On the Division Problem for the Wave Maps Equation
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We consider Wave Maps into the sphere and give a new proof of small data global well-posedness and scattering in the critical Besov space, in any space dimension \(n \geqslant 2\). We use an adapted version of the atomic space \(U^2\) as the single building block for the iteration space. Our approach to the so-called division problem is modular as it systematically uses two ingredients: atomic bilinear (adjoint) Fourier restriction estimates and an algebra property of the iteration space, both of which can be adapted to other phase functions.
KeywordsWave maps Division problem Bilinear Fourier restriction Atomic spaces
Mathematics Subject Classification35L15 35L52
The authors thank Kenji Nakanishi and Daniel Tataru for sharing their part in the story of the division problem with us. Also, the authors thank Daniel Tataru for providing a preliminary version of . In addition, we thank the referees for their helpful comments, in particular for the alternative strategy for the proof of Theorem 4.6 summarised in Remark 4.7. Financial support by the DFG through the CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications” is acknowledged.
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