Soliton resolution along a sequence of times for the focusing energy critical wave equation

Abstract

In this paper, we prove that any solution of the energy-critical wave equation in space dimensions 3, 4 or 5, which is bounded in the energy space decouples asymptotically, for a sequence of times going to its maximal time of existence, as a finite sum of modulated solitons and a dispersive term. This is an important step towards the full soliton resolution in the nonradial case and without any size restrictions. The proof uses a Morawetz estimate very similar to the one known for energy-critical wave maps, a virial type identity and a new channels of energy argument based on a lower bound of the exterior energy for well-prepared initial data.

On multi-solitons for the energy-critical wave equation in dimension 5

“…34)Finally, using(5.14),(5.15), (5.16), (5.34) and Cauchy-Schwarz inequality, we obtainI ≤ 4η 2 − 2|∇ε| 2 + 2(6 − 4δ)( χ · ∇ε)η + Cσ(|∇ε| 2 + η 2 ) − 4δ) |∇ε| 2 + η 2 + 2( χ · ∇ε)η + Cσ |∇ε| 2 + η 2 . Let x ∈ Ω 1,1 × Ω 1,2 × Ω 1,3 × Ω 1,4 × Ω 1,5 . From (5.17), we obtain…”

Multi-travelling waves for the nonlinear Klein-Gordon equation

“…where the vector-valued functions Φ 0 ℓ are defined in (15). However, unlike for JH, the eigenfunctions of H related to negative eigenvalues do not seem to be explicitly related to that of L. Nonetheless a key observation of this paper is that for any β ∈ R d , |β| < 1, the number of negative directions for the quadratic form H·, · is equal to the number k of negative eigenvalues of the operator L.…”

“…[42] for a proof in the case of the KdV equation. We refer to recent works of Duyckaerts, Kenig and Merle [16,15], and references therein for general soliton decomposition results in the nonintegrable situation of the energy critical wave equation.…”

Dynamics of the focusing critical wave equation

“…General case. The next theorem is due to Jia, Kenig, Merle and myself (see [12] and also the work of Jia [23] for a weaker version in the finite time blow-up case). …”

“…This note concerns results by Duyckaerts, Kenig and Merle [13], Jia [23] and Duyckaerts, Jia, Kenig and Merle [12] on the focusing energy critical wave equation:…”