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Modified Wave Operators for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

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Abstract

We prove the existence of the modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition. This is accomplished in three steps. First, we derive a new reduced asymptotic system for the quasilinear wave equation by modifying Hörmander’s method. Next, we construct an approximate solution, by solving our new reduced system given some scattering data at infinite time. Finally, we prove that the quasilinear wave equation has a global solution which agrees with the approximate solution at infinite time.

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Acknowledgements

The author would like to thank his advisor, Daniel Tataru, for suggesting this problem and for many helpful discussions. The author would also like to thank the anonymous reviewers for their valuable comments and suggestions on this paper. This research was partially supported by a James H. Simons Fellowship and by the NSF grant DMS-1800294.

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  1. Department of Mathematics, University of California at Berkeley, Berkeley, CA, USA

    Dongxiao Yu

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  1. Dongxiao Yu
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Correspondence to Dongxiao Yu.

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Communicated by P. Chrusciel

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Yu, D. Modified Wave Operators for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition. Commun. Math. Phys. 382, 1961–2013 (2021). https://doi.org/10.1007/s00220-021-03989-0

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