Title: Regularity and dynamics of weak solutions for one-dimensional compressible Navier-Stokes equations with vacuum Show Chinese title

Title: 一维可压缩纳维-斯托克斯方程弱解的正则性与动力学与真空

Abstract: In the spirit of D. Hoff's weak solution theory for the compressible Navier-Stokes equations (CNS) with bounded density, in this paper we establish the global existence and regularity properties of finite-energy weak solutions to an initial boundary value problem of one-dimensional CNS with general initial data and vacuum. The core of our proof is a global in time a priori estimate for one-dimensional CNS that holds for any $H^1$ initial velocity and bounded initial density not necessarily strictly positive: it could be a density patch or a vacuum bubble. We also establish that the velocity and density decay exponentially to equilibrium. As a by-product, we obtain the quantitative dynamics of aforementioned two vacuum states. Show Chinese abstract

Abstract: 在D. Hoff关于可压缩Navier-Stokes方程（CNS）有界密度的弱解理论的精神下，本文建立了具有任意初始数据和真空的一维CNS初边值问题的有限能量弱解的全局存在性和正则性性质。 我们证明的核心是一个针对一维CNS的全局时间先验估计，该估计适用于任何$H^1$初始速度和有界初始密度，而不要求初始密度严格为正：它可以是密度块或真空泡。 我们还建立了速度和密度指数衰减到平衡状态。 作为副产品，我们得到了上述两种真空状态的定量动力学。