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

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

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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.

Quasilinear wave equationGlobal-in-time solutionUniform energy boundsQuadratic null nonlinearityHyperboloidal foliation methodVector field method

Shijie Dong、Philippe G.LeFloch、Zhen Lei

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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

Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique,Sorbonne Université,4 Place Jussieu,Paris 75252,France

Shanghai Center for Mathematical Sciences,Shanghai 200433,China

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中国博士后科学基金国家自然科学基金Sino-German Center国家重点研发计划National Support Program for Young Top-Notch Talents上海市科技计划上海市科技计划

2021M69070211725102M-05482018AAA010030321JC140060019JC1420101

2024

10.1016/j.fmre.2022.06.010

自然科学基础研究(英文)

自然科学基础研究(英文)

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年,卷(期)：2024.(2)
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