Abstract. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrödinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation. Introduction.In this paper we shall prove a Strichartz estimate in the following abstract setting (see below for the concrete examples of the wave and Schrödinger equation): let (X, dx) be a measure space and H a Hilbert space.

We obtain global well-posedness, scattering, and global L 10 t,x spacetime bounds for energy-class solutions to the quintic defocusing Schrödinger equation in R 1+3 , which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain [4] and Grillakis [20], which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain [4], but we perform the induction analysis in both frequency space and physical space simultaneously, and replace the Morawetz inequality by an interaction variant (first used in [12], [13]). The principal advantage of the interaction Morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions. In particular, this interaction estimate, together with an almost-conservation argument controlling the movement of L 2 mass in frequency space, rules out the possibility of energy concentration.

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L 2 L^2 -based Sobolev spaces H s H^s where local well-posedness is presently known, apart from the H 1 4 ( R ) H^{\frac {1}{4}} (\mathbb {R} ) endpoint for mKdV and the H − 3 4 H^{-\frac {3}{4}} endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura’s transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ 1, K 1, s > 1, we construct smooth initial data u 0 with u 0 H s < δ, so that the corresponding time evolution u satisfies u(T ) H s > K at some time T . This growth occurs despite the Hamiltonian's bound on u(t) Ḣ 1 and despite the conservation of the quantity u(t) L 2 .

Abstract. We prove an "almost conservation law" to obtain global-in-time wellposedness for the cubic, defocussing nonlinear Schrödinger equation in H s (R n ) when n = 2, 3 and s > , respectively.