Remarks on Nonlinear Elastic Waves in the Radial Symmetry in 2-D

2015/12/04 by Dongbing Zha, Zha, Dongbing · 1 citation
Computer Science · Engineering · Mathematics · #35L52 #35Q74 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #math.AP #msc:35L52 #msc:35Q74

paper · pdf · doi:10.48550/arxiv.1512.01373

openalex publication_date 2015/12/04 · arxiv created 2015/12/24 · arxiv updated 2015/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this manuscript 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 [3] on 2-D quasilinear wave equations, we show that such null condition can ensure the globalexistence of smooth solutions with small initial data.

Citations

• On almost global existence for nonrelativistic wave equations in 3D
• Almost global existence for 2-D incompressible isotropic elastodynamics
• The null condition for quasilinear wave equations in two space dimensions I
• Global Well-posedness of Incompressible Elastodynamics in Two Dimensions

Cited by

• A Morawetz type energy estimate for wave equation in ℝ² and application to elastic waves

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