= Solution
Suppress hats for this calculation. Transverse <pressure> uniformity and zero surface <pressure> give $p=0$ throughout the film. Thus $u_{zz}=-S$. Integrating twice, using $u_z(h)=0$ and $u(0)=-1$, gives
$$
\boxed{u(z)=-1+S(hz-z^2/2).}
$$
The flux is $q=\int_0^h u\,dz=-h+Sh^3/3$. Integrate continuity across the layer and apply the wall and kinematic conditions to obtain $h_t+q_x=0$. Restoring hats yields
$$
\boxed{\widehat h_{\widehat t}+(S\widehat h^2-1)\widehat h_{\widehat x}=0.}
$$
The cubic flux contains downward <gravity>-driven transport and upward wall entrainment.
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