A Vector Field Method on the Distorted Fourier Side and Decay for Wave Equations with Potentials

Roland Donninger, Joachim Krieger (Autoren)

Buch | Softcover

80 Seiten

2016
American Mathematical Society (Verlag)
978-1-4704-1873-1 (ISBN)

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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).

Roland Donninger, and Joachim Krieger, Ecole Polytechnique Federale de Lausanne, Switzerland.

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

Erscheinungsdatum	05.05.2016
Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	155 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
ISBN-10	1-4704-1873-8 / 1470418738
ISBN-13	978-1-4704-1873-1 / 9781470418731
Zustand	Neuware