Hadamard Local Well-Posedness for Compressible Liquids with a Free Surface
Abstract
We prove local well-posedness in the Hadamard sense for the three-dimensional compressible Euler equations governing a liquid with a free surface and no surface tension. Under the Taylor sign condition and the boundary compatibility conditions, we establish existence, uniqueness, and strong continuous dependence in a hybrid Sobolev class for every s>3. The velocity and interface have Hs regularity, while the enthalpy and velocity divergence belong to Hs+1/2 and Hs−1/2, respectively. This separation reflects the coupling of a nondegenerate acoustic equation with the free-surface and vorticity dynamics. The proof uses regularization operators that smooth the moving domain and extend the nonlinear compatibility hierarchy, together with L2 and partial H2 distance estimates for solutions on different domains. Regular solutions are constructed by a regularized forward Euler scheme. High-order energy estimates and frequency envelopes then yield strong convergence of smooth approximations and continuity of the data-to-solution map. Integer Sobolev indices are treated using a Lions--Magenes endpoint compatibility condition.
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BibTeX
@misc{luo2026hadamard,
title = {{Hadamard Local Well-Posedness for Compressible Liquids with a Free Surface}},
author = {Tao Luo and Ruixi Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40163},
eprint = {2609.40163},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40163},
}
RIS
TY - UNPB
TI - Hadamard Local Well-Posedness for Compressible Liquids with a Free Surface
AU - Tao Luo
AU - Ruixi Zhang
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40163
UR - https://arxiv.org/abs/2609.40163
AB - We prove local well-posedness in the Hadamard sense for the three-dimensional compressible Euler equations governing a liquid with a free surface and no surface tension. Under the Taylor sign condition and the boundary compatibility conditions, we establish existence, uniqueness, and strong continuous dependence in a hybrid Sobolev class for every s>3. The velocity and interface have H^s regularity, while the enthalpy and velocity divergence belong to H^s+1/2 and H^s−1/2, respectively. This separation reflects the coupling of a nondegenerate acoustic equation with the free-surface and vorticity dynamics. The proof uses regularization operators that smooth the moving domain and extend the nonlinear compatibility hierarchy, together with L^2 and partial H^2 distance estimates for solutions on different domains. Regular solutions are constructed by a regularized forward Euler scheme. High-order energy estimates and frequency envelopes then yield strong convergence of smooth approximations and continuity of the data-to-solution map. Integer Sobolev indices are treated using a Lions--Magenes endpoint compatibility condition.
ER -