2019/05/01 by Pillai, Mohandas
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.00167
We consider the wave maps problem with domain ℝ2+1 and target \mathbbS2 in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from ℝ2 to \mathbbS2, with polar angle equal to Q1(r) = 2 \arctan(r). By applying the scaling symmetry of the equation, Qλ(r) = Q1(r λ) is also a harmonic map, and the family of all such Qλ 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 Qλ family. More precisely, for b>0, and for all λ0,0,b ∈ C∞([100,∞)) satisfying, for some Cl, Cm,k>0, \fracCllogb(t) ≤ λ0,0,b(t) ≤ \fracCmlogb(t), |λ0,0,b(k)(t)| ≤ \fracCm,ktk logb+1(t) , k≥ 1 t ≥ 100 there exists a wave map with the following properties. If ub denotes the polar angle of the wave map into \mathbbS2, we have ub(t,r) = Q_\frac1λb(t)(r) + v2(t,r) + ve(t,r), t ≥ T0 where -∂ttv2+∂rrv2+(1)/(r)∂rv2-\fracv2r2=0 ||∂t(Q_\frac1λb(t)+ve)||L2(r dr)2+||\fracver||L2(r dr)2 + ||∂rve||L2(r dr)2 ≤ \fracCt2 log2b(t), t ≥ T0 and λb(t) = λ0,0,b(t) + O(\frac1logb(t) √(log(log(t))))