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.

In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities.In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces, in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the L 2 topology, despite the existence of global weak solutions at this regularity.Finally, we investigate the KdV equation at the endpoint regularity H −3/4 , and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval [−T, T ] and a locally Lipschitz continuous map taking the initial data in H −3/4 to a distributional solution u ∈ C 0 ([−T, T ]; H −3/4 ) which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.

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.