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

About this Title

Mohandas Pillai

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 284, Number 1407
ISBNs: 978-1-4704-5993-2 (print); 978-1-4704-7445-4 (online)
DOI: https://doi.org/10.1090/memo/1407
Published electronically: March 21, 2023

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Overview of the proof
  • 3. Construction of the ansatz
  • 4. Solving the final equation
  • 5. The energy of the solution, and its decomposition as in Theorem
  • A. Proof of Theorem

Abstract

We consider the wave maps problem with domain $\mathbb {R}^{2+1}$ and target $\mathbb {S}^{2}$ in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from $\mathbb {R}^{2}$ to $\mathbb {S}^{2}$, with polar angle equal to $Q_{1}(r) = 2 \arctan (r)$. By applying the scaling symmetry of the equation, $Q_{\lambda }(r) = Q_{1}(r \lambda )$ is also a harmonic map, and the family of all such $Q_{\lambda }$ 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 $Q_{\lambda }$ family. More precisely, for $b>0$, and for all $\lambda _{0,0,b} \in C^{\infty }([100,\infty ))$ satisfying, for some $C_{l}, C_{m,k}>0$, \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 $u_{b}$ denotes the polar angle of the wave map into $\mathbb {S}^{2}$, we have \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 \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*} \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 \begin{equation*} \lambda _{b}(t) = \lambda _{0,0,b}(t) + O\left (\frac {1}{\log ^{b}(t) \sqrt {\log (\log (t))}}\right ) \end{equation*}

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