The energy-critical defocusing NLS on \({\mathbb{T}}^{3}\). (English) Zbl 1245.35119

Ionescu, Alexandru D.; Pausader, Benoit

Duke Math. J. 161, No. 8, 1581-1612 (2012).

The authors of the paper consider the initial value problem for the nonlinear Schrödinger (NLS) \[ (i\partial_t+\triangle )u=u|u|^4, \quad u(0)=\phi , \] where \(\partial_t u\equiv \partial u /\partial t\), \(x\in\mathbb{T}^3\), \(\mathbb{T}=\mathbb{R}/(2\pi \mathbb{Z})\). It is known that suitable solutions of this problem on a time interval \(I\) satisfy mass and energy conservation, that is, \[ M(u)(t)=\int_{\mathbb{T}^3}|u(t)|^2dx, \quad E(u)(t)=(1/2)\int_{\mathbb{T}^3} |\nabla u(t)|^2dx + (1/6)\int_{\mathbb{T}^3} |u(t)|^6dx \] are constant on the interval \(I\).
The main statement concerns global well-posedness in \(H^1(\mathbb{T}^3)\) for the problem under consideration. It is shown that if \(\phi \in H^1(\mathbb{T}^3)\), then there exists a unique global solution \(u\in X^1(\mathbb {R})\) of the above stated initial value problem. In addition the mapping \(\phi \to u\) extends to a continuous mapping from \(H^1(\mathbb{T}^3)\) to \(X^1([-T,T])\) for any \(T\in [0,\infty )\), and the quantities \(M(u)\) and \(E(u)\) are conserved along the flow.

Reviewer: Dimitar A. Kolev (Sofia)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
42B37	Harmonic analysis and PDEs
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Keywords:

nonlinear Schrödinger; well-posedness; energy-critical; defocusing; global solution

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