Smoothing effects and local existence for nonlinear Schrodinger equation with variable coefficients [French]

摘要

In this article. it is shown that the initial value problem for the nonlinear Schrodinger equation u(t) - iDelta(g)u + A(x, D-x)u = Q(u, del(x)u, (u) over bar, del(x)(u) over bar), t epsilon R, x epsilon R-n, is locally well posed for a class of "small" data under the geometrical assumption that there is no trapped bicharacteristics associated with Delta(g), Here Delta(g) is the Laplacian associated with an asymptotically flat metric, A is a pseudodifferential operator of order I and Q is a polynomial having no constant or linear terms. This generalizes a result from Kenig, Ponce and Vega which concerns the case of Euclidian metric. The proof is based on microlocal smoothing effects estimates as made Doi's. [References: 47]