Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.23091

Global well-posedness for the compressible Navier--Stokes equations in 3D bounded domains under a new smallness condition

Lin Xu·Xin Zhong
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate the three-dimensional barotropic compressible Navier--Stokes equations involving vacuum in a bounded domain with Navier-slip boundary conditions. It is shown that such a model admits a unique global strong solution provided the gradient of the initial velocity and the initial pressure deviation are suitably small in $L^2$. In particular, \textit{the smallness threshold depends only on the known parameters and the prescribed density bounds.} Our result provides a new analytical framework for the global well-posedness theory of the compressible Navier--Stokes equations.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lin Xu, Xin Zhong. 2026-10-16. Global well-posedness for the compressible Navier--Stokes equations in 3D bounded domains under a new smallness condition. https://arxiv.org/abs/2610.23091

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

KEEP EXPLORING

Related papers

arXiv2026

Quantitative Schauder estimates for hypoelliptic equations

We derive Schauder estimates using ideas from Campanato's approach for a general class of local hypoelliptic operators and non-local kinetic equations. The method covers equations in divergence and non-divergence form. In particular our results are applicable to the inhomogeneous Landau and to the non-cutoff Boltzmann equation. The paper is self-contained.

math.AP↗
arXiv2026

Nonlinear effects of incoming time-periodic boundary conditions on shock waves for scalar conservation laws

This paper is concerned with the large-time behavior of inviscid and viscous shocks for scalar conservation laws on the half-line $(-\infty,0)$ under an incoming \emph{time-periodic} boundary condition. This problem arises naturally from classical piston problems in fluid mechanics. Even for arbitrarily small periodic oscillations of the boundary data, the effect of the boundary manifests itself through a persistent time-periodic component near the boundary, which behaves as a \emph{boundary wave}. For inviscid shocks moving away from the boundary ($s<0$), this boundary wave repeatedly interacts with the shock over infinitely many periods. We show that the repeated interactions do not generate secular growth, but instead produce a finite \emph{cumulative} correction to the asymptotic shock location. For the inviscid stationary case $s=0$, the time-periodic boundary \emph{modulates} the background stationary shock into a genuinely time-periodic shock configuration, to which the solution converges. In the viscous case with $s<0$, the boundary wave couples with the viscous shock through a transitional region induced by diffusion, and the solution converges asymptotically to their superposition. Our results show that for scalar conservation laws, the persistent boundary wave modifies the shock without producing resonance.

math.AP↗
arXiv2026

A Liouville theorem for convex functions with periodic Monge-Ampère measure

We study global convex solutions of the Monge-Ampère equation \[ \det D^2 u = μ\quad \text{in } \mathbb{R}^n, \] where $μ\not\equiv 0$ is a nonnegative locally finite periodic Borel measure on $\mathbb{R}^n$. We prove a Liouville-type theorem showing that every such solution admits a unique decomposition, up to an additive constant, as the sum of a quadratic polynomial and a periodic function. This extends earlier results of Caffarelli-Li and Li-Lu, which required $μ$ to have a density with regular or bounded logarithm, to the full generality of periodic measures, allowing degeneracy and singularities. A key ingredient is a new dichotomous Harnack-type inequality for linearized Monge-Ampère equations with nonnegative periodic measures, which compensates for the failure of doubling and engulfing properties in the degenerate setting. In the extremal example where $μ$ is the periodic Dirac measure supported on the integer lattice, we show that the solutions, up to addition of a linear function, are in one-to-one correspondence with Dirichlet-Voronoi tilings of $\mathbb{R}^n$.

math.AP↗