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

Joachim Krieger École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
Available Formats:
Electronic ISBN: 978-1-4704-2877-8
Product Code: MEMO/241/1142.E
List Price: $73.00 MAA Member Price:$65.70
AMS Member Price: $43.80 Click above image for expanded view A Vector Field Method on the Distorted Fourier Side and Decay for Wave Equations with Potentials Joachim Krieger École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland Available Formats:  Electronic ISBN: 978-1-4704-2877-8 Product Code: MEMO/241/1142.E  List Price:$73.00 MAA Member Price: $65.70 AMS Member Price:$43.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 2412015; 80 pp
MSC: Primary 35; 42; Secondary 53;

The authors study the Cauchy problem for the one-dimensional wave equation $\partial_t^2 u(t,x)-\partial_x^2 u(t,x)+V(x)u(t,x)=0.$ The potential $V$ is assumed to be smooth with asymptotic behavior $V(x)\sim -\tfrac14 |x|^{-2}\mbox{ as } |x|\to \infty.$ They derive dispersive estimates, energy estimates, and estimates involving the scaling vector field $t\partial_t+x\partial_x$, 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).

• Chapters
• 1. Introduction
• 2. Weyl-Titchmarsh Theory for $A$
• 3. Dispersive Bounds
• 4. Energy Bounds
• 5. Vector Field Bounds
• 6. Higher Order Vector Field Bounds
• 7. Local Energy Decay
• 8. Bounds for Data in Divergence Form
• Requests

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Volume: 2412015; 80 pp
MSC: Primary 35; 42; Secondary 53;

The authors study the Cauchy problem for the one-dimensional wave equation $\partial_t^2 u(t,x)-\partial_x^2 u(t,x)+V(x)u(t,x)=0.$ The potential $V$ is assumed to be smooth with asymptotic behavior $V(x)\sim -\tfrac14 |x|^{-2}\mbox{ as } |x|\to \infty.$ They derive dispersive estimates, energy estimates, and estimates involving the scaling vector field $t\partial_t+x\partial_x$, 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).

• Chapters
• 1. Introduction
• 2. Weyl-Titchmarsh Theory for $A$
• 3. Dispersive Bounds
• 4. Energy Bounds
• 5. Vector Field Bounds
• 6. Higher Order Vector Field Bounds
• 7. Local Energy Decay
• 8. Bounds for Data in Divergence Form
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
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