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 $e_v=θp_θ-p$, 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 $(\bar v,0,\barθ)$, we further assume the mechanical stability condition $p_v(\bar v,\barθ)<0$. 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.