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 ) ≤ C m log b ⁡ ( t ) , | λ 0 , 0 , b ( k ) ( t ) | ≤ C m , k t k log b + 1 ⁡ ( t ) , k ≥ 1 t ≥ 100 \begin{equation*} \frac {C_{l}}{\log ^{b}(t)} \leq \lambda _{0,0,b}(t) \leq \frac {C_{m}}{\log ^{b}(t)}, \quad |\lambda _{0,0,b}^{(k)}(t)| \leq \frac {C_{m,k}}{t^{k} \log ^{b+1}(t) }, k\geq 1 \quad t \geq 100 \end{equation*} there exists a wave map with the following properties. If u b u_{b} denotes the polar angle of the wave map into S 2 \mathbb {S}^{2} , we have u b ( t , r ) = Q 1 λ b ( t ) ( r ) + v 2 ( t , r ) + v e ( t , r ) , t ≥ T 0 \begin{equation*} u_{b}(t,r) = Q_{\frac {1}{\lambda _{b}(t)}}(r) + v_{2}(t,r) + v_{e}(t,r), \quad t \geq T_{0} \end{equation*} where − ∂ t t v 2 + ∂ r r v 2 + 1 r ∂ r v 2 − v 2 r 2 = 0 \begin{equation*} -\partial _{tt}v_{2}+\partial _{rr}v_{2}+\frac {1}{r}\partial _{r}v_{2}-\frac {v_{2}}{r^{2}}=0 \end{equation*} | | ∂ t ( Q 1 λ b ( t ) + v e ) | | L 2 ( r d r ) 2 + | | v e r | | L 2 ( r d r ) 2 + | | ∂ r v e | | L 2 ( r d r ) 2 ≤ C t 2 log 2 b ⁡ ( t ) , t ≥ T 0 \begin{equation*} ||\partial _{t}(Q_{\frac {1}{\lambda _{b}(t)}}+v_{e})||_{L^{2}(r dr)}^{2}+||\frac {v_{e}}{r}||_{L^{2}(r dr)}^{2} + ||\partial _{r}v_{e}||_{L^{2}(r dr)}^{2} \leq \frac {C}{t^{2} \log ^{2b}(t)}, \quad t \geq T_{0} \end{equation*} and λ b ( t ) = λ 0 , 0 , b ( t ) + O ( 1 log b ⁡ ( t ) log ⁡ ( log ⁡ ( t ) ) ) \begin{equation*} \lambda _{b}(t) = \lambda _{0,0,b}(t) + O\left (\frac {1}{\log ^{b}(t) \sqrt {\log (\log (t))}}\right ) \end{equation*}