Remarks on nonlinear elastic waves in the radial symmetry in 2-D

Dongbing  Zha

doi:10.3934/dcds.2016.36.4051

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2016, Volume 36, Issue 7: 4051-4062. Doi: 10.3934/dcds.2016.36.4051

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Remarks on nonlinear elastic waves in the radial symmetry in 2-D

Dongbing Zha^1,

1.
School of Mathematical Sciences, Fudan University, Shanghai 200433

Received: May 31, 2015

Revised: October 31, 2015

Published: May 2017

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Abstract

In this paper, we first give the explicit variational structure of the nonlinear elastic waves for isotropic, homogeneous, hyperelastic materials in 2-D. Based on this variational structure, we suggest a null condition which is a kind of structural condition on the nonlinearity in order to stop the formation of finite time singularities of local smooth solutions. In the radial symmetric case, inspired by Alinhac's work on 2-D quasilinear wave equations [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001) 597--618], we show that such null condition can ensure the global existence of smooth solutions with small initial data.

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Mathematics Subject Classification: Primary: 35L52; Secondary: 35Q74.

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