A priori estimates for a relativistic liquid with free surface boundary

Abstract

We prove energy estimates for a relativistic free liquid body with sufficiently small fluid velocity in a general Lorentz spacetime. These estimates control Sobolev norms of the fluid velocity and enthalpy in the interior as well as Sobolev norms of the second fundamental form on the boundary. These estimates are generalizations of the energy estimates of Christodoulou and Lindblad [D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Commun. Pure Appl. Math. 53(12) (2000) 1536–1602] and rely on elliptic estimates which only require bounds for the second fundamental form of the time slices of the free boundary.