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Soliton resolution for equivariant wave maps on a wormhole. (English) Zbl 1437.83026

In the present paper, the author proves the soliton resolution conjecture for the equivariant wave maps on a wormhole. This conjecture states that the general globally defined solutions of (most) nonlinear dispersive partial differential equations decompose asymptotically into a superposition of a nonlinear bulk term and a radiation term. The model studied in this paper is of the equivariant maps \(U(\psi): \mathbb{R} \times \left( \mathbb{R} \times S^2 \right) \longrightarrow S^3 \) that are the critical point of the following action functional \[ S(U(\psi)) = \int_{\mathbb{R} \times \left( \mathbb{R} \times S^2 \right)} \langle \partial U(\psi), \partial U(\psi) \rangle, \tag{1} \] where \(\psi\) is an associated function to the wave map \(U\). The corresponding nonlinear equation and the initial value problem have the following form \[ \partial^{2}_{t} \psi - \partial^{2}_{r} \psi -\frac{2r}{r^2 +1 } \partial_r \psi + \frac{l(l+1)}{2(r^2 +1 )} \sin 2\psi =0, \qquad l \in \mathbb{N} , \tag{2} \] \[ \vec{\psi}(0) := (\left( \psi(0), \partial_t \psi(0) \right) ) = \left( \psi_0 , \psi_1 \right) , \tag{3} \] which conserves the energy defined by a functional \(\mathcal{E} (\psi, \partial \psi)\) along the flow. The finite energy pairs \(\left( \psi_0 , \psi_1 \right)\) of degree \(n \in \mathbb{N} \cup \{0\}\) have the finite energies \(\mathcal{E}_{nl} < \infty\). Then the author shows that for every \(l \in \mathbb{N}\), \(n \in \mathbb{N} \cup \{0\}\), there is a unique harmonic map \(Q_{l,n}(r)\) that satisfies the static equation from (2) and global solutions \(\phi^{\pm}_{L} (t,r)\) to the linearized equation from (2) such that \(Q_{l,n}\) and \(\phi^{\pm}_{L}\) correspond to the finite energy value \(\mathcal{E}_{nl}\). Moreover, the problem (2)–(3) has a unique global solution \(\vec{\psi}\) for which \[ \vec{\psi} (t) = \left( Q_{l,n}, 0 \right) + \phi^{\pm}_{L} (t) + O_{\mathcal{H}_{0}}(1), \qquad \text{for} \, \, t \rightarrow \pm \infty , \tag{4} \] where \(\mathcal{H}_{0} := \mathcal{H} ((-\infty, \infty);(r^2 + 1)dr)\) is energy space.
The proof of the results is based on the Kenig-Merle method from the reference [C. E. Kenig and F. Merle, Invent. Math. 166, No. 3, 645–675 (2006; Zbl 1115.35125)] and on the general arguments of the Strichartz-Strauss estimates from the small data theory applied to a constant \(t\) foliation of the wormhole space-time. The result is proved by demonstrating the following three affirmations: 1) that the small data problem admits global and scattering free waves; 2) that if the main result from the equation (4) were not true a non-scattering compact minimal solution exists; and 3) that the precompact solution vanishes. Thus, the author succeeds to prove the truth of the main result by contradiction.
The paper is written in a clear style, the proofs and the arguments are well presented and developed. There are 23 references and the reference [P. Bizoń et al., Nonlinearity 25, No. 5, 1299–1309 (2012; Zbl 1243.35117)] is anterior to the present paper within the same project.

MSC:

83C15 Exact solutions to problems in general relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
35L05 Wave equation
35Q51 Soliton equations
83C40 Gravitational energy and conservation laws; groups of motions
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References:

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