Global existence and uniqueness of Lions' weak solutions for the 2D inhomogeneous Navier-Stokes equations in bounded domains
Abstract
The global well-posedness for Lions' weak solutions to the inhomogeneous incompressible Navier-Stokes equations in two-dimensional bounded domains is concerned. When the initial density 0≤ ρ0∈ L∞(Om) (allowing for vacuum) and initial velocity u0∈ L2σ(Ω), the global existence, regularity, and exponential decay-in-time rates of weak solutions are established via a priori estimates, which rely crucially on the duality of BMO and Hardy space for the pressure term, and the classical Sobolev extension theorem. In particular, for the non-vacuous case where 0< ρ0∈ L∞(Om) is bounded away from zero, we further prove the uniqueness of weak solutions in the sense of approximation by strong solutions, based on the key estimate t1/2∇ u∈ L2(0,T;L∞(Ω)), which essentially follows from a delicate localized point-wise bound for ∇ u near the boundary. The latter bound is derived by the Green functions estimate for the nonhomogeneous Dirichlet Stokes problem, and the uniform C1 boundary estimate on the remaining homogeneous Stokes problem, through a Stokes decomposition on suitably constructed local C2 domains.
Related
- Existence and uniqueness of solutions for compressible magnetohydrodynamic equations with bounded density
- Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative α-stable processes: Existence, uniqueness and numerical approximation
- Uniqueness of solutions to the 2D compressible Navier-Stokes equations for bounded densities
- Existence and Uniqueness of Viscosity Solutions for Stochastic Hamilton--Jacobi--Bellman Equations
- Global existence and uniqueness for the planar four-velocity Broadwell model with arbitrary C1 bounded initial and boundary data in a rectangular domain
BibTeX
@misc{wang2026global,
title = {{Global existence and uniqueness of Lions' weak solutions for the 2D inhomogeneous Navier-Stokes equations in bounded domains}},
author = {Hongyu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41637},
eprint = {2610.41637},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41637},
}
RIS
TY - UNPB
TI - Global existence and uniqueness of Lions' weak solutions for the 2D inhomogeneous Navier-Stokes equations in bounded domains
AU - Hongyu Wang
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41637
UR - https://arxiv.org/abs/2610.41637
AB - The global well-posedness for Lions' weak solutions to the inhomogeneous incompressible Navier-Stokes equations in two-dimensional bounded domains is concerned. When the initial density 0≤ ρ_0∈ L^∞(Om) (allowing for vacuum) and initial velocity u_0∈ L^2_σ(Ω), the global existence, regularity, and exponential decay-in-time rates of weak solutions are established via a priori estimates, which rely crucially on the duality of BMO and Hardy space for the pressure term, and the classical Sobolev extension theorem. In particular, for the non-vacuous case where 0< ρ_0∈ L^∞(Om) is bounded away from zero, we further prove the uniqueness of weak solutions in the sense of approximation by strong solutions, based on the key estimate t^1/2∇ u∈ L^2(0,T;L^∞(Ω)), which essentially follows from a delicate localized point-wise bound for ∇ u near the boundary. The latter bound is derived by the Green functions estimate for the nonhomogeneous Dirichlet Stokes problem, and the uniform C^1 boundary estimate on the remaining homogeneous Stokes problem, through a Stokes decomposition on suitably constructed local C^2 domains.
ER -