On an equation describing the spreading of surfactants on thin films. (English) Zbl 0891.76029
Consider a thin fluid film whose bottom is rigid and the top free surface contains a monolayer of a surfactant. The hyperbolic-parabolic equation describing the mass conservation for the fluid and the surfactant, derived by M. S. Borgas and J. B. Grotberg [J. Fluid Mech. 193, 151-170 (1988; Zbl 0643.76032)], forms the basis of the paper. Existence of solutions locally in time for given initial data is investigated in detail. It is further shown that the smooth solutions generally do not exist globally in time, because the hyperbolic part of the equation leads to development of shocks. Finally, the author etablishes the stability of solution in a bounded domain with smooth boundary subject to the homogeneous Neumann boundary condition.
Reviewer: R.C.Gupta (Singapore)
Keywords:
mixed hyperbolic-parabolic equation; free surface; mass conservation; smooth solutions; stability of solution; homogeneous Neumann boundary condition; global existence in time; local existence in timeCitations:
Zbl 0643.76032References:
[1] | Borgas, M. S.; Grotberg, J. B., Monolayer flow on a thin film, J. Fluid Mech., 193, 151-170 (1988) · Zbl 0643.76032 |
[2] | Solonnikov, V. A., Estimates in \(L_p\) of solutions of elliptic and parabolic systems, (Proc. Steklov Inst. Math., 102 (1987)), 157-185 |
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