P. Bizon, D. Hunik-Kostyra, and D.E. Pelinovsky
Ground state of the conformal flow on S3
Communications in Pure and Applied Mathematics 72 (2019), 1123-1151
Abstract:
We consider the conformal flow model derived as a normal form for
the conformally invariant cubic wave equation on S3. We prove that the energy attains
a global constrained maximum at a family of particular stationary solutions which we
call the ground state family. Using this fact and spectral properties of the linearized
flow (which are interesting on their own due to a supersymmetric structure), we prove
nonlinear orbital stability of the ground state family. The main difficulty in the proof
is due to the degeneracy of the ground state family as a constrained maximizer of the
energy.
Keywords:
Conformal flow, resonant normal form, ground state, constrained maximization, spectral stability,
nonlinear stability.