Exterior Energy Bounds for the Critical Wave Equation Close to the Ground State
- PDF / 789,241 Bytes
- 63 Pages / 439.37 x 666.142 pts Page_size
- 80 Downloads / 209 Views
Communications in
Mathematical Physics
Exterior Energy Bounds for the Critical Wave Equation Close to the Ground State Thomas Duyckaerts1,2 , Carlos Kenig3 , Frank Merle4,5 1 2 3 4 5
Université Sorbonne Paris Nord, LAGA (UMR 7539), Villetaneuse, France Institut Universitaire de France, Paris, France University of Chicago, Chicago, USA Laboratoire de Mathématiques AGM (UMR 8088), Université de Cergy-Pontoise, Cergy-Pontoise, France Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France. E-mail: [email protected]
Received: 16 December 2019 / Accepted: 1 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: By definition, the exterior asymptotic energy of a solution to a wave equation on R1+N is the sum of the limits as t → ±∞ of the energy in the the exterior {|x| > |t|} of the wave cone. In our previous work Duyckaerts et al. (J Eur Math Soc 14(5):1389–1454, 2012), we have proved that the exterior asymptotic energy of a solution of the linear wave equation in odd space dimension N is bounded from below by the conserved energy of the solution. In this article, we study the analogous problem for the linear wave equation with a potential ∂t2 u + L W u = 0,
L W := − −
4 N +2 W N −2 N −2
(*)
obtained by linearizing the energy critical wave equation at the ground-state solution W , still in odd space dimension. This equation admits nonzero solutions of the form A + t B, where L W A = L W B = 0 with vanishing asymptotic exterior energy. We prove that the exterior energy of a solution of (*) is bounded from below by the energy of the projection of the initial data on the orthogonal complement of the space of initial data corresponding to these solutions. This will be used in a subsequent paper to prove soliton resolution for the energy-critical wave equation with radial data in all odd space dimensions. We also prove analogous results for the linearization of the energy-critical wave equation around a Lorentz transform of W , and give applications to the dynamics of the nonlinear equation close to the ground state in space dimensions 3 and 5. Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Duhamel formulation and Strichartz estimates . . . . . . . . . . . . . C. Kenig: Partially supported by NSF Grants DMS-14363746 and DMS-1800082.
2 8 8
T. Duyckaerts, C. Kenig, F. Merle
2.2 Profile decomposition . . . . . . . . . . . . . . . . . . . . . . 2.3 Wave equation with potential outside a wave cone . . . . . . . 3. Reduction to a Uniqueness Theorem . . . . . . . . . . . . . . . . 3.1 Lorentz transformation . . . . . . . . . . . . . . . . . . . . . 3.2 Reduction to a qualitative statement . . . . . . . . . . . . . . 4. Proof of the Uniqueness Theorem . . . . . . . . . . . . . . . . . . 4.1 Reduction to a radial problem . . . . . . . . . . . . . . . . . . 4.2 Preliminaries on a differential equation . . . . . . . . . . . . . 4.3 Compact support of the ini
Data Loading...
