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

Dong，Shijie1,2; LeFloch，Philippe G.3; Lei，Zhen2,4

2022

10.1016/j.fmre.2022.06.010

Fundamental Research   Impact Factor & Quartile

2667-3258

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.

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

English

First

China Postdoctoral Science Foundation[2021M690702];

2-s2.0-85133841430

Scopus

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

Dong，Shijie,LeFloch，Philippe G.,Lei，Zhen. The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded[J]. Fundamental Research,2022.

Dong，Shijie,LeFloch，Philippe G.,&Lei，Zhen.(2022).The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded.Fundamental Research.

Dong，Shijie,et al."The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded".Fundamental Research (2022).