Volume 26, 2020
|Number of page(s)||21|
|Published online||17 December 2020|
Remarks on nonlinear elastic waves with null conditions
Department of Mathematics and Institute for Nonlinear Sciences, Donghua University,
201620, P.R. China.
2 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, P.R. China.
* Corresponding author: firstname.lastname@example.org
Accepted: 29 June 2020
For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.
Mathematics Subject Classification: 35L52 / 35Q74
Key words: Nonlinear elastic waves / Helmholtz projection / null conditions / global existence
© EDP Sciences, SMAI 2020
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