Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.40163

Hadamard Local Well-Posedness for Compressible Liquids with a Free Surface

Tao Luo·Ruixi Zhang
View original source ↗Read source document ↗Export metadata ↓

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 $H^s$ 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.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tao Luo, Ruixi Zhang. 2026-09-30. Hadamard Local Well-Posedness for Compressible Liquids with a Free Surface. https://arxiv.org/abs/2609.40163

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Positive solutions to general semilinear overdetermined boundary problems

We establish the existence of positive solutions to a general class of overdetermined semilinear elliptic boundary problems on suitable bounded open sets $Ω\subset\mathbb{R}^n$. Specifically, for $n\leq 4$ and under mild technical hypotheses on the coefficients and the nonlinearity, we show that there exist open sets $Ω\subset\mathbb{R}^n$ with smooth boundary and of any prescribed volume where the overdetermined problem admits a positive solution. The proof builds on ideas of Alt and Caffarelli on variational problems for functions defined on a bounded region. In our case, we need to consider functions defined on the whole $\mathbb{R}^n$, so the key challenge is to obtain uniform bounds for the minimizer and for the diameter of its support. Our methods extend to higher dimensions, although in this case the free boundary $\partialΩ$ could have a singular set of codimension 5. The results are new even in the case of the Poisson equation $-Δv =g(x)$ with constant Neumann data.

math.AP↗
arXiv2026

The fuzzy Landau equation: global well-posedness and Fisher information

We investigate a fuzzy variant of the inhomogeneous Landau equation and establish global-in-time existence and uniqueness of smooth solutions for a range of soft potentials. The spatial delocalization introduced in the collision operator not only prevents singularity formation and enhances regularity, but also reveals additional structural properties of the model. In particular, we show that several forms of the Fisher information decay monotonically or remain uniformly bounded in time.

math.AP↗
arXiv2026

Sharp boundary asymptotics and monotonicity for singular quasilinear elliptic equations with gradient terms in half-spaces

We establish sharp boundary asymptotics for directional derivatives and monotonicity of positive weak solutions to $-Δ_p u+\vartheta|\nabla u|^q=u^{-γ}+f(u)$ in the half-space $\mathbb R^N_+$ with zero Dirichlet data, where $p>1$, $γ>0$, $0 1),\\ (1-\log t)^{-\frac1p}\,\partial_ηu(x',t) &\longrightarrow \left(\frac{p}{p-1}\right)^{\frac1p} \langleη,e_N\rangle && (γ=1), \end{aligned} \] where $C_{γ,p}=\bigl((γ+p-1)^p/[p^{p-1}(p-1)(γ-1)]\bigr)^{1/(γ+p-1)}$. The convergence is uniform in the tangential variable; in particular, the appropriately rescaled tangential derivatives vanish and the leading constants depend only on $p$ and $γ$, not on the gradient term or $f$. The normal-derivative limits imply the corresponding exact leading-order boundary profiles of $u$ by integration. In the weakly singular regime $0<γ<1$, we obtain linear boundary bounds and positive inward directional-derivative estimates. We also establish local counterparts of the boundary results for solutions merely locally bounded up to the boundary. Finally, under additional structural assumptions, the moving-plane method yields normal monotonicity for strip-bounded solutions in arbitrary dimension and for locally bounded solutions in dimension two. The boundary asymptotics hold throughout the range $0<q\le p$.

math.AP↗