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| We consider a sharp-interface problem for the flow of two incompressible Newtonian fluids in a bounded domain. The motion of the moving interface is governed by a stress balance which includes both surface tension and surface viscosity according to the Boussinesq-Scriven law. Here the stress balance on the interface is of second order with respect to the tangential velocity but only of first order in the normal velocity.
We prove that the problem is locally well-posed in an $L_p$-setting for a restricted class of initial data. By means of maximal regularity for linear problems, we obtain well-posedness by simply applying Banach's fixed-point theorem. We use a decomposition of the velocity and the stress balance into tangential and normal components and a suitable transformation to a fixed interface which respects this structure. Suitable function spaces are constructed by using the theory of parabolic mixed-order systems. |
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