Hart F. Smith

Global strichartz estimates for nonthapping perturbations of the laplacian

The authors prove global Strichartz estimates for compact perturbations of the wave operator in odd dimensions when a non-trapping assumption is satisfied.

Almost global existence for some semilinear wave equations

We prove almost global existence for semilinear wave equations outside of nontrapping obstacles. We use the vector field method, but only use the generators of translations and Euclidean rotations. Our method exploits 1/r decay of wave equations, as opposed to the much harder to prove 1/t decay.

Almost global existence for quasilinear wave equations in three space dimensions

This article studies almost global existence for solutions of quadratically quasi linear systems of wave equations in three space dimensions. The approach here uses only the classical invariance of the wave operator under translations, spatial rotations, and scaling. Using these techniques we can handle wave equations in Minkowski space or Dirichlet-wave equations in the exterior of a smooth, star shaped obstacle. We can also apply our methods to systems of quasilinear wave equations having different wave speeds. This extends our work [11] for the semilinear case. Previous almost global ex istence theorems for quasilinear equations in three space dimensions were for the non-obstacle case. In [9], John and Klainerman proved almost global existence on Minkowski space for quadratic, quasilinear equations using the Lorentz invariance of the wave operator in addition to the symmetries listed above. Subsequently, in [14], Klainerman and Sideris obtained the same result for a class of quadratic, divergence-form nonlinearities without relying on Lorentz invariance. This line of thought was refined and applied to prove global-in-time results for null-form equa tions related to the theory of elasticity in Sideris [22], [23], and for multiple-speed systems of null-form quasilinear equations in Sideris and Tu [24], and Yokoyama [29]. The main difference between our approach and the earlier ones is that we ex ploit the 0(|x|-1) decay of solutions of wave equations with sufficiently decaying initial data as much as we involve the stronger 0(t~l) decay. Here, of course, x = (x\,X2,x

On the critical semilinear wave equation outside convex obstacles

) is the spatial component, and t the time component, of a space time vector (t, x) G M+ x E3. Establishing 0(|x|_1) decay is considerably easier and can be achieved using only the invariance with respect to translations and spatial rotation. A weighted L2 space-time estimate for inhomogeneous wave equations (Proposition 3.1 below, from [11]) is important in making the spatial decay useful for the long-time existence argument. For semilinear systems, one can show almost global existence from small data using only this spatial decay [11]. For quasilinear systems, however, we also have to show that both first and second derivatives of u decay like 1/t. Fortunately, we can do this using a variant of some L1 ?> L?? estimates of John, H?rmander,

A Hardy space for Fourier integral operators

In this paper we shall show that certain estimates for the Euclidean wave equation also hold on Riemannian manifolds with smooth, strictly geodesically concave boundaries. By the last condition, we understand that the second fundamental form on the boundary of the manifold is strictly positive definite. We shall then give two applications of our estimates. First, we shall show that if n is the exterior of a smooth, compact, and strictly convex obstacle ~ c lle , then there exists a unique global, smooth solution to the critical wave equation in R+ x n:

ON STRICHARTZ ESTIMATES FOR SCHRÖDINGER OPERATORS IN COMPACT MANIFOLDS WITH BOUNDARY

ABSTRACTWe introduce a new function space, denoted by HFIO1(ℝn), which is preserved by the algebra of Fourier integral operators of order 0 associated to canonical transformations. A subspace of L1 (ℝn), this space in many aspects resembles the real Hardy space of Fefferman-Stein. In particular, we obtain an atomic characterization of HFIO1 (ℝn). In contrast to the standard Hardy space, these atoms are localized in frequency space as well as in real space.

Strichartz estimates for the wave equation on manifolds with boundary

We prove local Strichartz estimates with a loss of derivatives over compact manifolds with boundary. Our results also apply more generally to compact manifolds with Lipschitz metrics.

Seismic imaging with the generalized Radon transform: a curvelet transform perspective*

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.

A multi-scale approach to hyperbolic evolution equations with limited smoothness

A key challenge in the seismic imaging of reflectors using surface reflection data is the subsurface illumination produced by a given data set and for a given complexity of the background model (of wave speeds). The imaging is described here by the generalized Radon transform. To address the illumination challenge and enable (accurate) local parameter estimation, we develop a method for partial reconstruction. We make use of the curvelet transform, the structure of the associated matrix representation of the generalized Radon transform, which needs to be extended in the presence of caustics and phase linearization. We pair an image target with partial waveform reflection data, and develop a way to solve the matrix normal equations that connect their curvelet coefficients via diagonal approximation. Moreover, we develop an approximation, reminiscent of Gaussian beams, for the computation of the generalized Radon transform matrix elements only making use of multiplications and convolutions, given the underlying ray geometry; this leads to computational efficiency. Throughout, we exploit the (wave number) multi-scale features of the dyadic parabolic decomposition underlying the curvelet transform and establish approximations that are accurate for sufficiently fine scales. The analysis we develop here has its roots in and represents a unified framework for (double) beamforming and beam-stack imaging, parsimonious pre-stack Kirchhoff migration, pre-stack plane-wave (Kirchhoff) migration and delayed-shot pre-stack migration.

Strichartz estimates for Dirichlet-wave equations in two dimensions with applications

We discuss how techniques from multiresolution analysis and phase space transforms can be exploited in solving a general class of evolution equations with limited smoothness. We have wave propagation in media of limited smoothness in mind. The frame that appears naturally in this context belongs to the family of frames of curvelets. The construction considered here implies a full-wave description on the one hand but reveals the geometrical properties derived from the propagation of singularities on the other hand. The approach and analysis we present (i) aids in the understanding of the notion of scale in the wavefield and how this interacts with the configuration or medium, (ii) admits media of limited smoothness, viz. with Hölder regularity s ≥ 2, and (iii) suggests a novel computational algorithm that requires solving for the mentioned geometry on the one hand and solving a matrix Volterra integral equation of the second kind on the other hand. The Volterra equation can be solved by recursion—as in the computation of certain multiple scattering series—revealing a curvelet–curvelet interaction. We give precise estimates expressing the degree of concentration of curvelets following the propagation of singularities.