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Construction of infinite time bubble tower solutions to critical wave maps equation

Seunghwan Hwang, Kihyun Kim

Abstract

We construct infinite time bubble tower solutions to the critical wave maps equation taking values in the two-sphere. More precisely, for any integers $k\geq3$ and $J\geq1$, we construct a solution that is global in one time direction, has $k$-corotational symmetry, and asymptotically decomposes into $J$-many concentric bubbles of alternating sign with asymptotically vanishing radiation. The scales of each bubble are of order $t^{-α_{j}}$ with $α_{j}=(\frac{k}{k-2})^{j-1}-1$. This shows the existence of multi-bubble solutions with an arbitrary number of bubbles in soliton resolution, provided that $k\geq3$, global existence in one time direction, and alternating signs are considered. Our proof is based on modulation analysis with the method of backward construction. The key new ingredient is a Morawetz-type functional that provides suitable monotonicity estimates for solutions around multi-bubble configurations.

Construction of infinite time bubble tower solutions to critical wave maps equation

Abstract

We construct infinite time bubble tower solutions to the critical wave maps equation taking values in the two-sphere. More precisely, for any integers and , we construct a solution that is global in one time direction, has -corotational symmetry, and asymptotically decomposes into -many concentric bubbles of alternating sign with asymptotically vanishing radiation. The scales of each bubble are of order with . This shows the existence of multi-bubble solutions with an arbitrary number of bubbles in soliton resolution, provided that , global existence in one time direction, and alternating signs are considered. Our proof is based on modulation analysis with the method of backward construction. The key new ingredient is a Morawetz-type functional that provides suitable monotonicity estimates for solutions around multi-bubble configurations.
Paper Structure (4 sections, 1 theorem, 14 equations)

This paper contains 4 sections, 1 theorem, 14 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Notation
  4. Basic notation

Key Result

Theorem 1.1

Let $k\geq3$ and $J\in\mathbb{N}$. Then, there exists a solution $\bm{u}(t)=(u(t),\partial_{t}u(t))$ to ( ) with data in $\mathcal{E}_{0,J\mathrm{mod}2}\cap\dot{\mathcal{H}}^{2}$, defined for all large positive times $t$, such that where and $\gamma_{j}$ are the universal constants defined in ( ). By applying the time-reversal symmetry $u(t,r)\mapsto u(-t,r)$, one also obtains backward-in-time b

Theorems & Definitions (5)

  • Theorem 1.1: Construction of bubble towers
  • Remark 1.2
  • Remark 1.3: On strategy, difficulty, and novelty
  • Remark 1.4: Lower corotational index
  • Remark 1.5: Related works