Patrick Gérard

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We consider the following Hamiltonian equation on the L Hardy space on the circle, i∂tu = Π(|u|u) , where Π is the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and(More)
— We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere S and more generally on Zoll surfaces, we prove that, for s > 1/4, NLS is uniformly well-posed in H, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces. Résumé. — On étudie(More)
— We estimates the L norm (2 ≤ p ≤ +∞) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. All our estimates are shown to be optimal for the(More)
— We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold M . We prove global existence of strong H solutions on M = S and M = S × S as far as the nonlinearity is defocusing and sub-quintic and thus we extend results of Ginibre-Velo and Bourgain who treated the cases of the Euclidean space R and the torus T = R/Z(More)
We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schrödinger equation on compact Riemannian manifolds. New dispersive estimates on the linear Schrödinger group are used to get global existence in the energy space on arbitrary surfaces and three-dimensional manifolds, generalizing earlier results by Bourgain on tori. On the other(More)
The endoplasmic reticulum (ER) consists of a polygonal network of sheets and tubules interconnected by three-way junctions. This network undergoes continual remodeling through competing processes: the branching and fusion of tubules forms new three-way junctions and new polygons, and junction sliding and ring closure leads to polygon loss. However, little(More)
We consider the following degenerate half wave equation on the one dimensional torus i∂tu− |D|u = |u|u, u(0, ·) = u0. We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system– the cubic Szegö equation. As a consequence, we prove an instability result for large H norms of solutions of this(More)