The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded

Shijie Dong ¹Philippe G.LeFloch ²Zhen Lei³

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作者信息

1. Southern University of Science and Technology,SUSTech International Center for Mathematics,and Department of Mathematics,Shenzhen 518055,China;Fudan University,School of Mathematical Sciences,220 Handan Road,Shanghai 200433,China
2. Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique,Sorbonne Université,4 Place Jussieu,Paris 75252,France
3. Fudan University,School of Mathematical Sciences,220 Handan Road,Shanghai 200433,China;Shanghai Center for Mathematical Sciences,Shanghai 200433,China
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Abstract

Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions,provided that the initial data are compactly supported and sufficiently small in Sobolev norm.In this work,Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions.A natural question then arises whether the time-growth is a true phenomenon,despite the possible conservation of basic energy.In the present paper,we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time.

Key words

Quasilinear wave equation/Global-in-time solution/Uniform energy bounds/Quadratic null nonlinearity/Hyperboloidal foliation method/Vector field method

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基金项目

中国博士后科学基金(2021M690702)

国家自然科学基金(11725102)

Sino-German Center(M-0548)

国家重点研发计划(2018AAA0100303)

National Support Program for Young Top-Notch Talents()

上海市科技计划(21JC1400600)

上海市科技计划(19JC1420101)

出版年

2024

自然科学基础研究(英文)

CSTPCD

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参考文献量44

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