Infinite Time Blow-Up Solutions to the Energy Critical Wave Maps Equation

Abstract

We consider the wave maps problem with domain R 2 + 1 \mathbb {R}^{2+1} and target S 2 \mathbb {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 \mathbb {R}^{2} to S 2 \mathbb {S}^{2} , with polar angle equal to Q 1 ( r ) = 2 arctan ⁡ ( r ) Q_{1}(r) = 2 \arctan (r) . By applying the scaling symmetry of the equation, Q λ ( r ) = Q 1 ( r λ ) Q_{\lambda }(r) = Q_{1}(r \lambda ) is also a harmonic map, and the family of all such Q λ Q_{\lambda } 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 Q_{\lambda } family. More precisely, for b > 0 b>0 , and for all λ 0 , 0 , b ∈ C ∞ ( [ 100 , ∞ ) ) \lambda _{0,0,b} \in C^{\infty }([100,\infty )) satisfying, for some C l , C m , k > 0 C_{l}, C_{m,k}>0 , C l log b ⁡ ( t ) ≤ λ 0 , 0 , b ( t ) ≤