Vol. 2, No. 1, 2009

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Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation

Terence Tao

Vol. 2 (2009), No. 1, 61–81

We consider the focusing mass-critical NLS iut + Δu = |u|4du in high dimensions d 4, with initial data u(0) = u0 having finite mass M(u0) =d|u0(x)|2dx < . It is well known that this problem admits unique (but not global) strong solutions in the Strichartz class Ct,loc0Lx2 Lt,loc2Lx2d(d2), and also admits global (but not unique) weak solutions in LtLx2. In this paper we introduce an intermediate class of solution, which we call a semi-Strichartz class solution, for which one does have global existence and uniqueness in dimensions d 4. In dimensions d 5 and assuming spherical symmetry, we also show the equivalence of the Strichartz class and the strong solution class (and also of the semi-Strichartz class and the semi-strong solution class), thus establishing unconditional uniqueness results in the strong and semi-strong classes. With these assumptions we also characterise these solutions in terms of the continuity properties of the mass function tM(u(t)).

Strichartz estimates, nonlinear Schrodinger equation, weak solutions, unconditional uniqueness
Mathematical Subject Classification 2000
Primary: 35Q30
Received: 16 July 2008
Accepted: 17 February 2009
Published: 1 February 2009
Terence Tao
University of California, Los Angeles
Mathematics Department
Los Angeles CA 90095-1555
United States