Global well-posedness for Schrödinger equations with derivative. (English) Zbl 1002.35113

Colliander, James E.; Keel, Markus; Staffilani, Gigliola; Takaoka, Hideo; Tao, Terence C.

SIAM J. Math. Anal. 33, No. 3, 649-669 (2001).

Summary: We prove that the one-dimensional Schrödinger equation with derivative in the nonlinear term \[ \begin{aligned} & i\partial_t u+\partial^2_x u=i\lambda\partial_x(|u|^2u),\\ & u(x,0)=u_0(x),\qquad x\in\mathbb{R},\;t\in\mathbb{R},\end{aligned} \] is globally well-posed in \(H^{s}\) for \(s>2/3\), for small \(L^{2}\) data. The result follows from an application of the “I-method”. This method allows us to define a modification of the energy norm \(H^{1}\) that is “almost conserved” and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in \(H^{s}\), for \(s>2/3.\)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

nonlinear Schrödinger equations; globally well-posed; quintic nonlinear defocusing Schrödinger equation; Alfvén waves in magnetized plasma