Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations

Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations

Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds

We give a new proof of a theorem of Bourgain [4], asserting that solutions of linear Schrödinger equations on the torus, with smooth time dependent potential, have Sobolev norms growing at most like t when t→ +∞, for any > 0. Our proof extends to Schrödinger equations on other examples of compact riemannian manifolds.

Stochastic Volterra equations under perturbations

We study stochastic perturbed Volterra equations of convolution type in an infinite dimensional case. Our interest is directed towards the existence and regularity of stochastic convolutions connected to the equations considered under some kind of perturbations. We use an operator theoretical method for the representation of solutions.

Partial Differential Equations Γ -convergence and Sobolev norms

We study a Γ -convergence problem related to a new characterization of Sobolev spaces W1,p(RN) (p > 1) established in H.-M. Nguyen [H.-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689–720] and J. Bourgain and H.-M. Nguyen [J. Bourgain, H.-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 75–80]. We can also hand...

Spectral controllability for 2D and 3D linear Schrödinger equations

We consider a quantum particle in an infinite square potential well of Rn, n = 2, 3, subjected to a control which is a uniform (in space) electric field. Under the dipolar moment approximation, the wave function solves a PDE of Schrödinger type. We study the spectral controllability in finite time of the linearized system around the ground state. We characterize one necessary condition for spec...