Well-posedness for the Linearized Motion of the Free Surface of a Liquid

where ∂i = ∂/∂x i and D = ∪ 0≤t≤T {t}×Dt, Dt ⊂ R. Here V k = δvi = vk and we use the summation convention over repeated upper and lower indices. The velocity vector field of the fluid is V , p is the pressure and Dt is the domain the fluid occupies at time t. We also require boundary conditions on the free boundary ∂D; p = 0, on ∂D, (1.3) (∂t + V ∂k)|∂D ∈ T (∂D), (1.4) Condition (1.3) says that the pressure p vanishes outside the domain and condition (1.4) says that the boundary moves with the velocity V of the fluid particles at the boundary. Given a simply connected bounded domain D0 ⊂ R and initial data v0, satisfying the constraint (1.2), we want to find a set D ⊂ [0, T ]×Rn and a vector field v solving (1.1)-(1.4) with initial conditions (1.5) {x; (0, x) ∈ D} = D0, and v = v0, on {0} × D0 Let Dt={x; (t, x) ∈ D} and letN be the exterior unit normal to the free surface ∂Dt. Christodoulou[C2] conjectured that the initial value problem (1.1)-(1.5), is well posed in Sobolev spaces if (1.6) ∇N p ≤ −c0 < 0, on ∂D, where ∇N = N ∂xi .