Luc Robbiano

Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques

© Journées Équations aux dérivées partielles, 1991, tous droits réservés. L’accès aux archives de la revue « Journées Équations aux dérivées partielles » (http://www. math.sciences.univ-nantes.fr/edpa/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Microlocal analytic smoothing effect for the Schrödinger equation

0. Introduction and main results. The purpose of this work is to study the microlocal analytic smoothness of solutions of the initial value problem for the linear Schrödinger equation with variable coefficients. The aim is to relate the behavior at infinity of the initial data with the microlocal analytic smoothness; this phenomenon is known as themicrolocal smoothing effect. The results presented here are extensions to the analytic case of those of Craig-Kappeler-Strauss [CKS], [C], which concern theC∞ case. However, our method of proof, which relies on Sjöstrand theory [Sj], is entirely different from that of [CKS]. This question has also been investigated in recent years in the papers of Shananin [Sh], Kapitanski-Safarov [KS], and Wunsch [W]. Related results have also been obtained by Doi [D1], [D2], and we refer to the paper [CKS] for more references on the subject. Let us describe our main result. Let P = P(y,Dy), a second-order differential operator inR,

Local well-posedness and blow-up in the energy space for a class of L2 critical dispersion generalized Benjamin–Ono equations

Abstract We consider a family of dispersion generalized Benjamin–Ono equations (dgBO) u t − ∂ x | D | α u + | u | 2 α ∂ x u = 0 , ( t , x ) ∈ R × R , where | D | α u ˆ = | ξ | α u ˆ and 1 ⩽ α ⩽ 2 . These equations are critical with respect to the L 2 norm and global existence and interpolate between the modified BO equation ( α = 1 ) and the critical gKdV equation ( α = 2 ). First, we prove local well-posedness in the energy space for 1 α 2 , extending results in Kenig et al. (1991, 1993) [13] , [14] for the generalized KdV equations. Second, we address the blow-up problem in the spirit of Martel and Merle (2000) [19] and Merle (2001) [22] concerning the critical gKdV equation, by studying rigidity properties of the dgBO flow in a neighborhood of the solitons. We prove that for α close to 2, solutions of negative energy close to solitons blow up in finite or infinite time in the energy space H α 2 . The blow-up proof requires both extensions to dgBO of monotonicity results for local L 2 norms by pseudo-differential operator tools and perturbative arguments close to the gKdV case to obtain structural properties of the linearized flow around solitons.

DEGENERATE PARABOLIC OPERATORS OF KOLMOGOROV TYPE WITH A GEOMETRIC CONTROL CONDITION

We consider Kolmogorov-type equations on a rectangle domain (x, v) ∈ Ω = T × (−1, 1), that combine diffusion in variable v and transport in variable x at speed v γ , γ ∈ N ∗ , with Dirichlet boundary conditions in v. We study the null controllability of this equation with a distributed control as source term, localized on a subset ω of Ω. When the control acts on a horizontal strip ω = T × (a, b) with 0 0w henγ =1 , and only in large time T> T min > 0w henγ = 2 (see (K. Beauchard, Math. Control Signals Syst. 26 (2014) 145-176)). In this article, we prove that, when γ> 3, the system is not null controllable (whatever T is) in this configuration. This is due to the diffusion weakening produced by the first order term. When the control acts on a vertical strip ω = ω1 × (−1, 1) with ω1 ⊂ T, we investigate the null controllability on a toy model, where (∂x ,x ∈ T )i s replaced by (i(−Δ) 1/2 ,x ∈ Ω1), and Ω1 is an open subset of R N.A s the original system, this toy model satisfies the controllability properties listed above. We prove that, for γ =1 , 2 and for appropriate domains (Ω1 ,ω 1), then null controllability does not hold (whatever T> 0 is), when the control acts on a vertical strip ω = ω1 × (−1, 1) with ω1 ⊂ Ω1. Thus, a geometric control condition is required for the null controllability of this toy model. This indicates that a geometric control condition may be necessary for the original model too.

Remark on the Kato Smoothing Effect for Schrödinger Equation with Superquadratic Potentials

The aim of this note is to extend recent results of Yajima and Zhang (2001, 2004) on the -smoothing effect for Schrödinger equation with potential growing at infinity faster than quadratically.

Carleman estimates for the Zaremba Boundary Condition and Stabilization of Waves

In this paper, we shall prove a Carleman estimate for the so-called Zaremba problem. Using some techniques of interpolation and spectral estimates, we deduce a result of stabilization for the wave equation by means of a linear Neumann feedback on the boundary. This extends previous results from the literature: indeed, our logarithmic decay result is obtained while the part where the feedback is applied contacts the boundary zone driven by an homogeneous Dirichlet condition.We also derive a controllability result for the heat equation with the Zaremba boundary condition.

Controllability of a parabolic system with a diffuse interface

We consider a linear parabolic transmission problem across an interface of codimension one in a bounded domain or on a Riemannian manifold, where the transmission conditions involve an additional parabolic operator on the interface. This system is an idealization of a three-layer model in which the central layer has a small thickness

Linear distribution processes

\delta

Stabilization of the wave equation by the boundary

. We prove a Carleman estimate in the neighborhood of the interface for an associated elliptic operator by means of partial estimates in several microlocal regions. In turn, from the Carleman estimate, we obtain a spectral inequality that yields the null-controllability of the parabolic system. These results are uniform with respect to the small parameter

Exposant critique de Sobolev et régularité des solutions d’équations elliptiques

\delta