On the justification of the quasistationary approximation in the problem of motion of a viscous capillary drop (Q1971101)

The author studies the following free boundary problem, which describes the motion of an isolated liquid drop: \[ \begin{aligned} \varepsilon(v_t+ (v\cdot\nabla) u)- \Delta v+\nabla p & =0,\qquad x\in \Omega_t,\;t>0,\tag{1}\\ \text{div }v & =0,\qquad x\in \Omega_t,\;t>0,\tag{2}\\ v\cdot n=V_n,\quad T(v,p)n- Hn &=0,\qquad x\in\Gamma_t,\;t>0,\tag{3}\\ v(x, 0) &= v_0(x),\;x\in\Omega_0.\tag{4}\end{aligned} \] Here \(\Omega_t\) is the domain occupied by the drop at time \(t\) with boundary \(\Gamma_t\), \(v\) denotes the velocity, \(p\) the pressure of the liquid, and \(T(v,p)\) is the stress tensor. Furthermore, \(V_n\) and \(H\) are normal velocity and mean curvature of \(\Gamma_t\) and \(\varepsilon\) denotes the Reynolds number, which is supposed to be small. A quasistationary approximation \((w,q)\) of \((v,p)\) is obtained by setting \(\varepsilon= 0\) in (1) and considering the resulting system (1)--(3). The main result of the paper states that the differences \(v-w\) and \(p-q\) (expressed in Lagrangean coordinates) are of order \(O(\varepsilon)\) on intervals \((t_0,T]\) for all \(t_0> 0\), where \([0,T]\) is the existence interval for \((w,q)\).