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A Local and Global Well-Posedness Results for the General Stress-Assisted Diffusion Systems

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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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Correspondence to Marta Lewicka.

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:

$$DW_{1}(\phi, F) : (\tilde{\phi},\tilde{F}) = \phi\tilde{\phi}+ \bigl\langle DW_{0}\bigl(FB(\phi)\bigr) : \tilde{\phi}F B'( \phi)\bigr\rangle + \bigl\langle DW_{0}\bigl(FB(\phi)\bigr) : \tilde{F} B(\phi)\bigr\rangle , $$

which implies the second condition in (1.7). Further:

$$\begin{aligned} D^{2}W_{1}(0,\mbox{Id}) : (\tilde{\phi},\tilde{F})^{\otimes2} =& |\tilde{\phi}|^{2} + D^{2}W_{0}( \mbox{Id}) : \bigl(\tilde{\phi}B'(0)\bigr)^{\otimes2} \\ &{}+ 2 D^{2}W_{0}(\mbox{Id}) : \bigl(\tilde{B}'(0)\otimes\tilde{F}\bigr) + D^{2}W_{0}( \mbox{Id}) : {\tilde{F}}^{\otimes2} \\ =& |\tilde{\phi}|^{2} + D^{2}W_{0}(\mbox{Id}) : \bigl(\tilde{F} +\tilde{\phi}B'(0)\bigr)^{\otimes2} \\ \geq&|\tilde{\phi}|^{2} + c\bigl\vert \mbox{sym } \tilde{F} + \tilde{\phi}B'(0)\bigr\vert ^{2}, \end{aligned}$$

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:

$$ |\tilde{\phi}|^{2} + \bigl|\mbox{sym } \tilde{F} + \tilde{\phi}B'(0)|^{2} \geq c \bigl(|\tilde{\phi}\bigr|^{2} + | \mbox{sym } \tilde{F}|^{2} \bigr), $$
(6.1)

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:

$$ \bigl(1-c + \bigl|B'(0)\bigr|^{2}\bigr) + (1-c) \biggl|\mbox{sym} \biggl(\frac{1}{\tilde{\phi}}\tilde{F}\biggr)\biggr|^{2} + 2\biggl\langle \mbox{sym} \biggl(\frac{1}{\tilde{\phi}}\tilde{F}\biggr) : B'(0)\biggr\rangle \geq0, $$

which becomes:

$$ 1-c -\frac{c}{1-c} \bigl|B'(0)\bigr|^{2} + \biggl|\sqrt{1-c} \mbox{ sym} \biggl(\frac{1}{\tilde{\phi}}\tilde{F} \biggr) + \frac{1}{\sqrt {1-c}} B'(0)\biggr|^{2} \geq0. $$

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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