Local Well-posedness of Free Boundary Problems for the Euler--Monge--Amp\`ere Equations

Abstract

We study the free-boundary Euler--Monge--Amp\`ere system as a conservative relaxation of ideal incompressible fluid motion. The force is determined on the actual fluid domain by a Monge--Amp\`ere equation coupled to a nonlinear boundary condition. For each fixed relaxation parameter, we establish local well-posedness for uniformly convex initial domains of class $C^{2,\alpha}$, densities in $C^{1,\alpha}$, and arbitrary initial velocities in $C^{2,\alpha}$. The solution retains these spatial regularities, and the force potential belongs to $C^{3,\alpha}$ up to the moving boundary. Within uniformly bounded data families, the solution depends continuously on the initial data at every lower H\"older exponent.