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3 Piechór K.

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Reaction-diffusion equation modelling calcium waves with fast buffering in visco-elastic environment

Piechór K.

Archives of Mechanics

2012

Vol. 64, nr 5

477-477

The model we consider treats a cell or a group of cells as a viscoelastic medium whose stress tensor has a term - the traction- representing the stresses generated in the medium by the actomyosin molecules. We consider three kinds of domains (“shapes” of cells): the thin circular cylinder mimicking a long cell, the thin slab being a cari-cature of a tissue, and the unbounded space. We assume that the viscous effects are much weaker than the elastic ones and consider two extreme cases: either the body force is negligible or it is strong. This leads to three pairs, one pair for each domain, of approximations for the dilatation. We interpolate between the approximated ex-pressions forming one pair and as the result we obtain a single calcium conservation equation and a system of buffer equations. Using the rapid buffering approximation we reduce the problem to a single reaction-diffusion equation. We study the travelling wave solutions to these equations. We show that not only the high affinity buffers but also the mechanical effects alone can prevent the formation and propagation of the waves if the supply of calcium is not sufficiently substantial.

Non-local Korteweg stresses from kinetic theory point of view

Piechór K.

Archives of Mechanics

2008

Vol. 60, nr 1

23-58

The aim of the paper is to elaborate a kinetic theory having the non-local Korteweg equations for non-isothermal liquid-vapour systems as the hydrodynamic limit. This is the topic of the second part of the paper. The first part of our paper is devoted to presentation of the Korteweg equations. We write the non-local Korteweg equations in a conservative form and we discuss the relations between them and the local ones.

The dependence of dynamic phase transitions on parameters

Piechór K.

Archives of Mechanics

1998

Vol. 50, nr 5

811-827

We consider phase changes described by a second order ordinary differential equation. The equation depends parametrically on the states of rest and the speed of the wave. We prove that, under some additional conditions, the solution is differentiable with respect to any of these parameters. As an application of the general theory we discuss the case when the data are close to the Maxwell line and obtain results generalising those of the previous authors.