The Problem of Symmetric Equilibrium in Compressible Barotropic Fluids with a Free Surface for a General Body Force

摘要

The problem under study is to determine a pair {p, Vs} that satisfies the nonlinear integro-differential equation [p(ρ)]'=ρh[m] with m(v) = m[ρ](v) :=∫_(v_0)~vρ(v)dv (1) in Ω := (v_0, v_S), where ρ(v)≥0 is the density of the fluid, v_S with v_0 < V_s < ∞ is the normalized volume, and the pressure on the unknown boundary and total mass of the fluid are given: p(ρ)|_(v=v_s) = p_Γ>0, ∫_Ω ρ dv = M > 0. (2)