  This paper is motivated by the study of Lyapunov functionals for four
equations describing free surface flows in fluid dynamics: the Hele-Shaw and
Mullins-Sekerka equations together with their lubrication approximations, the
Boussinesq and thin-film equations. We identify new Lyapunov functionals,
including some which decay in a convex manner (these are called strong Lyapunov
functionals). For the Hele-Shaw equation and the Mullins-Sekerka equation, we
prove that the $L^2$-norm of the free surface elevation and the area of the
free surface are Lyapunov functionals, together with parallel results for the
thin-film and Boussinesq equations. The proofs combine exact identities for the
dissipation rates with functional inequalities. For the thin-film and
Boussinesq equations, we introduce a Sobolev inequality of independent interest
which revisits some known results and exhibits strong Lyapunov functionals. For
the Hele-Shaw and Mullins-Sekerka equations, we introduce a functional which
controls the $L^2$-norm of three-half spatial derivative. Under a mild
smallness assumption on the initial data, we show that the latter quantity is
also a Lyapunov functional for the Hele-Shaw equation, implying that the area
functional is a strong Lyapunov functional. Precise lower bounds for the
dissipation rates are established, showing that these Lyapunov functionals are
in fact entropies. Other quantities are also studied such as Lebesgue norms or
the Boltzmann's entropy.
