Local and Global Existence of Strong Solutions to the One-Dimensional Compressible Navier--Stokes Equations with General Pressure Law
Abstract
We consider the Cauchy problem for the one-dimensional full compressible Navier--Stokes equations with general constitutive laws p=p(v,θ) and e=e(v,θ). The pressure and internal energy are assumed to be sufficiently smooth, thermodynamically compatible in the sense that ev=θpθ−p, and to satisfy eθ>0. We first establish the local-in-time existence and uniqueness of strong solutions for large initial data without imposing monotonicity conditions on the pressure. For perturbations around a constant equilibrium (v̅,0,θ̅), we further assume the mechanical stability condition pv(v̅,θ̅)<0. By introducing a relative thermodynamic energy through the Gibbs relation, we derive a basic energy identity and prove its local quadratic coercivity near the equilibrium. Combining this estimate with higher-order a priori bounds, we obtain the global existence and uniqueness of strong solutions for sufficiently small H1(ℝ) perturbations. Moreover, the specific volume and temperature remain uniformly bounded away from zero and infinity for all time.
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BibTeX
@misc{zhao2026local,
title = {{Local and Global Existence of Strong Solutions to the One-Dimensional Compressible Navier--Stokes Equations with General Pressure Law}},
author = {Qingsong Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15317},
eprint = {2609.15317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15317},
}
RIS
TY - UNPB
TI - Local and Global Existence of Strong Solutions to the One-Dimensional Compressible Navier--Stokes Equations with General Pressure Law
AU - Qingsong Zhao
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15317
UR - https://arxiv.org/abs/2609.15317
AB - We consider the Cauchy problem for the one-dimensional full compressible Navier--Stokes equations with general constitutive laws p=p(v,θ) and e=e(v,θ). The pressure and internal energy are assumed to be sufficiently smooth, thermodynamically compatible in the sense that e_v=θp_θ−p, and to satisfy e_θ>0. We first establish the local-in-time existence and uniqueness of strong solutions for large initial data without imposing monotonicity conditions on the pressure. For perturbations around a constant equilibrium (v̅,0,θ̅), we further assume the mechanical stability condition p_v(v̅,θ̅)<0. By introducing a relative thermodynamic energy through the Gibbs relation, we derive a basic energy identity and prove its local quadratic coercivity near the equilibrium. Combining this estimate with higher-order a priori bounds, we obtain the global existence and uniqueness of strong solutions for sufficiently small H^1(ℝ) perturbations. Moreover, the specific volume and temperature remain uniformly bounded away from zero and infinity for all time.
ER -