Jason Metcalfe

Strichartz Estimates on Schwarzschild Black Hole Backgrounds

We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of [29], to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.

LONG-TIME EXISTENCE OF QUASILINEAR WAVE EQUATIONS EXTERIOR TO STAR-SHAPED OBSTACLES VIA ENERGY METHODS ∗

We establish long-time existence results for quasilinear wave equations in the exterior of star-shaped obstacles. To do so, we prove an analogue of the mixed-norm estimates of Keel, Smith, and Sogge for the perturbed wave equation. The arguments that are presented rely only upon the invariance of the wave operator under translations and spatial rotations.

Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle

hhIn this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimensions. In order to prove the global estimates, we explore weighted Strichartz estimates for solutions of the wave equation when the Cauchy data and forcing term are compactly supported.

Hyperbolic trapped rays and global existence of quasilinear wave equations

We prove global existence for quasilinear wave equations outside of a wide class of obstacles. The obstacles may contain trapped hyperbolic rays as long as there is local exponential energy decay for the associated linear wave equation. Thus, we can handle all non-trapping obstacles. We are also able to handle non-diagonal systems satisfying the appropriate null condition.

Global Existence of Solutions to Multiple Speed Systems of Quasilinear Wave Equations in Exterior Domains

Abstract In this paper we prove global existence for certain multispeed Dirichlet-wave equations with quadratic nonlinearities outside of obstacles. We assume the natural null condition for systems of quasilinear wave equations with multiple speeds. The null condition only puts restrictions on the self-interactions of each wave family. We use the method of commuting vector fields and weighted space-time L 2 estimates.

Concerning the Strauss Conjecture and Almost Global Existence for Nonlinear Dirichlet-Wave Equations in 4-Dimensions

We show the obstacle version of the Strauss conjecture holds when the spatial dimension is equal to 4. We also show that an almost global existence theorem of Hörmander for (4 + 1)-dimensional Minkowski space holds in the obstacle setting. We use weighed space-time variants of the energy inequality and a variant of the classical Hardy inequality.

Paraproducts in one and several parameters

Abstract For multiparameter bilinear paraproduct operators B we prove the estimate Here, 1/p + 1/q = 1/r and special attention is paid to the case of 0 < r < 1. (Note that the families of multiparameter paraproducts are much richer than in the one parameter case.) These estimates are the essential step in the version of the multiparameter Coifman-Meyer theorem proved by C. Muscalu, J. Pipher, T. Tao, and C. Thiele [Mucalu Camil, Pipher Jill, Tao Terrance, and Thiele Christoph: Bi-parameter paraproducts. Acta Math. 193 (2004), 269–296, Mucalu Camil, Pipher Jill, Tao Terrance, and Thiele Christoph: Multi-parameter paraproducts. arxiv:math.CA/0411607]. We offer a different proof of these inequalities.

The Strauss conjecture on Kerr black hole backgrounds

We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the Strauss conjecture on the Schwarzschild and Kerr, with small angular momentum, black hole backgrounds. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and a localized energy estimate on the black hole background, which handles the behavior in the remaining compact set.

NONLINEAR WAVES ON 3D HYPERBOLIC SPACE

In this article, global-in-time dispersive estimates and Strichartz estimates are explored for the wave equation on three dimensional hyperbolic space. Due to the negative curvature, extra dispersion is noted, as compared to the Euclidean case, and a wider range of Strichartz estimates is proved. Using these, small data global existence to semilinear wave equations is shown for a range of powers that is broader than that known for Euclidean space.

Nonlinear Hyperbolic Equations in Infinite Homogeneous Waveguides

ABSTRACT In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neumann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results and in terms of the lifespan for the almost global results. For nonlinear wave equations, in the case where the infinite part of the waveguide has spatial dimension three, the hypotheses in the theorem concern whether or not the Laplacian for the compact base of the waveguide has a zero mode or not.