Local and Global Existence of Strong Solutions to the One-Dimensional Compressible Navier--Stokes Equations with General Pressure Law

We consider the Cauchy problem for the one-dimensional full compressible Navier–Stokes equations with general constitutive laws $p=p(v,\theta )$ and $e=e(v,\theta )$. The pressure and internal energy are assumed to be sufficiently smooth, thermodynamically compatible in the sense that $e_v=\theta p_{\theta }-p$, and to satisfy $e_{\theta }>0$. 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 $(\bar{v},0,\bar{\theta })$, we further assume the mechanical stability condition $p_v(\bar{v},\bar{\theta })<0$. 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 $H^1(\mathbb{R})$ perturbations. Moreover, the specific volume and temperature remain uniformly bounded away from zero and infinity for all time.