Global Solutions for Small Nonlinear Long Range Perturbations of Two Dimensional Schrödinger Equations
Author: Jean-Marc Delort
Publisher:
Published: 2002
Total Pages: 110
ISBN-13:
DOWNLOAD EBOOK →Here the author presents the following: Let $Q_1, Q_2$ be two quadratic forms, and $u$ a local solution of the two-dimensional Schrodinger equation $(i\partial _t + \Delta )u = Q_1(u,\nabla _x u) + Q_2(\bar {u},\nabla _x \bar {u})$. He proves that if $Q_1$ and $Q_2$ do depend on the derivatives of $u$, and if the Cauchy datum is small enough and decaying enough at infinity, the solution exists for all times. The difficulty of the problem originates in the fact that the nonlinear perturbation is a long range one: This means that it can be written as the product of (a derivative of) $u$ and of a potential whose $L^\infty$ space-norm is not time integrable at infinity.
