一般压力律下一维可压缩Navier--Stokes方程强解的局部与整体存在性

中文摘要 AI 辅助

我们考虑具有一般本构律$p=p(v,\theta)$和$e=e(v,\theta)$的一维完全可压缩Navier--Stokes方程的Cauchy问题。假设压力和内能足够光滑，在热力学上相容，即$e_v=\theta p_\theta-p$，并满足$e_\theta>0$。我们首先在不施加压力单调性条件的情况下，对大初始数据建立了强解的局部时间存在性和唯一性。对于常数平衡态$(\bar v,0,\bar\theta)$附近的扰动，我们进一步假设力学稳定性条件$p_v(\bar v,\bar\theta)<0$。通过Gibbs关系引入相对热力学能，我们推导出基本能量恒等式，并证明其在平衡态附近的局部二次强制性。将此估计与高阶先验界相结合，我们获得了足够小的$H^1(\mathbb R)$扰动的强解的整体存在性和唯一性。此外，比容和温度在所有时间内一致地远离零和无穷大。

英文摘要

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.

发表机构

• Nanyang Institute of Technology（南阳理工学院）

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