The basic model for incompressible two-phase flows with phase transitions where the interface is nearly flat in the case of non-equal densities is considered. The local well-posedness of the model in $L_p$-setting was proved in Prüss and Shimizu . In this paper, we prove local well-posedness of the model $L_p$ in time $L_q$ in space settting.
Senjoi Shimizu. Shintaro Yagi. "On local $L_p$-$L_q$ well-posedness of incompressible two-phase flows with phase transitions: the case of non equal densities." Differential Integral Equations 28 (1/2) 29 - 58, January/February 2015. https://doi.org/10.57262/die/1418310420