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Transactions of the American Mathematical Society

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A class of large global solutions for the wave-map equation


Authors: Elisabetta Chiodaroli and Joachim Krieger
Journal: Trans. Amer. Math. Soc. 369 (2017), 2747-2773
MSC (2010): Primary 35L05
DOI: https://doi.org/10.1090/tran/6805
Published electronically: June 20, 2016
MathSciNet review: 3592527
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Abstract: In this paper we consider the equation for equivariant wave maps from $ \mathbb{R}^{3+1}$ to $ \mathbb{S}^3$ and we prove global in forward time existence of certain $ C^\infty $-smooth solutions which have infinite critical Sobolev norm $ \dot {H}^{\frac {3}{2}}(\mathbb{R}^3)\times \dot {H}^{\frac {1}{2}}(\mathbb{R}^3)$. Our construction provides solutions which can moreover satisfy the additional size condition $ \Vert u(0, \cdot )\Vert _{L^\infty (\vert x\vert\geq 1)}>M$ for arbitrarily chosen $ M>0$. These solutions are also stable under suitable perturbations. Our method, strongly inspired by work of Krieger and Schlag, is based on a perturbative approach around suitably constructed approximate self-similar solutions.


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Additional Information

Elisabetta Chiodaroli
Affiliation: EPFL Lausanne, Station 8, CH-1015 Lausanne, Switzerland

Joachim Krieger
Affiliation: EPFL Lausanne, Station 8, CH-1015 Lausanne, Switzerland

DOI: https://doi.org/10.1090/tran/6805
Received by editor(s): April 22, 2015
Published electronically: June 20, 2016
Article copyright: © Copyright 2016 American Mathematical Society