A Vector Field Method on the Distorted Fourier Side and Decay for Wave Equations with Potentials (Memoirs of the American Mathematical Society)

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A Vector Field Method on the Distorted Fourier Side and Decay for Wave Equations with Potentials (Memoirs of the American Mathematical Society)

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  • 製本 Paperback:紙装版/ペーパーバック版／ページ数 80 p.
  • 言語 ENG
  • 商品コード 9781470418731
  • DDC分類 515.7246

Full Description

The authors study the Cauchy problem for the one-dimensional wave equation ∂ 2 t u (t , x) - ∂ 2 x u (t , x) V (x)u (t , x) = 0. The potential V is assumed to be smooth with asymptotic behavior V (x) ∼ - 1 4 |x|-2 as |x| →∞. They derive dispersive estimates, energy estimates, and estimates involving the scaling vector field t ∂t x∂x , where the latter are obtained by employing a vector field method on the "distorted" Fourier side. In addition, they prove local energy decay estimates. Their results have immediate applications in the context of geometric evolution problems. The theory developed in this paper is funda­mental for the proof of the co-dimension 1 stability of the catenoid under the vanishing mean curvature flow in Minkowski space; see Donninger, Krieger, Szeftel, and Wong, "Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space", preprint arXiv:1310.5606 (2013).

Contents

Introduction
Weyl-Titchmarsh theory for $A$
Dispersive bounds
Energy bounds
Vector field bounds
Higher order vector field bounds
Local energy decay
Bounds for data in divergence form
Bibliography