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Local Well-posedness of Free Boundary Problems for the Euler--Monge--Ampère Equations
Abstract
We study the free-boundary Euler--Monge--Ampère system as a conservative relaxation of ideal incompressible fluid motion. The force is determined on the actual fluid domain by a Monge--Ampère equation coupled to a nonlinear boundary condition. For each fixed relaxation parameter, we establish local well-posedness for uniformly convex initial domains of class C2,α, densities in C1,α, and arbitrary initial velocities in C2,α. The solution retains these spatial regularities, and the force potential belongs to C3,α up to the moving boundary. Within uniformly bounded data families, the solution depends continuously on the initial data at every lower Hölder exponent.
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BibTeX
@misc{liu2026local,
title = {{Local Well-posedness of Free Boundary Problems for the Euler--Monge--Ampère Equations}},
author = {Sicheng Liu and Tao Luo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16717},
eprint = {2609.16717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16717},
}
RIS
TY - UNPB
TI - Local Well-posedness of Free Boundary Problems for the Euler--Monge--Ampère Equations
AU - Sicheng Liu
AU - Tao Luo
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16717
UR - https://arxiv.org/abs/2609.16717
AB - We study the free-boundary Euler--Monge--Ampère system as a conservative relaxation of ideal incompressible fluid motion. The force is determined on the actual fluid domain by a Monge--Ampère 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,α, densities in C^1,α, and arbitrary initial velocities in C^2,α. The solution retains these spatial regularities, and the force potential belongs to C^3,α up to the moving boundary. Within uniformly bounded data families, the solution depends continuously on the initial data at every lower Hölder exponent.
ER -