348 research outputs found
Global Strichartz Estimates for Solutions to the Wave Equation Exterior to a Convex Obstacle
In 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.Comment: 17 pages. Exposition simplified, error concerning the range of
admissible corrected, and the two dimensional argument was removed
due to errors. To appear, Trans. Amer. Math. So
Global existence for semilinear wave equations exterior to nontrapping obstacles
In this paper, we show global existence, in spatial dimensions greater than
or equal to four, for semilinear wave equations with quadratic nonlinearities
exterior to a nontrapping obstacle. This extends the previous work of
Shibata-Tsutsumi and Hayashi. Shibata-Tsutsumi showed the result for spatial
dimensions greater than or equal to 6, and Hayashi proved the result exterior
to a ball
Localized energy estimates for wave equations on high dimensional Schwarzschild space-times
The localized energy estimate for the wave equation is known to be a fairly
robust measure of dispersion. Recent analogs on the -dimensional
Schwarzschild space-time have played a key role in a number of subsequent
results, including a proof of Price's law. In this article, we explore similar
localized energy estimates for wave equations on -dimensional
hyperspherical Schwarzschild space-times.Comment: 15 pages. (updated References to the August 27, 2010 posting
The lifespan for 3-dimensional quasilinear wave equations in exterior domains
This article focuses on long-time existence for quasilinear wave equations
with small initial data in exterior domains. The nonlinearity is permitted to
fully depend on the solution at the quadratic level, rather than just the first
and second derivatives of the solution. The corresponding lifespan bound in the
boundaryless case is due to Lindblad, and Du and Zhou first proved such
long-time existence exterior to star-shaped obstacles. Here we relax the
hypothesis on the geometry and only require that there is a sufficiently rapid
decay of local energy for the linear homogeneous wave equation, which permits
some domains that contain trapped rays. The key step is to prove useful energy
estimates involving the scaling vector field for which the approach of the
second author and Sogge provides guidance.Comment: 26 page
Global parametrices and dispersive estimates for variable coefficient wave equations
In this article we consider variable coefficient, time-dependent wave
equations. Using phase space methods we construct outgoing parametrices and
prove Strichartz-type estimates globally in time. This is done in the context
of C^2 metrics which satisfy a weak aymptotic flatness condition at infinity.Comment: 52 pages. Several typos corrected, and the exposition was expanded in
Sections 8 and beyon
Elastic waves in exterior domains, Part II: Global existence with a null structure
In this article, we prove that solutions to a problem in nonlinear elasticity
corresponding to small initial displacements exist globally in the exterior of
a nontrapping obstacle. The medium is assumed to be homogeneous, isotropic, and
hyperelastic, and the nonlinearity is assumed to satisfy a null condition. The
techniques contained herein would allow for more complicated geometries
provided that there is a sufficient decay of local energy for the linearized
problem
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 for Dirichlet-wave equations with quadratic nonlinearties in high dimensions
We prove global existence of solutions to quasilinear wave equations with
quadratic nonlinearities exterior to nontrapping obstacles in spatial
dimensions four and higher. This generalizes a result of Shibata and Tsutsumi
in spatial dimensions greater than or equal to six. The technique of proof
would allow for more complicated geometries provided that an appropriate local
energy decay exists for the associated linear wave equation.Comment: Some corrections (per the referee's suggestions) were made in Section
3. 24 page
Global existence of null-form wave equations in exterior domains
We provide a proof of global existence of solutions to quasilinear wave
equations satisfying the null condition in certain exterior domains. In
particular, our proof does not require estimation of the fundamental solution
for the free wave equation. We instead rely upon a class of Keel-Smith-Sogge
estimates for the perturbed wave equation. Using this, a notable simplification
is made as compared to previous works concerning wave equations in exterior
domains: one no longer needs to distinguish the scaling vector field from the
other admissible vector fields.Comment: 29 pages. To appear in Math.
Paraproducts in One and Several Parameters
For multiparameter bilinear paraproduct operators we prove the estimate
Here, and special
attention is paid to the case of . (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.
We offer a different proof of these inequalities.Comment: Minor corrections mad
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