Infinite time blow-up solutions to the energy critical wave maps equation

Abstract

We consider the wave maps problem with domain R 2`1 and target S 2 in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from R 2 to S 2 , with polar angle equal to Q1prq " 2 arctanprq. By applying the scaling symmetry of the equation, Q λ prq " Q1prλq is also a harmonic map, and the family of all such Q λ are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps. In this work, we construct infinite time blowup solutions along the Q λ family. More precisely, for b ą 0, and for all λ 0,0,b P C 8 pr100, 8qq satisfying, for some C l , C m,k ą 0, 5 4. Construction of the ansatz 9 4.1. Outline 9 4.2. Correcting the large r behavior of Q 1 λptq 11 4.3. The free wave correction 18 4.4. Further improvement of the soliton error term 19 4.5. The linear error terms for large r 23 4.6. The nonlinear error terms involving v 1 , v 2 , v 3 , v 4 25 4.7. The equation resulting from u ansatz 25 4.8. Choosing λptq 26 4.9. Estimates on B k r B j t F 4 131 4.10. Estimates on F 5 138 4.11. Estimates on F 6 142 4.12. Estimates on v corr -dependent quantities 142 5. Solving the final equation 144 5.1. The equation for F puq 144 5.2. Estimates on F 2 145 5.3. F 3 Estimates 147 2010 Mathematics Subject Classification. Primary 35L05, 35Q75. 1 2 MOHANDAS PILLAI 5.4. Estimates on F 4 -related oscillatory integrals 156 5.5. Setup of the final iteration 177 6. The energy of the solution, and its decomposition as in Theorem 1.1 188 Appendix A. Proof of Theorem 1.2 191 References 193

Global, Non-Scattering Solutions to the Energy Critical Yang-Mills Problem

“…As mentioned earlier, the method is similar to that used by the author in [18]. The argument can be split broadly into two steps: constructing an ansatz, and then completing this ansatz to an exact solution.…”

“…Compared with [18], we do not need a correction analogous to the one denoted by v 1 in that paper, since B tt Q 1 λptq P L 2 pp0, 8q, rdrq in this setting, and we therefore have a simpler modulation equation.…”

“…As previously mentioned, our procedure in this paper is similar to that used in the previous work of the author, [18]. That work constructed infinite time blow-up solutions to the energy critical, 1-equivariant wave maps problem with S 2 target, with a symbol class of possible asymptotic behaviors of the soliton length scale, λptq.…”

“…As remarked in Appendix A of [20], and remark 4 of [21], the methods used in this classification result also apply to (1.1). Our procedure will then be similar to that of the previous work of the author, [18]. In particular, our solutions will involve a modulated soliton Q 1 λptq coupled to radiation, and our procedure to construct these solutions will be to find a precise relation between the radiation and the dynamics of λptq.…”

“…For estimating the radiation profile, we use λ 2 0 ptq λ 0 ptq " f 1 ptq t , but for the entirety of the rest of the argument, we only use the symbol-type estimates on λ 1 0 ptq λ 0 ptq (which can be satisfied by non-asymptotically constant λ 0 ). In particular, our solutions are constructed using a similar argument to the previous work of the author regarding wave maps, [18], rather than assuming apriori that λptq is asymptotically constant.…”