June 2023 Global existence for null-form wave equations with data in a Sobolev space of lower regularity and weight
Kunio HIDANO, Kazuyoshi YOKOYAMA
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Hokkaido Math. J. 52(2): 197-251 (June 2023). DOI: 10.14492/hokmj/2021-523


Assuming initial data have small weighted $H^4\times H^3$ norm, we prove global existence of solutions to the Cauchy problem for systems of quasi-linear wave equations in three space dimensions satisfying the null condition of Klainerman. Compared with the work of Christodoulou, our result assumes smallness of data with respect to $H^4\times H^3$ norm having a lower weight. Our proof uses the space-time $L^2$ estimate due to Alinhac for some special derivatives of solutions to variable-coefficient wave equations. It also uses the conformal energy estimate for inhomogeneous wave equation $\Box u=F$. A new observation made in this paper is that, in comparison with the proofs of Klainerman and Hörmander, we can limit the number of occurrences of the generators of hyperbolic rotations or dilations in the bootstrap argument. This limitation allows us to obtain global solutions for radially symmetric data, when a certain norm with considerably lower weight is small enough.


The authors are very grateful to the referee for careful reading of the manuscript and helpful comments. The first author was supported by JSPS KAKENHI Grant Number JP18K03365.


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Kunio HIDANO. Kazuyoshi YOKOYAMA. "Global existence for null-form wave equations with data in a Sobolev space of lower regularity and weight." Hokkaido Math. J. 52 (2) 197 - 251, June 2023. https://doi.org/10.14492/hokmj/2021-523


Received: 5 April 2021; Revised: 23 August 2021; Published: June 2023
First available in Project Euclid: 9 July 2023

Digital Object Identifier: 10.14492/hokmj/2021-523

Primary: 35L15 , 35L52
Secondary: 35L72

Keywords: global existence , null condition , Quasi-linear wave equations

Rights: Copyright c 2023 Hokkaido University, Department of Mathematics


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Vol.52 • No. 2 • June 2023
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