Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in \(\mathbb R^{1+4}\).

*(English)*Zbl 1160.35067The paper is dealing with the initial-value problem for the cubic nonlinear Schrödinger equation with the self-defocusing nonlinearity in the four-dimensional space:
\[
iu_t + \Delta u - |u|^2u = 0.
\]
This case is considered as the “energy-critical” one, because the conserved energy corresponding to the equation,
\[
E = \int d^4x (\frac 12 |\nabla u(x,t)|^2+\frac 14 |u|^4),
\]
is invariant with respect to the scaling transformation: \(t \to t/\lambda^2,x\to x/\lambda, u\to \lambda u\). Another integral quantity which is invariant, in the same case, with respect to the scaling transformation is the sixth-power spatiotemporal norm,
\[
\|u\|\equiv \int_{-\infty}^{+\infty}dt\int d^4x |u(x,t)|^6.
\]
The main result of the work is a proof of the theorem stating that any initial condition with finite energy \(E\) generates a unique global solution, which obeys the following restriction: \(\|u\| < C(E)\), with \(C\) depending only on the energy. In fact, the solution is of the scattering type, decaying at \(t\to\infty\) into quasi-linear waves. The proof closely follows the lines of analysis developed in another recent work [J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in \(\mathbb{R}^3\)”, to appear in Ann. of Math., cf. arXiv:math/0402129]. The main idea of the proof is to develop a frequency-localized estimate for the mass of the solution (i.e., its usual quadratic norm, which, as well as the energy, is a dynamical invariant of the equation, but not a scale-invariant one) that allows one to check that the equation does not pump energy into high-frequency modes.

Reviewer: Boris A. Malomed (Tel Aviv)