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We consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay estimates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions to the wave equation on the domain of outer communications of the Schwarzschild spacetime manifold (where is the spatial dimension, and is the mass of the black hole) the associated energy flux through a foliation of hypersurfaces (terminating at future null infinity and to the future of the bifurcation sphere) decays, , where is a constant depending on and , and is a suitable higher-order initial energy on ; moreover we improve the decay rate for the first-order energy to for any , where denotes the hypersurface truncated at an arbitrarily large fixed radius provided the higher-order energy on is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate . In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski (2010).
We consider the Schrödinger map initial value problem
with a smooth map from the Euclidean space to the sphere with subthreshold () energy. Assuming an a priori boundedness condition on the solution , we prove that the Schrödinger map system admits a unique global smooth solution provided that the initial data is sufficiently energy-dispersed, i.e., sufficiently small in the critical Besov space . Also shown are global-in-time bounds on certain Sobolev norms of . Toward these ends we establish improved local smoothing and bilinear Strichartz estimates, adapting the Planchon–Vega approach to such estimates to the nonlinear setting of Schrödinger maps.
We prove new bilinear dispersive estimates. They are obtained and described via a bilinear time-frequency analysis following the space-time resonances method, introduced by Masmoudi, Shatah, and the second author. They allow us to understand the large time behavior of solutions of quadratic dispersive equations.
We consider the Zakharov system with periodic boundary conditions in dimension one. In the first part of the paper, it is shown that for fixed initial data in a Sobolev space, the difference of the nonlinear and the linear evolution is in a smoother space for all times the solution exists. The smoothing index depends on a parameter distinguishing the resonant and nonresonant cases. As a corollary, we obtain polynomial-in-time bounds for the Sobolev norms with regularity above the energy level. In the second part of the paper, we consider the forced and damped Zakharov system and obtain analogous smoothing estimates. As a corollary we prove the existence and smoothness of global attractors in the energy space.