This work is devoted to the description of bounded energy sequences of solutions to the equation (1) □ u + | u | 4 = 0 in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], up to remainder terms small in energy norm and in every Strichartz norm. The proof relies on scattering theory for (1) and on a structure theorem for bounded energy sequences of solutions to the linear wave equation. In particular, we infer the existence of an a priori estimate of Strichartz norms of solutions to (1) in terms of their energy.

We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any Riemannian compact manifold. As a consequence we infer low regularity local well-posedness results in any dimension, as well as global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of cubic defocusing nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.

We present a theory for carrying out homogenization limits for quadratic functions (called "energy densities") of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure.The very general theory is illustrated by typical examples like (semi)classical limits of Schrödinger equations (with or without a periodic potential), the homogenization limit of the acoustic equation in a periodic medium, and the classical limit of the Dirac equation. IntroductionWe consider the following type of initial value problems:where ε is a small parameter, u ε (t, x) is a vector-valued L 2 -function on R m x , and P ε is an anti-self-adjoint, matrix-valued (pseudo)-differential operator with a Weyl symbol given by P 0 (x, x/ε, εξ) + O(ε). Here P 0 = P 0 (x, y, ξ) is a smooth function that is periodic with respect to y. By ξ we denote the conjugate variable to the position x; that is, ξ = −i∇ x .The main assumptions are that the data u ε I are bounded in L 2 as ε goes to 0 and that u ε I oscillates at most at frequency 1/ε; for instance,A more general formulation of the assumptions on u ε I is given in definitions (1.26) and (1.27) below.

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