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Session 24. Nonlinear PDEs with applications in materials science and biology
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A swelling model: existence, basic properties |
Piotr Bogusław Mucha, University of Warsaw, Poland
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The talk is based on a joint work with Marta Lewicka (University of Pittsburgh)
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We study the evolutionary model in the description of
morphogenesis of an elastic body \(\Omega\) exhibiting residual
strain at free equilibria. The model consists of the balance of
linear momentum and the diffusion law:
\[
u_{tt} - {\rm div $\,$ }(\frac{1}{f(\phi)} DW(\frac{1}{f(\phi)}\nabla
u)) = 0
\]\[
\phi_t = \Delta (-\frac{f'(\phi)}{f(\phi)^2}\langle
DW(\frac{1}{f(\phi)} \nabla u) : \nabla u\rangle + \frac{\partial
F}{\partial \phi})
\]
written in terms of the deformation u of \(\Omega\) (\(u:\Omega \to
\mathbb{R}^N\)), and the growth (swelling) agent density \(\phi\).
The basic structure assumptions are:
\[ W(X)\geq c|X-Id|^2, \quad |DW(X)|\leq c|X-Id|, \quad D^2W(X)
M\otimes M \geq c|Sym \, M|^2
\]
for \(X\) close to \(Id\). Functions \(f\) and \(F\) are sufficiently smooth
and obey natural conditions. The basic information is given by the
energy inequality
\[ \frac{d}{dt} \int \left( \frac 12 u_t^2 + W(\frac{1}{f(\phi)}
\nabla u) + F(\phi)\right) dx \leq 0. \] We investigate the issue
of well posedness of the system. The key difficulty is located in
the hyperbolic character of the system and very high
nonlinearities of the second equation. The straightforward
approaches do not lead to the a priori estimate. To find the
required information about the solutions we have to analyze the
structure of the nonlinearities.
We plan to point several interesting properties and possible
generalizations of the studied system, too.
The talk will base on results of the joint work with Marta Lewicka [1].
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References- M. Lewicka, P.B. Mucha, The evolutionary swelling
model, in preparation.
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