On the soliton resolution conjecture for wave maps

Thesis

Broadly speaking, the research presented in this thesis is centered around the study of the Soliton Resolution Conjecture (SRC) for the wave maps equation in dimension 2+1, which is rooted in a belief held by the physics community since the 1970's predicting that for a large class of non-linear dispersive/hyperbolic evolution equations in mathematical physics (of which wave maps are an example), the solution should decompose into a decoupled sum of rescaled solitons plus a regular term, up to an error of asymptotically vanishing energy, as one evolves towards its maximal time of existence.

To be more precise, we consider large energy wave maps as in the resolution of the threshold conjecture by Sterbenz and Tataru, but more specifically into the unit round sphere 𝕊^n-1 ⊂ ℝⁿ with n ≥ 2 (although parts of our argument work for general targets). We prove that, on a suitably chosen sequence of time slices approaching maximal existence, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm. For the latter, we further describe it as a linear gauge co-variant wave, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space.

Similar and stronger results have been recently obtained in the equivariant setting by several authors, where better control on the dispersive term concentrating on the null boundary of the cone is provided and in some cases the asymptotic decomposition is shown to hold for all time. Here however, we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to Lin and Rivière, which we make work here in the hyperbolic context of. A large part of the work presented in this thesis has appeared in author's.

Authors: Grinis, R

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford