PreprintCC BY 4.0arXiv
Endpoint Compactness and Finite-Energy Weak Solutions for the 2D Isothermal Compressible Navier-Stokes
Abstract
We prove global existence of finite-energy renormalized weak solutions to the 2D isothermal compressible Navier--Stokes equations in bounded C2 domains with no-slip boundary conditions. The initial data are arbitrary in the natural energy class, allowing vacuum and unbounded density. This resolves the isothermal endpoint γ=1 of the classical 2D finite-energy renormalized existence theory for power-law pressures. We develop a new endpoint compactness argument based on a truncation-independent integrable bound for centered density oscillations. This argument yields strong L1 compactness of the approximating densities at the natural energy level.
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BibTeX
@misc{yu2026endpoint,
title = {{Endpoint Compactness and Finite-Energy Weak Solutions for the 2D Isothermal Compressible Navier-Stokes}},
author = {Cheng Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12387},
eprint = {2610.12387},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12387},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Endpoint Compactness and Finite-Energy Weak Solutions for the 2D Isothermal Compressible Navier-Stokes
AU - Cheng Yu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12387
UR - https://arxiv.org/abs/2610.12387
AB - We prove global existence of finite-energy renormalized weak solutions to the 2D isothermal compressible Navier--Stokes equations in bounded C^2 domains with no-slip boundary conditions. The initial data are arbitrary in the natural energy class, allowing vacuum and unbounded density. This resolves the isothermal endpoint γ=1 of the classical 2D finite-energy renormalized existence theory for power-law pressures. We develop a new endpoint compactness argument based on a truncation-independent integrable bound for centered density oscillations. This argument yields strong L^1 compactness of the approximating densities at the natural energy level.
ER -