Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics

Abstract

This paper is motivated by the study of Lyapunov functionals for the Hele-Shaw and Mullins-Sekerka equations describing free surface flows in fluid dynamics. We prove that the $L^2$-norm of the free surface elevation and the area of the free surface are Lyapunov functionals. The proofs combine exact identities for the dissipation rates with functional inequalities. 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.