Jean-Marc Delort

Existence globale et comportement asymptotique pour l'équation de Klein–Gordon quasi linéaire à données petites en dimension 1

Resume Soit v une solution de lequation de Klein–Gordon quasi lineaire en dimensionxa01 despace □ v+v=F(v,∂ t v,∂ x v,∂ t ∂ x v,∂ x 2 v) a donnees de Cauchy regulieres a support compact, de taille e→0. Supposons que F sannule au moins a lordrexa02 enxa00. On sait alors que la solution v existe sur un intervalle de temps de longueur superieure ou egale a ec/e2 pour une constante positive c, et que pour une non-linearite generale F elle explose en temps fini de lordre de ec′/e2 (c′>0). Nous avons conjecture dansxa0[7] une condition necessaire et suffisante sur F sous laquelle la solution devrait exister globalement en temps, pour e assez petit. Nous prouvons dans cet article la suffisance de cette condition. De plus, nous obtenons le premier terme dun developpement asymptotique de v lorsque t→+∞.

Almost global existence for solution of semilinear klein-gordon equations with small weakly decaying cauchy data: Almost global existance

Let T∈ be the time of exisstence of a semilinear Klein-Gordon equation with small,smooth,Cauchy data of size ∈ in space dimension d≥2 If the Cauchy data are decaying rapidly enough at infinity,and the nonlinearity vanishes at least at order 2 at 0,it is well known that T∈=+∞ for∈ small enough. The aim of this paper is to show that if one assumes only a weak decay of the Cauchy data at infinity,one has a lower bound T∈≤Cexp(c∈−μ)(μ=2/3 if d =2,μ=1 id d≤3) when the nonlinearity satisfies a convenient null condition

Une remarque sur le problème des nappes de tourbillon axisymétriques sur R3

Resume On a prouve dans “Existence de nappes de tourbillon en dimension deux” ( J. Amer. Math. Soc. , a paraitre) lexistence dune solution au probleme des nappes de tourbillon en dimension 2, lorsque le tourbillon initial est une mesure de signe fixe appartenant a lespace de Sobolev H −1 comp ( R 2 ). Pour cela, on a etabli que si une suite ( v e ) e de solutions de lequation dEuler, correspondant a une famille de regularisees de la condition initiale, converge faiblement vers une limite v , cette limite continue a satisfaire lequation. Le but de cet article est de montrer que, dans le cas du probleme analogue pour des flots axisymetriques en dimension 3, la situation est radicalement differente: en effet, on prouve que soit la suite dapproximations est fortement convergente, soit sa limite faible nest pas solution de lequation.

Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle

We prove that higher Sobolev norms of solutions of quasi-linear Klein-Gordon equations with small Cauchy data on S 1 remain small over intervals of time longer than the ones given by local existence theory. This result extends previous ones obtained by several authors in the semi-linear case. The main new difficulty one has to cope with is the loss of one derivative coming from the quasi-linear character of the problem. The main tool used to overcome it is a global paradifferential calculus adapted to the Sturm-Liouville operator with periodic boundary conditions.

Growth of Sobolev Norms for Solutions of Time Dependent Schrödinger Operators with Harmonic Oscillator Potential

It has been known for some time that solutions of linear Schrödinger operators on the torus, with bounded, smooth, time dependent (order zero pseudo-differential) potential, have Sobolev norms growing at most like t ε when t → + ∞ for any ε > 0. This property is proved exploiting the fact that, on the circle, successive eigenvalues of the laplacian are separated by increasing gaps (and a more involved, but similar property, for clusters of eigenvalues in higher dimension). We study here the case of solutions of , where V is a time periodic pseudo-differential order zero perturbation. In this case, the gap between successive eigenvalues of the stationary operator is constant. We show that there are order zero potentials V for which some solutions u have Sobolev norms of order s growing like t s/2 when t → + ∞, i.e. . The idea of the proof is to construct a potential which, at the classical level, pulls frequencies to higher modes, so that they will be of size at time t. One constructs then the wanted solution passing from the classical level to the quantum one.