages des simplexes, schémas de graphes recollés et compactification des PGLrn+1/PGLr

Some equivariant compactifications of the quotients PGL _r ^{n +1}/PGL _r are constructed. Each one is decomposed into locally closed strata which are smooth, are indexed by the entire convex pavings of the simplex of dimension n and admit a modular interpretation deduced from that of the Grassmann varieties. Together, they form a simplicial scheme which “compactifies” the classifying simplicial scheme of PGL _r consisting of all the quotients PGL _r ^{n +1}/PGL _r , n≥0.

Oblatum 8-IV-1998 & 8-X-1998 / Published online: 28 January 1999     

Sur le caractère entropique des schémas de relaxation appliqués à une équation d'état non classique

In this Note, we show that the relaxation scheme theory applied to a non-classical equation of state allows to build in a very easy way a family of entropic schemes. It is very easy to build this kind of scheme because there exists a kinetic model whose fluid limit gives this non-classical equation of state.     

Vitesse de convergence en norme p-intégrale et normalité asymptotique de l'estimateur crible de l'opérateur d'un ARB(1)

The autoregressive model in a Banach space (ARB) allows to represent many continuous time processes used in practice (see, for example, D. Bosq, Linear Processes in Function Spaces: Theory and Applications, 2000, Springer, p. 150). In this Note we study an estimator of the operator in ARB(1) by the least squares method, when the operator is strictly p-integral, $p\in ]1,\infty [$, and we use Grenander's method of sieves (From U. Grenander, Abstract Inference, Wiley, 1981). We show consistency of the sieve estimator and we derive a central limit theorem for this estimator. To cite this article: F. Rachedi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).