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

Tao, Terence

doi:10.2140/apde.2009.2.61

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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

DOI: 10.2140/apde.2009.2.61

Abstract

We consider the focusing mass-critical NLS $i u_{t} + Δ u = - | u |^{4 ∕ d} u$ in high dimensions $d \geq 4$ , with initial data $u (0) = u_{0}$ having finite mass $M (u_{0}) = \int_{ℝ^{d}} | u_{0} (x) |^{2} d x < \infty$ . It is well known that this problem admits unique (but not global) strong solutions in the Strichartz class $C_{t, loc}^{0} L_{x}^{2} \cap L_{t, loc}^{2} L_{x}^{2 d ∕ (d - 2)}$ , and also admits global (but not unique) weak solutions in $L_{t}^{\infty} L_{x}^{2}$ . 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 $d \geq 4$ . In dimensions $d \geq 5$ 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 $t \mapsto M (u (t))$ .

Keywords

Strichartz estimates, nonlinear Schrodinger equation, weak solutions, unconditional uniqueness

Mathematical Subject Classification 2000

Primary: 35Q30

Milestones

Received: 16 July 2008

Accepted: 17 February 2009

Published: 1 February 2009

Authors

Terence Tao
University of California, Los Angeles Mathematics Department Los Angeles CA 90095-1555 United States
http://ftp.math.ucla.edu/~tao/

Vol. 2, No. 1, 2009