Abstract
We prove the local and global in time existence of the classical solutions to two general classes of the stress-assisted diffusion systems. Our results are applicable in the context of the non-Euclidean elasticity and liquid crystal elastomers.
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Acknowledgements
M.L. was partially supported by the NSF DMS grants 0846996 and 1406730. P.B.M. was partly supported by the NCN grant No. 2011/01/B/ST1/01197. The authors are grateful to the editor for many helpful reference suggestions. The authors declare that they have no conflict of interest.
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Appendix: Proof of Proposition 1.3
Appendix: Proof of Proposition 1.3
The first condition in (1.7) is obvious. A direct calculation shows that:
which implies the second condition in (1.7). Further:
where we concluded from (1.9) that \(DW_{0}(\mbox{Id}) = 0\) and that \(D^{2}W_{0}(\mbox{Id})\) is positive definite on symmetric matrices. We also note that: \(D^{2}W_{2}(0,\mbox{Id}) = D^{2}W_{1}(0,\mbox{Id})\). To conclude the proof, it is hence enough to show that:
for all \(\tilde{\phi}\) and \(\tilde{F}\). Expanding the square in the left hand side, dividing by \(|\tilde{\phi}|\) and collecting terms, this is equivalent to:
which becomes:
The above inequality follows from: \(1-c - \frac{c}{1-c} |B'(0)|^{2} >0\), which is true whenever \(c>0\) is sufficiently small.
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Lewicka, M., Mucha, P.B. A Local and Global Well-Posedness Results for the General Stress-Assisted Diffusion Systems. J Elast 123, 19–41 (2016). https://doi.org/10.1007/s10659-015-9545-2
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DOI: https://doi.org/10.1007/s10659-015-9545-2