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Almost global existence for 2-D incompressible isotropic elastodynamics


Authors: Zhen Lei, Thomas C. Sideris and Yi Zhou
Journal: Trans. Amer. Math. Soc. 367 (2015), 8175-8197
MSC (2010): Primary 35L60, 74B20
DOI: https://doi.org/10.1090/tran/6294
Published electronically: April 9, 2015
MathSciNet review: 3391913
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Abstract: We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval $ [0,{T}/{\epsilon }]$ for initial data of the form $ \epsilon U_0$, where $ T$ depends only on some Sobolev norm of $ U_0$. We show that for such data there exists a unique solution on a time interval $ [0, \exp {T}/{\epsilon }]$, provided that $ \epsilon $ is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies a null condition. This is essential for time decay estimates. The pressure, which arises as a Lagrange multiplier to enforce the incompressibility constraint, is estimated in a novel way as a nonlocal nonlinear term with null structure. The proof employs the generalized energy method of Klainerman, enhanced by weighted $ L^2$ estimates and the ghost weight introduced by Alinhac.


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Additional Information

Zhen Lei
Affiliation: School of Mathematical Sciences, LMNS and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
Email: leizhn@gmail.com, zlei@fudan.edu.cn

Thomas C. Sideris
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: sideris@math.ucsb.edu

Yi Zhou
Affiliation: School of Mathematical Sciences, LMNS and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
Email: yizhou@fudan.edu.cn

DOI: https://doi.org/10.1090/tran/6294
Keywords: Incompressible elastodynamics, almost global existence, generalized energy method, null condition, ghost weight
Received by editor(s): April 11, 2013
Received by editor(s) in revised form: September 10, 2013
Published electronically: April 9, 2015
Additional Notes: The authors would like to thank Professor Fang-hua Lin of the Courant Institute for some helpful discussions
The first author was supported in part by NSFC (grants No. 11171072, 11421061 and 11222107), Shanghai Talent Development Fund, and SGST 09DZ2272900
The second author was partially supported by the National Science Foundation
The first and third authors were supported by the Foundation for Innovative Research Groups of NSFC (grant No. 11121101) and SGST 09DZ2272900.
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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