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| We study a model that describes the quasi-stationary evolution of a membrane formed by a lipid bilayer in a surrounding incompressible Newtonian fluid. The membrane is thereby modeled as a sharp interface that encloses a part of the liquid - the vesicle. In contrast to fluid interfaces the dynamics of such a membrane are rather driven by elasticity than by surface tension. Moreover, the model incorporates the conservation not only of the vesicle volume but also of the membrane area, which leads to an incompressibility constraint on the interface. After a short introduction of the model we prove its (local-in-time) well-posedness in an $L_p$-setting. Our approach is based on the direct mapping technique (i.e.\ a Hanzawa transformation), maximal regularity results for a suitable linearization and a fixed-point argument. |
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